From 85427600c83b160ded645abfbb24c1664377702d Mon Sep 17 00:00:00 2001
From: bernborgess <62393097+bernborgess@users.noreply.github.com>
Date: Tue, 21 May 2024 11:00:11 -0300
Subject: [PATCH 01/93] Update README.md for KaTeX

---
 README.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/README.md b/README.md
index a193070..673996e 100644
--- a/README.md
+++ b/README.md
@@ -27,7 +27,7 @@ Run `mdbook test` to test all `lean` code blocks.
 ## Example
 
 We want to represent Pseudo-Boolean formulas to decide whether they are _satisfiable_. For instance,
-$$ \overline x_1 + 2x_2 + 3\overline x_3 + 4x_4 + 5\overline x_5 \ge 1 $$
+$$\overline x_1 + 2x_2 + 3\overline x_3 + 4x_4 + 5\overline x_5 \ge 1$$
 will be represented as
 
 ```lean

From 8409859af73067837245d5287376d572832f7872 Mon Sep 17 00:00:00 2001
From: bernborgess <62393097+bernborgess@users.noreply.github.com>
Date: Tue, 21 May 2024 11:08:27 -0300
Subject: [PATCH 02/93] Update README.md with TODO list

---
 README.md | 9 +++++++++
 1 file changed, 9 insertions(+)

diff --git a/README.md b/README.md
index 673996e..bc187e2 100644
--- a/README.md
+++ b/README.md
@@ -24,6 +24,15 @@ You can read the [docs](https://bernborgess.github.io/lean-cutting-planes/) here
 
 Run `mdbook test` to test all `lean` code blocks.
 
+## Work Progress
+
+- [ ] Addition proof
+- [x] Multiplication proof
+- [ ] Saturation proof
+- [ ] Division proof
+- [ ] Test cases
+- [ ] Documentation
+
 ## Example
 
 We want to represent Pseudo-Boolean formulas to decide whether they are _satisfiable_. For instance,

From 9003b8d730c1a1e603530cda3951ede36060734b Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 21 May 2024 15:25:35 +0000
Subject: [PATCH 03/93] Partial lemma `min_sum_le_sum_min`

---
 LeanCuttingPlanes/Rules/Saturation.lean | 33 ++++++++++++++++++++-----
 1 file changed, 27 insertions(+), 6 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index 256bfd1..f5991bb 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -16,16 +16,37 @@ lemma min_elim
   exact min_eq_right_of_lt h
   done
 
+-- https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Order/LinearOrder.html
+
 lemma min_sum_le_sum_min (A B C : ℕ)
   : min A (B + C) ≤ (min A B) + (min A C) := by
   simp
   by_cases h : A ≤ B + C
-  . sorry
-  simp at h
-  right
-  have hca : C < A := by sorry
-  rw [min_eq_right_of_lt hca]
-  sorry
+  . left
+    /-
+    h : A ≤ B + C
+    ⊢ A ≤ min A B + min A C
+    -/
+    sorry
+
+  . right
+    rw [not_le,←Nat.succ_le,Nat.succ_eq_add_one] at h
+
+    have hca : C ≤ A := by
+      apply le_of_add_le_left at h
+      apply le_of_add_le_right at h
+      exact h
+
+    rw [min_eq_right hca]
+
+    have hba : B ≤ A := by
+      apply le_of_add_le_left at h
+      apply le_of_add_le_left at h
+      exact h
+
+    rw [min_eq_right hba]
+    done
+
 
 -- Saturation
 -- ∑i (a i * l i) ≥ A

From 141b79aa399644eb84b0c82467fc6803f9cbf861 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 21 May 2024 15:43:09 -0300
Subject: [PATCH 04/93] examples with le sum min

---
 LeanCuttingPlanes/Rules/Saturation.lean | 33 +++++++++++++++++++++++++
 1 file changed, 33 insertions(+)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index f5991bb..bc0a621 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -27,6 +27,17 @@ lemma min_sum_le_sum_min (A B C : ℕ)
     h : A ≤ B + C
     ⊢ A ≤ min A B + min A C
     -/
+    by_cases h₁ : A ≤ B
+    . rw [min_eq_left h₁]
+      exact Nat.le_add_right A (min A C)
+    . simp at h₁
+      rw [min_eq_right_of_lt h₁]
+      by_cases h₂ : A ≤ C
+      . rw [min_eq_left h₂]
+        exact Nat.le_add_left A B
+      . simp at h₂
+        rw [min_eq_right_of_lt h₂]
+        exact h
     sorry
 
   . right
@@ -47,6 +58,28 @@ lemma min_sum_le_sum_min (A B C : ℕ)
     rw [min_eq_right hba]
     done
 
+example (A B C D : ℕ)
+  (h : A ≤ B + C + D)
+  : A ≤ min A B + min A C + min A D := by
+  by_cases h₁ : A ≤ B
+  . rw [min_eq_left h₁]
+    rw [add_assoc]
+    exact Nat.le_add_right A (min A C + min A D)
+  . simp at h₁
+    rw [min_eq_right_of_lt h₁]
+    by_cases h₂ : A ≤ C
+    . rw [min_eq_left h₂]
+      rw [add_comm B A,add_assoc]
+      exact Nat.le_add_right A (B + min A D)
+    . simp at h₂
+      rw [min_eq_right_of_lt h₂]
+      by_cases h₃ : A ≤ D
+      . rw [min_eq_left h₃]
+        exact Nat.le_add_left A (B + C)
+      . simp at h₃
+        rw [min_eq_right_of_lt h₃]
+        exact h
+
 
 -- Saturation
 -- ∑i (a i * l i) ≥ A

From 0a6e9efb45f87fcafb6a0f1ed990f9a8917af1c8 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 21 May 2024 18:46:27 +0000
Subject: [PATCH 05/93] Cleaning sorry, adding comment

---
 LeanCuttingPlanes/Rules/Saturation.lean | 6 +-----
 1 file changed, 1 insertion(+), 5 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index bc0a621..4973957 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -23,10 +23,6 @@ lemma min_sum_le_sum_min (A B C : ℕ)
   simp
   by_cases h : A ≤ B + C
   . left
-    /-
-    h : A ≤ B + C
-    ⊢ A ≤ min A B + min A C
-    -/
     by_cases h₁ : A ≤ B
     . rw [min_eq_left h₁]
       exact Nat.le_add_right A (min A C)
@@ -38,7 +34,6 @@ lemma min_sum_le_sum_min (A B C : ℕ)
       . simp at h₂
         rw [min_eq_right_of_lt h₂]
         exact h
-    sorry
 
   . right
     rw [not_le,←Nat.succ_le,Nat.succ_eq_add_one] at h
@@ -58,6 +53,7 @@ lemma min_sum_le_sum_min (A B C : ℕ)
     rw [min_eq_right hba]
     done
 
+-- Pattern keeps going for each new term
 example (A B C D : ℕ)
   (h : A ≤ B + C + D)
   : A ≤ min A B + min A C + min A D := by

From d224a213c370ea9a3ef4389eec068bb185e63eaa Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 21 May 2024 20:06:21 +0000
Subject: [PATCH 06/93] Splitting steps of Saturation

---
 LeanCuttingPlanes/Rules/Saturation.lean | 28 +++++++++++++++++++------
 1 file changed, 22 insertions(+), 6 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index 4973957..6ccb36d 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -76,6 +76,14 @@ example (A B C D : ℕ)
         rw [min_eq_right_of_lt h₃]
         exact h
 
+example (A : ℕ) (xs : Fin n → ℕ)
+  (h : A ≤ min A (∑i,xs i))
+  : A ≤ ∑i,min A (xs i) := by sorry
+
+example (A B C : ℕ) (i : Bool)
+  : min A (if i then B else C) = (if i then min A B else min A C) := by
+  exact apply_ite (min A) (i = true) B C
+  done
 
 -- Saturation
 -- ∑i (a i * l i) ≥ A
@@ -86,13 +94,21 @@ theorem Saturation
   {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
   : PBIneq (map (mapBoth (min A)) as) xs A := by
   unfold PBIneq PBSum FinVec.map mapBoth at *
-  simp at *
+  simp only [Fin.isValue, ge_iff_le, Prod_map, seq_eq] at *
+  apply le_min_self_of_le at ha
+
+  have h1 : min A (∑ x : Fin n, if xs x = 1 then (as x).1 else (as x).2) ≤ (∑ x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2)) := sorry
+
+  have h2 : A ≤ (∑ x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2)) := by
+    apply le_trans ha h1
+
+  have r1 : ∀x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2) = (if xs x = 1 then min A (as x).1 else min A (as x).2) := by exact fun x => apply_ite (min A) (xs x = 1) (as x).1 (as x).2
+
+  -- rw [r1] at h2
+
   /-
-  A ≤
-    ∑ x : Fin n,
-      if xs x = 1
-      then min A (as x).1
-      else min A (as x).2
+  h2 : A ≤ ∑ x : Fin n, min A (if xs x = 1 then       (as x).1 else       (as x).2)
+  ⊢    A ≤ ∑ x : Fin n,       (if xs x = 1 then min A (as x).1 else min A (as x).2)
   -/
   sorry
   done

From 314d9fcca32a5a70aa0bb4178a30700f52ca32d8 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 21 May 2024 19:46:56 -0300
Subject: [PATCH 07/93] Gathering References

---
 LeanCuttingPlanes/Rules/Saturation.lean | 45 +++++++++++++++++++++++--
 1 file changed, 43 insertions(+), 2 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index 6ccb36d..aef0240 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -85,6 +85,43 @@ example (A B C : ℕ) (i : Bool)
   exact apply_ite (min A) (i = true) B C
   done
 
+-- set_option autoImplicit false
+
+#check apply_ite (min 3) true 3 4
+
+-- Zulip Question: how to rewrite inside BigOperators
+-- https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/.E2.9C.94.20Expanding.20a.20sum
+-- https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/patterns.20with.20coerced.20var
+-- https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/finset.20sum.20function.2Eextend
+-- https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Finset.2Euniv.2Esum.20split
+-- https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/.E2.9C.94.20Can't.20rewrite.20seemingly.20indentical.20terms
+-- https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Point.26Click.20library.20rewrite.20tactic
+example (A B : ℕ)
+  (xs : Fin n → ℕ)
+  (h : A = B)
+  (g : B ≤ ∑i,(min A (if xs i = 1 then xs i else 3)))
+  : B ≤ ∑i, (if xs i = 1 then min A (xs i) else min A 3) := by
+  have r : ∀ i : Fin n, (min A (if xs i = 1 then xs i else 3)) = ((if xs i = 1 then min A (xs i) else min A 3)) := by
+    intro i
+    exact apply_ite (min A) (xs i = 1) (xs i) 3
+    done
+  sorry
+  -- rw [r] at g
+  -- assumption
+  done
+
+example (A B : ℕ)
+  (xs : Fin n → ℕ)
+  (h : A = B)
+  (g : B ≤ ∑i,(min A (if xs i = 1 then xs i else 3)))
+  : B ≤ ∑i, (if xs i = 1 then min A (xs i) else min A 3) := by
+  have r : (λ i : Fin n => (min A (if xs i = 1 then xs i else 3)))
+         = (λ i : Fin n => (if xs i = 1 then min A (xs i) else min A 3)) := by sorry
+  rw [r] at g
+  assumption
+  done
+
+
 -- Saturation
 -- ∑i (a i * l i) ≥ A
 -- ⊢
@@ -102,9 +139,13 @@ theorem Saturation
   have h2 : A ≤ (∑ x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2)) := by
     apply le_trans ha h1
 
-  have r1 : ∀x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2) = (if xs x = 1 then min A (as x).1 else min A (as x).2) := by exact fun x => apply_ite (min A) (xs x = 1) (as x).1 (as x).2
+  have r1 {b : Bool} {x y : ℕ} : min A (if b then x else y) = (if b then min A x else min A y) := by exact apply_ite (min A) (b = true) x y
+
+  -- rw [apply_ite] at h2
+
+  -- h2 : A ≤ ∑ x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2)
+  -- h2 : A ≤ ∑ x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2)
 
-  -- rw [r1] at h2
 
   /-
   h2 : A ≤ ∑ x : Fin n, min A (if xs x = 1 then       (as x).1 else       (as x).2)

From 272975e0d1a7dc8b62991606623d428c265d0bdb Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 22 May 2024 15:19:38 +0000
Subject: [PATCH 08/93] Breaking down Saturation proof steps

---
 LeanCuttingPlanes/Rules/Saturation.lean | 109 +++++++-----------------
 1 file changed, 33 insertions(+), 76 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index aef0240..f76997c 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -3,22 +3,8 @@ import «LeanCuttingPlanes».Data.PBO
 namespace PseudoBoolean
 open FinVec Matrix BigOperators
 
-lemma le_min_self_of_le
-  {A B : ℕ}
-  (h : A ≤ B)
-  : A ≤ min A B := by
-  simp only [ge_iff_le, h, min_eq_left, le_refl]
-
-lemma min_elim
-  (A C : ℕ)
-  (h : C < A)
-  : min A C = C := by
-  exact min_eq_right_of_lt h
-  done
-
 -- https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Order/LinearOrder.html
-
-lemma min_sum_le_sum_min (A B C : ℕ)
+lemma min_add_le_add_min (A B C : ℕ)
   : min A (B + C) ≤ (min A B) + (min A C) := by
   simp
   by_cases h : A ≤ B + C
@@ -76,51 +62,34 @@ example (A B C D : ℕ)
         rw [min_eq_right_of_lt h₃]
         exact h
 
-example (A : ℕ) (xs : Fin n → ℕ)
-  (h : A ≤ min A (∑i,xs i))
-  : A ≤ ∑i,min A (xs i) := by sorry
-
-example (A B C : ℕ) (i : Bool)
-  : min A (if i then B else C) = (if i then min A B else min A C) := by
-  exact apply_ite (min A) (i = true) B C
-  done
-
--- set_option autoImplicit false
-
-#check apply_ite (min 3) true 3 4
-
--- Zulip Question: how to rewrite inside BigOperators
--- https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/.E2.9C.94.20Expanding.20a.20sum
--- https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/patterns.20with.20coerced.20var
--- https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/finset.20sum.20function.2Eextend
--- https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Finset.2Euniv.2Esum.20split
--- https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/.E2.9C.94.20Can't.20rewrite.20seemingly.20indentical.20terms
--- https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Point.26Click.20library.20rewrite.20tactic
-example (A B : ℕ)
-  (xs : Fin n → ℕ)
-  (h : A = B)
-  (g : B ≤ ∑i,(min A (if xs i = 1 then xs i else 3)))
-  : B ≤ ∑i, (if xs i = 1 then min A (xs i) else min A 3) := by
-  have r : ∀ i : Fin n, (min A (if xs i = 1 then xs i else 3)) = ((if xs i = 1 then min A (xs i) else min A 3)) := by
-    intro i
-    exact apply_ite (min A) (xs i = 1) (xs i) 3
-    done
+-- # Need to achieve this proof for BigOperators
+lemma min_sum_le_sum_min
+  (A n B : ℕ)
+  : min A (∑i:Fin n,B) ≤ ∑i:Fin n,min A B := by sorry
+
+lemma min_sum_le_sum_min_ite
+  (A : ℕ)
+  (xs : Fin n → Fin 2)
+  (as : Fin n → ℕ × ℕ)
+  : min A (∑ x : Fin n, if xs x = 1 then (as x).1 else (as x).2)
+  ≤ (∑ x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2)) := by
+  -- apply min_sum_le_sum_min A n (B:=if xs _ = 1 then (as _).1 else (as _).2)
   sorry
-  -- rw [r] at g
-  -- assumption
-  done
-
-example (A B : ℕ)
-  (xs : Fin n → ℕ)
-  (h : A = B)
-  (g : B ≤ ∑i,(min A (if xs i = 1 then xs i else 3)))
-  : B ≤ ∑i, (if xs i = 1 then min A (xs i) else min A 3) := by
-  have r : (λ i : Fin n => (min A (if xs i = 1 then xs i else 3)))
-         = (λ i : Fin n => (if xs i = 1 then min A (xs i) else min A 3)) := by sorry
-  rw [r] at g
-  assumption
+  /-
+  tactic 'apply' failed, failed to unify
+    min A (∑ i : Fin n, if xs ?m.12973 = 1 then (as ?m.13003).1 else (as ?m.13008).2) ≤
+      ∑ i : Fin n, min A (if xs ?m.12973 = 1 then (as ?m.13003).1 else (as ?m.13008).2)
+  with
+    min A (∑ x : Fin n, if xs x = 1 then (as x).1 else (as x).2) ≤
+      ∑ x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2)
+  -/
   done
 
+lemma le_min_self_of_le
+  {A B : ℕ}
+  (h : A ≤ B)
+  : A ≤ min A B := by
+  simp only [h, min_eq_left, le_refl]
 
 -- Saturation
 -- ∑i (a i * l i) ≥ A
@@ -134,29 +103,17 @@ theorem Saturation
   simp only [Fin.isValue, ge_iff_le, Prod_map, seq_eq] at *
   apply le_min_self_of_le at ha
 
-  have h1 : min A (∑ x : Fin n, if xs x = 1 then (as x).1 else (as x).2) ≤ (∑ x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2)) := sorry
-
-  have h2 : A ≤ (∑ x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2)) := by
-    apply le_trans ha h1
+  have h1 := by exact min_sum_le_sum_min_ite A xs as
+  -- min A (∑i, if _ then _ else _) ≤ ∑i, min A (if _ then _ else _)
 
-  have r1 {b : Bool} {x y : ℕ} : min A (if b then x else y) = (if b then min A x else min A y) := by exact apply_ite (min A) (b = true) x y
-
-  -- rw [apply_ite] at h2
-
-  -- h2 : A ≤ ∑ x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2)
-  -- h2 : A ≤ ∑ x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2)
-
-
-  /-
-  h2 : A ≤ ∑ x : Fin n, min A (if xs x = 1 then       (as x).1 else       (as x).2)
-  ⊢    A ≤ ∑ x : Fin n,       (if xs x = 1 then min A (as x).1 else min A (as x).2)
-  -/
-  sorry
+  have h2 := by apply le_trans ha h1 -- A ≤ ∑i, min A (if _ then _ else _)
+  simp_rw [apply_ite (min A) ((xs _ = 1)) ((as _).1) ((as _).2)] at h2
+  exact h2
   done
 
 example
-  (ha : PBIneq ![3,4] xs 3)
-  : PBIneq ![3,3] xs 3 := by
+  (ha : PBIneq ![(3,0),(4,0)] xs 3)
+  : PBIneq ![(3,0),(3,0)] xs 3 := by
   apply Saturation ha
   done
 

From a92f8980f1395b557d74259d0cbda2e909a473dc Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 22 May 2024 16:37:14 +0000
Subject: [PATCH 09/93] Simplified lemma `min_sum_le_sum_min`

---
 LeanCuttingPlanes/Rules/Saturation.lean | 26 +++++--------------------
 1 file changed, 5 insertions(+), 21 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index f76997c..2f53516 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -64,26 +64,9 @@ example (A B C D : ℕ)
 
 -- # Need to achieve this proof for BigOperators
 lemma min_sum_le_sum_min
-  (A n B : ℕ)
-  : min A (∑i:Fin n,B) ≤ ∑i:Fin n,min A B := by sorry
-
-lemma min_sum_le_sum_min_ite
   (A : ℕ)
-  (xs : Fin n → Fin 2)
-  (as : Fin n → ℕ × ℕ)
-  : min A (∑ x : Fin n, if xs x = 1 then (as x).1 else (as x).2)
-  ≤ (∑ x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2)) := by
-  -- apply min_sum_le_sum_min A n (B:=if xs _ = 1 then (as _).1 else (as _).2)
-  sorry
-  /-
-  tactic 'apply' failed, failed to unify
-    min A (∑ i : Fin n, if xs ?m.12973 = 1 then (as ?m.13003).1 else (as ?m.13008).2) ≤
-      ∑ i : Fin n, min A (if xs ?m.12973 = 1 then (as ?m.13003).1 else (as ?m.13008).2)
-  with
-    min A (∑ x : Fin n, if xs x = 1 then (as x).1 else (as x).2) ≤
-      ∑ x : Fin n, min A (if xs x = 1 then (as x).1 else (as x).2)
-  -/
-  done
+  (B : Fin n → ℕ)
+  : min A (∑i:Fin n,B i) ≤ ∑i:Fin n,min A (B i) := by sorry
 
 lemma le_min_self_of_le
   {A B : ℕ}
@@ -103,10 +86,11 @@ theorem Saturation
   simp only [Fin.isValue, ge_iff_le, Prod_map, seq_eq] at *
   apply le_min_self_of_le at ha
 
-  have h1 := by exact min_sum_le_sum_min_ite A xs as
-  -- min A (∑i, if _ then _ else _) ≤ ∑i, min A (if _ then _ else _)
+  have h1 := by
+    exact min_sum_le_sum_min A (B:=λi=> if xs i = 1 then(as i).1 else (as i).2)
 
   have h2 := by apply le_trans ha h1 -- A ≤ ∑i, min A (if _ then _ else _)
+
   simp_rw [apply_ite (min A) ((xs _ = 1)) ((as _).1) ((as _).2)] at h2
   exact h2
   done

From 5fbc02cec81f6f4a46cac05c3f9aebd3e31d78bf Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 22 May 2024 16:49:24 +0000
Subject: [PATCH 10/93] Formatting lemma

---
 LeanCuttingPlanes/Rules/Saturation.lean | 14 ++++++++------
 1 file changed, 8 insertions(+), 6 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index 2f53516..ed640d4 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -3,7 +3,9 @@ import «LeanCuttingPlanes».Data.PBO
 namespace PseudoBoolean
 open FinVec Matrix BigOperators
 
--- https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Order/LinearOrder.html
+/--
+# Convincing myself of the validity
+ https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Order/LinearOrder.html
 lemma min_add_le_add_min (A B C : ℕ)
   : min A (B + C) ≤ (min A B) + (min A C) := by
   simp
@@ -61,12 +63,12 @@ example (A B C D : ℕ)
       . simp at h₃
         rw [min_eq_right_of_lt h₃]
         exact h
+-/
 
--- # Need to achieve this proof for BigOperators
-lemma min_sum_le_sum_min
-  (A : ℕ)
-  (B : Fin n → ℕ)
-  : min A (∑i:Fin n,B i) ≤ ∑i:Fin n,min A (B i) := by sorry
+lemma min_sum_le_sum_min (A : ℕ) (B : Fin n → ℕ)
+  : min A (∑i,B i) ≤ ∑i, min A (B i) := by
+  sorry
+  done
 
 lemma le_min_self_of_le
   {A B : ℕ}

From b04f3ba97424eca43ede8983b802f1ffe10ac43d Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 22 May 2024 17:24:44 +0000
Subject: [PATCH 11/93] Removing unneeded lemmas

---
 LeanCuttingPlanes/Rules/Saturation.lean | 21 +++++++--------------
 1 file changed, 7 insertions(+), 14 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index ed640d4..0359638 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -65,17 +65,16 @@ example (A B C D : ℕ)
         exact h
 -/
 
-lemma min_sum_le_sum_min (A : ℕ) (B : Fin n → ℕ)
-  : min A (∑i,B i) ≤ ∑i, min A (B i) := by
-  sorry
-  done
-
 lemma le_min_self_of_le
   {A B : ℕ}
   (h : A ≤ B)
   : A ≤ min A B := by
   simp only [h, min_eq_left, le_refl]
 
+lemma le_sum_min_of_le_sum {n A : ℕ} {B : Fin n → ℕ}
+  (h : A ≤ ∑i, B i)
+  : A ≤ ∑i, min A (B i) := sorry
+
 -- Saturation
 -- ∑i (a i * l i) ≥ A
 -- ⊢
@@ -86,15 +85,9 @@ theorem Saturation
   : PBIneq (map (mapBoth (min A)) as) xs A := by
   unfold PBIneq PBSum FinVec.map mapBoth at *
   simp only [Fin.isValue, ge_iff_le, Prod_map, seq_eq] at *
-  apply le_min_self_of_le at ha
-
-  have h1 := by
-    exact min_sum_le_sum_min A (B:=λi=> if xs i = 1 then(as i).1 else (as i).2)
-
-  have h2 := by apply le_trans ha h1 -- A ≤ ∑i, min A (if _ then _ else _)
-
-  simp_rw [apply_ite (min A) ((xs _ = 1)) ((as _).1) ((as _).2)] at h2
-  exact h2
+  have h := le_sum_min_of_le_sum ha
+  simp_rw [apply_ite (min A) ((xs _ = 1)) ((as _).1) ((as _).2)] at h
+  exact h
   done
 
 example

From 6452aaa4b384ca01fdfc395f3cc49e703acdf7b5 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 22 May 2024 20:02:24 +0000
Subject: [PATCH 12/93] Partial proof of `le_sum_min_of_le_sum`

---
 LeanCuttingPlanes/Rules/Saturation.lean | 90 +++++++------------------
 1 file changed, 26 insertions(+), 64 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index 0359638..eacb7ce 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -3,77 +3,39 @@ import «LeanCuttingPlanes».Data.PBO
 namespace PseudoBoolean
 open FinVec Matrix BigOperators
 
-/--
-# Convincing myself of the validity
- https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Order/LinearOrder.html
-lemma min_add_le_add_min (A B C : ℕ)
-  : min A (B + C) ≤ (min A B) + (min A C) := by
-  simp
-  by_cases h : A ≤ B + C
-  . left
-    by_cases h₁ : A ≤ B
-    . rw [min_eq_left h₁]
-      exact Nat.le_add_right A (min A C)
-    . simp at h₁
-      rw [min_eq_right_of_lt h₁]
-      by_cases h₂ : A ≤ C
-      . rw [min_eq_left h₂]
-        exact Nat.le_add_left A B
-      . simp at h₂
-        rw [min_eq_right_of_lt h₂]
-        exact h
-
-  . right
-    rw [not_le,←Nat.succ_le,Nat.succ_eq_add_one] at h
+lemma le_min_self_of_le
+  {A B : ℕ}
+  (h : A ≤ B)
+  : A ≤ min A B := by
+  simp only [h, min_eq_left, le_refl]
 
-    have hca : C ≤ A := by
-      apply le_of_add_le_left at h
-      apply le_of_add_le_right at h
-      exact h
+lemma le_sum_min_of_le_sum {n A : ℕ} {as : Fin n → ℕ}
+  (h : A ≤ ∑i, as i)
+  : A ≤ ∑i, min A (as i) := by
+  by_cases ha : ∀i, as i ≤ A
+  . -- Assume all elements of as are ≤ A
+    simp_rw [@Nat.min_eq_right A (as _) (ha _)]
+    -- rewrite min A (as i) to (as i)
+    exact h
 
-    rw [min_eq_right hca]
+  . -- Otherwise, ∃x, (as x) > A
 
-    have hba : B ≤ A := by
-      apply le_of_add_le_left at h
-      apply le_of_add_le_left at h
-      exact h
+    -- Then: hxA : min A (as x) = A
 
-    rw [min_eq_right hba]
-    done
+    -- Split goal from
+    -- ⊢ A ≤ ∑i, min A (as i)
+    -- to
+    -- ⊢ A ≤ (∑ i ≠ x, min A (as i)) + min A (as x)
 
--- Pattern keeps going for each new term
-example (A B C D : ℕ)
-  (h : A ≤ B + C + D)
-  : A ≤ min A B + min A C + min A D := by
-  by_cases h₁ : A ≤ B
-  . rw [min_eq_left h₁]
-    rw [add_assoc]
-    exact Nat.le_add_right A (min A C + min A D)
-  . simp at h₁
-    rw [min_eq_right_of_lt h₁]
-    by_cases h₂ : A ≤ C
-    . rw [min_eq_left h₂]
-      rw [add_comm B A,add_assoc]
-      exact Nat.le_add_right A (B + min A D)
-    . simp at h₂
-      rw [min_eq_right_of_lt h₂]
-      by_cases h₃ : A ≤ D
-      . rw [min_eq_left h₃]
-        exact Nat.le_add_left A (B + C)
-      . simp at h₃
-        rw [min_eq_right_of_lt h₃]
-        exact h
--/
+    -- rw [hxA]
+    -- ⊢ A ≤ (∑ i ≠ x, min A (as i)) + A
 
-lemma le_min_self_of_le
-  {A B : ℕ}
-  (h : A ≤ B)
-  : A ≤ min A B := by
-  simp only [h, min_eq_left, le_refl]
+    -- > Just for the demo
+    have oracle : (A ≤ ∑ i : Fin n, min A (as i)) = (A ≤ (∑i, min A (as i)) + A) := sorry
+    rw [oracle]
+    -- >
 
-lemma le_sum_min_of_le_sum {n A : ℕ} {B : Fin n → ℕ}
-  (h : A ≤ ∑i, B i)
-  : A ≤ ∑i, min A (B i) := sorry
+    exact Nat.le_add_left A _
 
 -- Saturation
 -- ∑i (a i * l i) ≥ A

From 71f416077823064a34720aeb219447a155a131ab Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 22 May 2024 20:35:53 +0000
Subject: [PATCH 13/93] Lemma `extract_x_sum` in Saturation

---
 LeanCuttingPlanes/Rules/Saturation.lean | 26 +++++++++++++++----------
 1 file changed, 16 insertions(+), 10 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index eacb7ce..0622ca6 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -1,7 +1,7 @@
 import «LeanCuttingPlanes».Data.PBO
 
 namespace PseudoBoolean
-open FinVec Matrix BigOperators
+open FinVec Matrix BigOperators Finset
 
 lemma le_min_self_of_le
   {A B : ℕ}
@@ -9,6 +9,14 @@ lemma le_min_self_of_le
   : A ≤ min A B := by
   simp only [h, min_eq_left, le_refl]
 
+lemma extract_x_sum (x : Fin n) (as : Fin n → ℕ)
+  : (∑i,as i) = (∑i with i≠x,as i) + as x := by
+  sorry
+  done
+
+--   (A ≤ ∑ i : Fin n, min A (as i))
+-- = (A ≤ ∑ i ∈ filter (fun i => i ≠ x) univ, min A (as i) + min A (as x))
+
 lemma le_sum_min_of_le_sum {n A : ℕ} {as : Fin n → ℕ}
   (h : A ≤ ∑i, as i)
   : A ≤ ∑i, min A (as i) := by
@@ -19,22 +27,20 @@ lemma le_sum_min_of_le_sum {n A : ℕ} {as : Fin n → ℕ}
     exact h
 
   . -- Otherwise, ∃x, (as x) > A
-
-    -- Then: hxA : min A (as x) = A
+    simp only [not_forall, not_le] at ha
+    obtain ⟨x,hx⟩ := ha
+    have hxA : min A (as x) = A := by exact min_eq_left_of_lt hx
 
     -- Split goal from
     -- ⊢ A ≤ ∑i, min A (as i)
     -- to
     -- ⊢ A ≤ (∑ i ≠ x, min A (as i)) + min A (as x)
+    have r_sum_x : (A ≤ ∑i, min A (as i))
+                   = (A ≤ (∑i with i≠x, min A (as i)) + min A (as x))
+                   := by simp_rw [extract_x_sum x _]
 
-    -- rw [hxA]
+    rw [r_sum_x,hxA]
     -- ⊢ A ≤ (∑ i ≠ x, min A (as i)) + A
-
-    -- > Just for the demo
-    have oracle : (A ≤ ∑ i : Fin n, min A (as i)) = (A ≤ (∑i, min A (as i)) + A) := sorry
-    rw [oracle]
-    -- >
-
     exact Nat.le_add_left A _
 
 -- Saturation

From 1acdd8afbb84a37dde47bd5f0d1900b07c8d8351 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 22 May 2024 22:03:47 +0000
Subject: [PATCH 14/93] Understanding `Finset.sum_apply_dite`

---
 LeanCuttingPlanes/Rules/Saturation.lean | 25 +++++++++++++++++++++++++
 1 file changed, 25 insertions(+)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index 0622ca6..a6cc68d 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -9,8 +9,32 @@ lemma le_min_self_of_le
   : A ≤ min A B := by
   simp only [h, min_eq_left, le_refl]
 
+def mp : ℕ → Prop := λ i => i ≠ 3
+/-
+@[simp]
+theorem Finset.sum_cons{α : Type u_3} {β : Type u_4} {s : Finset α} {a : α} {f : α → β}
+[AddCommMonoid β] (h : a ∉ s) :
+  ((Finset.cons a s h).sum fun (x : α) => f x) = f a + s.sum fun (x : α) => f x
+-/
+
+def myFinset : Finset ℕ := {0,1,2,3}
+def p : ℕ → Prop := λ i => i ≠ 1
+
+#check DecidablePred
+def hp : DecidablePred p := fun _ => sorry -- decidable_of_iff _ _
+
+#check @Finset.sum_apply_dite ℕ ℕ ℕ _ myFinset p hp
+/-
+theorem Finset.sum_apply_dite.{u_5, u_4, u_3} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [AddCommMonoid β] {s : Finset α}
+  {p : α → Prop} {hp : DecidablePred p} [DecidablePred fun x => ¬p x] (f : (x : α) → p x → γ) (g : (x : α) → ¬p x → γ)
+  (h : γ → β) :
+  ∑ x ∈ s, h (if hx : p x then f x hx else g x hx) =
+    ∑ x ∈ (filter p s).attach, h (f ↑x ⋯) + ∑ x ∈ (filter (fun x => ¬p x) s).attach, h (g ↑x ⋯)
+-/
 lemma extract_x_sum (x : Fin n) (as : Fin n → ℕ)
   : (∑i,as i) = (∑i with i≠x,as i) + as x := by
+  rw [@sum_filter]
+
   sorry
   done
 
@@ -29,6 +53,7 @@ lemma le_sum_min_of_le_sum {n A : ℕ} {as : Fin n → ℕ}
   . -- Otherwise, ∃x, (as x) > A
     simp only [not_forall, not_le] at ha
     obtain ⟨x,hx⟩ := ha
+
     have hxA : min A (as x) = A := by exact min_eq_left_of_lt hx
 
     -- Split goal from

From bf5e2859b2479b5138369b2cfc2fb763ad46daab Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 23 May 2024 12:30:24 +0000
Subject: [PATCH 15/93] =?UTF-8?q?Reduce=20`le=5Fsum=5Fmin=5Fof=5Fle=5Fsum`?=
 =?UTF-8?q?=20down=20to=20`min=20A=20(as=20k)=20=E2=89=A4=20=E2=88=91i,min?=
 =?UTF-8?q?=20A=20(as=20i)`?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 LeanCuttingPlanes/Rules/Saturation.lean | 60 +++++--------------------
 1 file changed, 10 insertions(+), 50 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index a6cc68d..25745ad 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -3,44 +3,6 @@ import «LeanCuttingPlanes».Data.PBO
 namespace PseudoBoolean
 open FinVec Matrix BigOperators Finset
 
-lemma le_min_self_of_le
-  {A B : ℕ}
-  (h : A ≤ B)
-  : A ≤ min A B := by
-  simp only [h, min_eq_left, le_refl]
-
-def mp : ℕ → Prop := λ i => i ≠ 3
-/-
-@[simp]
-theorem Finset.sum_cons{α : Type u_3} {β : Type u_4} {s : Finset α} {a : α} {f : α → β}
-[AddCommMonoid β] (h : a ∉ s) :
-  ((Finset.cons a s h).sum fun (x : α) => f x) = f a + s.sum fun (x : α) => f x
--/
-
-def myFinset : Finset ℕ := {0,1,2,3}
-def p : ℕ → Prop := λ i => i ≠ 1
-
-#check DecidablePred
-def hp : DecidablePred p := fun _ => sorry -- decidable_of_iff _ _
-
-#check @Finset.sum_apply_dite ℕ ℕ ℕ _ myFinset p hp
-/-
-theorem Finset.sum_apply_dite.{u_5, u_4, u_3} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [AddCommMonoid β] {s : Finset α}
-  {p : α → Prop} {hp : DecidablePred p} [DecidablePred fun x => ¬p x] (f : (x : α) → p x → γ) (g : (x : α) → ¬p x → γ)
-  (h : γ → β) :
-  ∑ x ∈ s, h (if hx : p x then f x hx else g x hx) =
-    ∑ x ∈ (filter p s).attach, h (f ↑x ⋯) + ∑ x ∈ (filter (fun x => ¬p x) s).attach, h (g ↑x ⋯)
--/
-lemma extract_x_sum (x : Fin n) (as : Fin n → ℕ)
-  : (∑i,as i) = (∑i with i≠x,as i) + as x := by
-  rw [@sum_filter]
-
-  sorry
-  done
-
---   (A ≤ ∑ i : Fin n, min A (as i))
--- = (A ≤ ∑ i ∈ filter (fun i => i ≠ x) univ, min A (as i) + min A (as x))
-
 lemma le_sum_min_of_le_sum {n A : ℕ} {as : Fin n → ℕ}
   (h : A ≤ ∑i, as i)
   : A ≤ ∑i, min A (as i) := by
@@ -50,22 +12,20 @@ lemma le_sum_min_of_le_sum {n A : ℕ} {as : Fin n → ℕ}
     -- rewrite min A (as i) to (as i)
     exact h
 
-  . -- Otherwise, ∃x, (as x) > A
+  . -- Otherwise, ∃k, (as k) > A
     simp only [not_forall, not_le] at ha
-    obtain ⟨x,hx⟩ := ha
+    obtain ⟨k,hk⟩ := ha
+
+    have hask_le_sum : min A (as k) ≤ ∑i,min A (as i) := sorry
 
-    have hxA : min A (as x) = A := by exact min_eq_left_of_lt hx
+    rw [Nat.sub_eq_iff_eq_add hask_le_sum |>.mp rfl]
+    -- Split goal from  ⊢ A ≤  ∑i, min A (as i)
+    -- to               ⊢ A ≤ (∑i, min A (as i) - min A (as k)) + min A (as k)
 
-    -- Split goal from
-    -- ⊢ A ≤ ∑i, min A (as i)
-    -- to
-    -- ⊢ A ≤ (∑ i ≠ x, min A (as i)) + min A (as x)
-    have r_sum_x : (A ≤ ∑i, min A (as i))
-                   = (A ≤ (∑i with i≠x, min A (as i)) + min A (as x))
-                   := by simp_rw [extract_x_sum x _]
+    -- min A (as k) = A
+    rw [min_eq_left_of_lt hk]
 
-    rw [r_sum_x,hxA]
-    -- ⊢ A ≤ (∑ i ≠ x, min A (as i)) + A
+    -- ⊢ A ≤ (∑i,min A (as i) - A) + A
     exact Nat.le_add_left A _
 
 -- Saturation

From 4b6f252d7de599c7fde01524a529c12bb5039832 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 23 May 2024 13:26:38 +0000
Subject: [PATCH 16/93] Finalise Saturation Proof

---
 LeanCuttingPlanes/Rules/Saturation.lean | 13 +++++++++----
 1 file changed, 9 insertions(+), 4 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index 25745ad..f724547 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -3,6 +3,13 @@ import «LeanCuttingPlanes».Data.PBO
 namespace PseudoBoolean
 open FinVec Matrix BigOperators Finset
 
+-- @collares
+lemma split_summation (n : ℕ) (as : Fin n → ℕ) (k : Fin n) :
+    (∑i with i≠k, as i) + as k = (∑i, as i) := by
+  have : (∑i with i=k, as i) = as k := by rw [Finset.sum_eq_single_of_mem] <;> simp
+  rw [← this, ← Finset.sum_filter_add_sum_filter_not Finset.univ (· ≠ k)]
+  simp only [ne_eq, Decidable.not_not]
+
 lemma le_sum_min_of_le_sum {n A : ℕ} {as : Fin n → ℕ}
   (h : A ≤ ∑i, as i)
   : A ≤ ∑i, min A (as i) := by
@@ -16,11 +23,9 @@ lemma le_sum_min_of_le_sum {n A : ℕ} {as : Fin n → ℕ}
     simp only [not_forall, not_le] at ha
     obtain ⟨k,hk⟩ := ha
 
-    have hask_le_sum : min A (as k) ≤ ∑i,min A (as i) := sorry
-
-    rw [Nat.sub_eq_iff_eq_add hask_le_sum |>.mp rfl]
+    rw [←split_summation]
     -- Split goal from  ⊢ A ≤  ∑i, min A (as i)
-    -- to               ⊢ A ≤ (∑i, min A (as i) - min A (as k)) + min A (as k)
+    -- to               ⊢ A ≤ (∑i with i ≠ k, min A (as i)) + min A (as k)
 
     -- min A (as k) = A
     rw [min_eq_left_of_lt hk]

From a582521ece5f1ab404675973e822fccf0425ca13 Mon Sep 17 00:00:00 2001
From: bernborgess <62393097+bernborgess@users.noreply.github.com>
Date: Thu, 23 May 2024 10:29:00 -0300
Subject: [PATCH 17/93] Saturation Rule check

---
 README.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/README.md b/README.md
index bc187e2..36ffc93 100644
--- a/README.md
+++ b/README.md
@@ -28,7 +28,7 @@ Run `mdbook test` to test all `lean` code blocks.
 
 - [ ] Addition proof
 - [x] Multiplication proof
-- [ ] Saturation proof
+- [x] Saturation proof
 - [ ] Division proof
 - [ ] Test cases
 - [ ] Documentation

From 22593491fc982468548b31ea6386889db444ea84 Mon Sep 17 00:00:00 2001
From: bernborgess <62393097+bernborgess@users.noreply.github.com>
Date: Thu, 23 May 2024 11:18:07 -0300
Subject: [PATCH 18/93] KaTex Format fix

---
 README.md | 6 +++++-
 1 file changed, 5 insertions(+), 1 deletion(-)

diff --git a/README.md b/README.md
index 36ffc93..998e90c 100644
--- a/README.md
+++ b/README.md
@@ -36,7 +36,11 @@ Run `mdbook test` to test all `lean` code blocks.
 ## Example
 
 We want to represent Pseudo-Boolean formulas to decide whether they are _satisfiable_. For instance,
-$$\overline x_1 + 2x_2 + 3\overline x_3 + 4x_4 + 5\overline x_5 \ge 1$$
+
+$$
+\overline x_1 + 2x_2 + 3\overline x_3 + 4x_4 + 5\overline x_5 \ge 1
+$$
+
 will be represented as
 
 ```lean

From b821c125612cc2d7c9d27deb982cb181e650eb92 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Fri, 24 May 2024 02:16:13 -0300
Subject: [PATCH 19/93] lake update

---
 lake-manifest.json | 8 ++++----
 lean-toolchain     | 2 +-
 2 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/lake-manifest.json b/lake-manifest.json
index 04434de..61f6ac9 100644
--- a/lake-manifest.json
+++ b/lake-manifest.json
@@ -4,7 +4,7 @@
  [{"url": "https://github.com/leanprover-community/batteries",
    "type": "git",
    "subDir": null,
-   "rev": "56d2e4ee226603eb6b90b05f6b944bde42672cd5",
+   "rev": "7b3c48b58fa0ae1c8f27889bdb086ea5e4b27b06",
    "name": "batteries",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -22,7 +22,7 @@
   {"url": "https://github.com/leanprover-community/aesop",
    "type": "git",
    "subDir": null,
-   "rev": "2225b0e4a3528da20499e2304b521e0c4c2a4563",
+   "rev": "e8c8a42642ceb5af33708b79ae8a3148b681c236",
    "name": "aesop",
    "manifestFile": "lake-manifest.json",
    "inputRev": "master",
@@ -49,7 +49,7 @@
   {"url": "https://github.com/leanprover-community/import-graph.git",
    "type": "git",
    "subDir": null,
-   "rev": "c46d22bbe4f8363c0829ce0eb48a95012cdc0d79",
+   "rev": "35e38eb320982cfd2fcc864e0e0467ca223c8cdb",
    "name": "importGraph",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -58,7 +58,7 @@
   {"url": "https://github.com/leanprover-community/mathlib4",
    "type": "git",
    "subDir": null,
-   "rev": "690d741a2041d5670fd10338b772280cca9b47cd",
+   "rev": "e76a76c36a1c2792438b62656bb94234bc4e5317",
    "name": "mathlib",
    "manifestFile": "lake-manifest.json",
    "inputRev": null,
diff --git a/lean-toolchain b/lean-toolchain
index d8a6d7e..78f39e2 100644
--- a/lean-toolchain
+++ b/lean-toolchain
@@ -1 +1 @@
-leanprover/lean4:v4.8.0-rc1
+leanprover/lean4:v4.8.0-rc2

From 3a58ced33c474a0b828201973e08fca89b4ec28a Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Sat, 25 May 2024 20:59:02 -0300
Subject: [PATCH 20/93] Move definitions to `Basic.lean`

---
 LeanCuttingPlanes.lean                      |  1 -
 LeanCuttingPlanes/Basic.lean                | 41 +++++++++++++++++++--
 LeanCuttingPlanes/Data/PBO.lean             | 34 -----------------
 LeanCuttingPlanes/Rules/Addition.lean       |  2 +-
 LeanCuttingPlanes/Rules/Division.lean       |  2 +-
 LeanCuttingPlanes/Rules/Multiplication.lean |  2 +-
 LeanCuttingPlanes/Rules/Saturation.lean     |  2 +-
 7 files changed, 42 insertions(+), 42 deletions(-)
 delete mode 100644 LeanCuttingPlanes/Data/PBO.lean

diff --git a/LeanCuttingPlanes.lean b/LeanCuttingPlanes.lean
index 3988262..2808d6c 100644
--- a/LeanCuttingPlanes.lean
+++ b/LeanCuttingPlanes.lean
@@ -1,7 +1,6 @@
 -- This module serves as the root of the `LeanCuttingPlanes` library.
 -- Import modules here that should be built as part of the library.
 import «LeanCuttingPlanes».Basic
-import «LeanCuttingPlanes».Data.PBO
 import «LeanCuttingPlanes».Rules.Addition
 import «LeanCuttingPlanes».Rules.Multiplication
 import «LeanCuttingPlanes».Rules.Division
diff --git a/LeanCuttingPlanes/Basic.lean b/LeanCuttingPlanes/Basic.lean
index 64507df..913866a 100644
--- a/LeanCuttingPlanes/Basic.lean
+++ b/LeanCuttingPlanes/Basic.lean
@@ -1,3 +1,11 @@
+/-
+Copyright (c) 2024 Bernardo Borges. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Bernardo Borges
+-/
+import Mathlib.Data.Fin.Tuple.Reflection
+import Mathlib.Data.Matrix.Notation
+
 /-! # Representation of a Linear Inequality
 
   ## Basic Definition
@@ -31,11 +39,38 @@
   pseudo-Boolean PBO format (only linear objective and constraints)
 
   * ⋈ is always ≥
-  * aᵢ is a list of ℤ
-  * A is a ℤ
+  * aᵢ is a list of ℕ
+  * A is a ℕ
 
+-/
+namespace PseudoBoolean
 
+open FinVec BigOperators
 
+abbrev Coeff (n : ℕ) := Fin n → (ℕ × ℕ)
 
+def PBSum (cs : Coeff n) (xs : Fin n → Fin 2) :=
+  ∑ i, let (p,n) := cs i;
+    if xs i = 1 then p else n
 
--/
+def PBIneq (cs : Coeff n) (xs : Fin n → Fin 2) (const : ℕ) :=
+  PBSum cs xs ≥ const
+
+example : PBIneq ![(1,0)] ![1] 1 := by
+  reduce                -- Expand the goal to 1 * 1 ≥ 1
+  exact Nat.le_refl 1   -- Prove that 1 * 1 ≥ 1
+  done
+
+example : PBIneq ![(1,0),(2,0)] ![0,1] 2 := by
+  reduce                -- Change goal to 1 * 0 + 2 * 1 ≥ 2
+  exact Nat.le_refl 2   -- Prove 1 * 0 + 2 * 1 ≥ 2
+  done
+
+example : PBIneq ![(3,0),(4,0)] ![0,1] 2 := by
+  reduce
+  simp
+  done
+
+def mapBoth (f : α → β) (t : α × α) : β × β := Prod.map f f t
+
+end PseudoBoolean
diff --git a/LeanCuttingPlanes/Data/PBO.lean b/LeanCuttingPlanes/Data/PBO.lean
deleted file mode 100644
index fb34f26..0000000
--- a/LeanCuttingPlanes/Data/PBO.lean
+++ /dev/null
@@ -1,34 +0,0 @@
-import Mathlib.Data.Fin.Tuple.Reflection
-import Mathlib.Data.Matrix.Notation
-
-namespace PseudoBoolean
-
-open FinVec BigOperators
-
-abbrev Coeff (n : ℕ) := Fin n → (ℕ × ℕ)
-
-def PBSum (cs : Coeff n) (xs : Fin n → Fin 2) :=
-  ∑ i, let (p,n) := cs i;
-    if xs i = 1 then p else n
-
-def PBIneq (cs : Coeff n) (xs : Fin n → Fin 2) (const : ℕ) :=
-  PBSum cs xs ≥ const
-
-example : PBIneq ![(1,0)] ![1] 1 := by
-  reduce                -- Expand the goal to 1 * 1 ≥ 1
-  exact Nat.le_refl 1   -- Prove that 1 * 1 ≥ 1
-  done
-
-example : PBIneq ![(1,0),(2,0)] ![0,1] 2 := by
-  reduce                -- Change goal to 1 * 0 + 2 * 1 ≥ 2
-  exact Nat.le_refl 2   -- Prove 1 * 0 + 2 * 1 ≥ 2
-  done
-
-example : PBIneq ![(3,0),(4,0)] ![0,1] 2 := by
-  reduce
-  simp
-  done
-
-def mapBoth (f : α → β) (t : α × α) : β × β := Prod.map f f t
-
-end PseudoBoolean
diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index 13f17aa..1db1945 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -1,4 +1,4 @@
-import «LeanCuttingPlanes».Data.PBO
+import «LeanCuttingPlanes».Basic
 
 namespace PseudoBoolean
 
diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index 28c8726..bc74bb3 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -1,4 +1,4 @@
-import «LeanCuttingPlanes».Data.PBO
+import «LeanCuttingPlanes».Basic
 
 namespace PseudoBoolean
 open FinVec Matrix BigOperators
diff --git a/LeanCuttingPlanes/Rules/Multiplication.lean b/LeanCuttingPlanes/Rules/Multiplication.lean
index 8802f26..fff10c8 100644
--- a/LeanCuttingPlanes/Rules/Multiplication.lean
+++ b/LeanCuttingPlanes/Rules/Multiplication.lean
@@ -1,4 +1,4 @@
-import «LeanCuttingPlanes».Data.PBO
+import «LeanCuttingPlanes».Basic
 
 namespace PseudoBoolean
 
diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index f724547..e949e74 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -1,4 +1,4 @@
-import «LeanCuttingPlanes».Data.PBO
+import «LeanCuttingPlanes».Basic
 
 namespace PseudoBoolean
 open FinVec Matrix BigOperators Finset

From 17dc91ded793d88f7ace92585ca54674734c7a09 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Mon, 27 May 2024 11:36:54 -0300
Subject: [PATCH 21/93] Split Division proof into lemmas

---
 LeanCuttingPlanes/Rules/Division.lean | 33 ++++++++++++++++++---------
 1 file changed, 22 insertions(+), 11 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index bc74bb3..1cf88a8 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -1,9 +1,24 @@
 import «LeanCuttingPlanes».Basic
+import Mathlib.Algebra.Order.Floor.Div
 
 namespace PseudoBoolean
-open FinVec Matrix BigOperators
+open FinVec Matrix BigOperators Finset
 
-def ceildiv (c : ℕ) (a : ℕ) := (a+c-1) / c
+def ceildiv (c : ℕ) (a : ℕ) := a ⌈/⌉ c
+
+lemma ceildiv_le_ceildiv_left
+  {a b : ℕ}
+  (c : ℕ)
+  (hab : a ≤ b)
+  : a ⌈/⌉ c ≤ b ⌈/⌉ c := sorry
+
+lemma sum_ceildiv (as : Fin n → ℕ) (c : ℕ)
+  : (∑i, as i) ⌈/⌉ c = ∑i,(as i ⌈/⌉ c)
+  := sorry
+
+lemma ceildiv_ite (P : Prop) [Decidable P] (a b c : ℕ) :
+  (if P then b else c) ⌈/⌉ a = if P then b ⌈/⌉ a else c ⌈/⌉ a
+  := sorry
 
 -- Division
 -- ∑i (a i * l i) ≥ A
@@ -17,15 +32,11 @@ theorem Division
   {c : ℕ} (hc0 : c > 0)
   : PBIneq (map (mapBoth (ceildiv c)) as) xs (ceildiv c A) := by
   unfold PBIneq PBSum ceildiv mapBoth at *
-  simp at *
-  /-
-  (A + c - 1) / c ≤
-    ∑ x : Fin n,
-      if xs x = 1
-      then ((as x).1 + c - 1) / c
-      else ((as x).2 + c - 1) / c
-  -/
-  sorry
+  simp only [Fin.isValue, ge_iff_le, gt_iff_lt, Prod_map, map_eq, Function.comp_apply] at *
+  apply ceildiv_le_ceildiv_left c at ha
+  rw [sum_ceildiv] at ha
+  simp only [ceildiv_ite] at ha
+  exact ha
   done
 
 example

From 9361c07a3615765e0c3424c6e4973913326553cb Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Mon, 27 May 2024 17:19:01 +0000
Subject: [PATCH 22/93] Refined Division lemmas

---
 LeanCuttingPlanes/Rules/Division.lean | 34 +++++++++++++--------------
 Main.lean                             |  3 +--
 2 files changed, 18 insertions(+), 19 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index 1cf88a8..8cf10b6 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -2,23 +2,24 @@ import «LeanCuttingPlanes».Basic
 import Mathlib.Algebra.Order.Floor.Div
 
 namespace PseudoBoolean
-open FinVec Matrix BigOperators Finset
+open FinVec BigOperators
 
 def ceildiv (c : ℕ) (a : ℕ) := a ⌈/⌉ c
 
-lemma ceildiv_le_ceildiv_left
-  {a b : ℕ}
-  (c : ℕ)
-  (hab : a ≤ b)
-  : a ⌈/⌉ c ≤ b ⌈/⌉ c := sorry
+lemma ceildiv_le_ceildiv_left {a b : ℕ} (c : ℕ) (hab : a ≤ b)
+  : a ⌈/⌉ c ≤ b ⌈/⌉ c := by
+  sorry
+  done
 
 lemma sum_ceildiv (as : Fin n → ℕ) (c : ℕ)
-  : (∑i, as i) ⌈/⌉ c = ∑i,(as i ⌈/⌉ c)
-  := sorry
+  : (∑i, as i) ⌈/⌉ c ≤ ∑i,(as i ⌈/⌉ c) := by
+  sorry
+  done
 
-lemma ceildiv_ite (P : Prop) [Decidable P] (a b c : ℕ) :
-  (if P then b else c) ⌈/⌉ a = if P then b ⌈/⌉ a else c ⌈/⌉ a
-  := sorry
+lemma ceildiv_ite (P : Prop) [Decidable P] (a b c : ℕ)
+  : (if P then b else c) ⌈/⌉ a = if P then (b ⌈/⌉ a) else (c ⌈/⌉ a) := by
+  sorry
+  done
 
 -- Division
 -- ∑i (a i * l i) ≥ A
@@ -29,21 +30,20 @@ lemma ceildiv_ite (P : Prop) [Decidable P] (a b c : ℕ) :
 theorem Division
   {xs : Fin n → Fin 2}
   {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
-  {c : ℕ} (hc0 : c > 0)
+  (c : ℕ)
   : PBIneq (map (mapBoth (ceildiv c)) as) xs (ceildiv c A) := by
   unfold PBIneq PBSum ceildiv mapBoth at *
   simp only [Fin.isValue, ge_iff_le, gt_iff_lt, Prod_map, map_eq, Function.comp_apply] at *
   apply ceildiv_le_ceildiv_left c at ha
-  rw [sum_ceildiv] at ha
-  simp only [ceildiv_ite] at ha
-  exact ha
+  apply le_trans ha
+  simp only [←ceildiv_ite]
+  apply sum_ceildiv
   done
 
 example
   (ha : PBIneq ![3,4] xs 3)
   : PBIneq ![2,2] xs 2 := by
-  let h2z : 2 > 0 := Nat.zero_lt_succ 1
-  apply Division ha h2z
+  apply Division ha 2
   done
 
 end PseudoBoolean
diff --git a/Main.lean b/Main.lean
index f312240..e0ef49b 100644
--- a/Main.lean
+++ b/Main.lean
@@ -15,12 +15,11 @@ def derived
   (c3 : PBIneq ![(0,0),(0,0),(0,0),(0,1)] xs 0)
   : PBIneq ![(1,0),(2,0),(2,0),(0,0)] xs 3
   := by
-  let h3z : 3 > 0 := Nat.zero_lt_succ 2
   let t1 : PBIneq ![(2,0),(4,0),(2,0),(0,0)] xs 4  := by apply Multiplication c1 2
   let t2 : PBIneq ![(3,0),(6,0),(6,0),(2,0)] xs 9  := by apply Addition t1 c2
   let t3 : PBIneq ![(0,0),(0,0),(0,0),(0,2)] xs 0  := by apply Multiplication c3 2
   let t4 : PBIneq ![(3,0),(6,0),(6,0),(0,0)] xs 7  := by apply Addition t2 t3
-  exact Division t4 h3z
+  exact Division t4 3
   done
 
 def resolved : ∃xs, PBIneq ![(1,0),(2,0),(2,0),(0,0)] xs 3 :=

From 6c7ce457a1a544c89be8effa890fa54c1bc1493e Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Mon, 27 May 2024 17:37:46 +0000
Subject: [PATCH 23/93] Closed `ceildiv_ite` lemma

---
 LeanCuttingPlanes/Rules/Division.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index 8cf10b6..03dbc4e 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -18,7 +18,7 @@ lemma sum_ceildiv (as : Fin n → ℕ) (c : ℕ)
 
 lemma ceildiv_ite (P : Prop) [Decidable P] (a b c : ℕ)
   : (if P then b else c) ⌈/⌉ a = if P then (b ⌈/⌉ a) else (c ⌈/⌉ a) := by
-  sorry
+  split_ifs <;> rfl
   done
 
 -- Division

From 82beb88f3e278bf0e0db6a3ede0917ad96765295 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Mon, 27 May 2024 20:53:59 +0000
Subject: [PATCH 24/93] Proved lemma `ceildiv_le_ceildiv_right`

---
 LeanCuttingPlanes/Rules/Division.lean | 10 +++++++---
 1 file changed, 7 insertions(+), 3 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index 03dbc4e..97799ac 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -6,9 +6,13 @@ open FinVec BigOperators
 
 def ceildiv (c : ℕ) (a : ℕ) := a ⌈/⌉ c
 
-lemma ceildiv_le_ceildiv_left {a b : ℕ} (c : ℕ) (hab : a ≤ b)
+lemma ceildiv_le_ceildiv_right {a b : ℕ} (c : ℕ) (hab : a ≤ b)
   : a ⌈/⌉ c ≤ b ⌈/⌉ c := by
-  sorry
+  repeat rw [Nat.ceilDiv_eq_add_pred_div]
+  apply Nat.div_le_div_right
+  apply Nat.sub_le_sub_right
+  apply Nat.add_le_add_right
+  exact hab
   done
 
 lemma sum_ceildiv (as : Fin n → ℕ) (c : ℕ)
@@ -34,7 +38,7 @@ theorem Division
   : PBIneq (map (mapBoth (ceildiv c)) as) xs (ceildiv c A) := by
   unfold PBIneq PBSum ceildiv mapBoth at *
   simp only [Fin.isValue, ge_iff_le, gt_iff_lt, Prod_map, map_eq, Function.comp_apply] at *
-  apply ceildiv_le_ceildiv_left c at ha
+  apply ceildiv_le_ceildiv_right c at ha
   apply le_trans ha
   simp only [←ceildiv_ite]
   apply sum_ceildiv

From 7c08d06d716bde36a8116f19a8f966187a5a156b Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Mon, 27 May 2024 21:12:19 +0000
Subject: [PATCH 25/93] Thinking on `sum_ceildiv` lemma

---
 LeanCuttingPlanes/Rules/Division.lean | 6 ++++++
 1 file changed, 6 insertions(+)

diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index 97799ac..fabf43f 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -17,6 +17,12 @@ lemma ceildiv_le_ceildiv_right {a b : ℕ} (c : ℕ) (hab : a ≤ b)
 
 lemma sum_ceildiv (as : Fin n → ℕ) (c : ℕ)
   : (∑i, as i) ⌈/⌉ c ≤ ∑i,(as i ⌈/⌉ c) := by
+  -- by_cases hnz : n = 0 -- TODO
+  simp_rw [Nat.ceilDiv_eq_add_pred_div]
+  /-
+  ⊢ (∑i,  as i + c - 1) / c
+  ≤ ∑ i,((as i + c - 1)/c)
+  -/
   sorry
   done
 

From 003cbd20b572f5243d4ed8682530f9d12261d206 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 28 May 2024 17:54:03 +0000
Subject: [PATCH 26/93] Fixed division example

---
 LeanCuttingPlanes/Rules/Division.lean | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index fabf43f..686d6ad 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -51,8 +51,8 @@ theorem Division
   done
 
 example
-  (ha : PBIneq ![3,4] xs 3)
-  : PBIneq ![2,2] xs 2 := by
+  (ha : PBIneq ![(3,0),(4,0)] xs 3)
+  : PBIneq ![(2,0),(2,0)] xs 2 := by
   apply Division ha 2
   done
 

From d0b5bd5d22d1967ae72b6a0da6bcfe64bd1495a6 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 28 May 2024 21:52:38 +0000
Subject: [PATCH 27/93] lemma `add_ceildiv` to understand `sum_ceildiv`

---
 LeanCuttingPlanes/Rules/Division.lean | 54 +++++++++++++++++++++++++++
 1 file changed, 54 insertions(+)

diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index 686d6ad..d06772d 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -1,4 +1,5 @@
 import «LeanCuttingPlanes».Basic
+import Mathlib.Data.Nat.ModEq
 import Mathlib.Algebra.Order.Floor.Div
 
 namespace PseudoBoolean
@@ -15,6 +16,59 @@ lemma ceildiv_le_ceildiv_right {a b : ℕ} (c : ℕ) (hab : a ≤ b)
   exact hab
   done
 
+-- same argument
+lemma add_ceildiv (a b c : ℕ)
+  : (a + b) ⌈/⌉ c ≤ (a ⌈/⌉ c) + (b ⌈/⌉ c) := by
+  rw [←Nat.div_add_mod a c,←Nat.div_add_mod b c]
+
+  let ma := a % c
+  have rma : ma = a % c := by exact rfl
+  rw [←rma]
+
+  let da := a / c
+  have rda : da = a / c := by exact rfl
+  rw [←rda]
+
+  let mb := b % c
+  have rmb : mb = b % c := by exact rfl
+  rw [←rmb]
+
+  let db := b / c
+  have rdb : db = b / c := by exact rfl
+  rw [←rdb]
+
+  clear rda rdb rma rmb
+
+  -- ⊢ (c * da + ma + (c * db + mb)) ⌈/⌉ c ≤ (c * da + ma) ⌈/⌉ c + (c * db + mb) ⌈/⌉ c
+  -- ⊢ (c * (da + db) + ma + mb) ⌈/⌉ c ≤ (c * da + ma) ⌈/⌉ c + (c * db + mb) ⌈/⌉ c
+  -- lemma factor_out_ceildiv : (c * a + b) ⌈/⌉ c = a + b ⌈/⌉ c
+  -- ⊢ da + db + (ma + mb) ⌈/⌉ c ≤ da + ma ⌈/⌉ c + db + mb ⌈/⌉ c
+  -- ⊢ (ma + mb) ⌈/⌉ c ≤ ma ⌈/⌉ c + mb ⌈/⌉ c
+  have oracle : ((c * da + ma + (c * db + mb)) ⌈/⌉ c ≤ (c * da + ma) ⌈/⌉ c + (c * db + mb) ⌈/⌉ c) = ((ma + mb) ⌈/⌉ c ≤ ma ⌈/⌉ c + mb ⌈/⌉ c) := sorry
+  rw [oracle]; clear oracle
+
+  by_cases hma0 : ma = 0
+  . simp only [hma0, zero_add, zero_ceilDiv, le_refl]
+  by_cases hmb0 : mb = 0
+  . simp only [hmb0, add_zero, zero_ceilDiv, le_refl]
+
+  have hmac : 0 ≤ ma ∧ ma < c := sorry
+  have hmac1 : ma ⌈/⌉ c = 1 := sorry -- hma0, hmac
+
+  have hmbc : 0 ≤ mb ∧ mb < c := sorry
+  have hmbc1 : mb ⌈/⌉ c = 1 := sorry -- hmb0, hmbc
+
+  rw [hmac1,hmbc1]
+  simp only [Nat.reduceAdd, ge_iff_le]
+  -- ⊢ (ma + mb) ⌈/⌉ c ≤ 2
+  have hmamb2c : ma + mb < 2 * c := sorry -- hmac, hmbc
+
+  have oracle : ((ma + mb) ⌈/⌉ c ≤ 2) = (ma + mb ≤ 2 * c) := sorry
+  rw [oracle]; clear oracle
+
+  exact Nat.le_of_succ_le hmamb2c
+  done
+
 lemma sum_ceildiv (as : Fin n → ℕ) (c : ℕ)
   : (∑i, as i) ⌈/⌉ c ≤ ∑i,(as i ⌈/⌉ c) := by
   -- by_cases hnz : n = 0 -- TODO

From 16ae71d5d802af4763a7d349332061890bc6068a Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 28 May 2024 19:14:16 -0300
Subject: [PATCH 28/93] Using `set` for clarity

---
 LeanCuttingPlanes/Rules/Division.lean | 27 +++++++++------------------
 1 file changed, 9 insertions(+), 18 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index d06772d..63b31a5 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -1,6 +1,10 @@
 import «LeanCuttingPlanes».Basic
-import Mathlib.Data.Nat.ModEq
 import Mathlib.Algebra.Order.Floor.Div
+import Mathlib.Data.Nat.Defs
+import Mathlib.Data.Nat.ModEq
+import Mathlib.Logic.Equiv.Fin
+-- Init.Data.DivMod.Lemmas
+-- Init.Data.Nat.Div
 
 namespace PseudoBoolean
 open FinVec BigOperators
@@ -21,23 +25,10 @@ lemma add_ceildiv (a b c : ℕ)
   : (a + b) ⌈/⌉ c ≤ (a ⌈/⌉ c) + (b ⌈/⌉ c) := by
   rw [←Nat.div_add_mod a c,←Nat.div_add_mod b c]
 
-  let ma := a % c
-  have rma : ma = a % c := by exact rfl
-  rw [←rma]
-
-  let da := a / c
-  have rda : da = a / c := by exact rfl
-  rw [←rda]
-
-  let mb := b % c
-  have rmb : mb = b % c := by exact rfl
-  rw [←rmb]
-
-  let db := b / c
-  have rdb : db = b / c := by exact rfl
-  rw [←rdb]
-
-  clear rda rdb rma rmb
+  set ma := a % c
+  set da := a / c
+  set mb := b % c
+  set db := b / c
 
   -- ⊢ (c * da + ma + (c * db + mb)) ⌈/⌉ c ≤ (c * da + ma) ⌈/⌉ c + (c * db + mb) ⌈/⌉ c
   -- ⊢ (c * (da + db) + ma + mb) ⌈/⌉ c ≤ (c * da + ma) ⌈/⌉ c + (c * db + mb) ⌈/⌉ c

From fcb9a7835b17de8db8dfd85eeb1417d0270be014 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Mon, 3 Jun 2024 17:15:16 -0300
Subject: [PATCH 29/93] Working out `Nat.add_ceildiv_le_add_ceildiv`

---
 LeanCuttingPlanes/Basic.lean          |  1 -
 LeanCuttingPlanes/Rules/Division.lean | 63 +++++++++------------------
 2 files changed, 21 insertions(+), 43 deletions(-)

diff --git a/LeanCuttingPlanes/Basic.lean b/LeanCuttingPlanes/Basic.lean
index 913866a..18b3993 100644
--- a/LeanCuttingPlanes/Basic.lean
+++ b/LeanCuttingPlanes/Basic.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Bernardo Borges
 -/
 import Mathlib.Data.Fin.Tuple.Reflection
-import Mathlib.Data.Matrix.Notation
 
 /-! # Representation of a Linear Inequality
 
diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index 63b31a5..e74388f 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -1,10 +1,6 @@
 import «LeanCuttingPlanes».Basic
 import Mathlib.Algebra.Order.Floor.Div
-import Mathlib.Data.Nat.Defs
 import Mathlib.Data.Nat.ModEq
-import Mathlib.Logic.Equiv.Fin
--- Init.Data.DivMod.Lemmas
--- Init.Data.Nat.Div
 
 namespace PseudoBoolean
 open FinVec BigOperators
@@ -20,47 +16,30 @@ lemma ceildiv_le_ceildiv_right {a b : ℕ} (c : ℕ) (hab : a ≤ b)
   exact hab
   done
 
--- same argument
-lemma add_ceildiv (a b c : ℕ)
-  : (a + b) ⌈/⌉ c ≤ (a ⌈/⌉ c) + (b ⌈/⌉ c) := by
-  rw [←Nat.div_add_mod a c,←Nat.div_add_mod b c]
-
-  set ma := a % c
-  set da := a / c
-  set mb := b % c
-  set db := b / c
-
-  -- ⊢ (c * da + ma + (c * db + mb)) ⌈/⌉ c ≤ (c * da + ma) ⌈/⌉ c + (c * db + mb) ⌈/⌉ c
-  -- ⊢ (c * (da + db) + ma + mb) ⌈/⌉ c ≤ (c * da + ma) ⌈/⌉ c + (c * db + mb) ⌈/⌉ c
-  -- lemma factor_out_ceildiv : (c * a + b) ⌈/⌉ c = a + b ⌈/⌉ c
-  -- ⊢ da + db + (ma + mb) ⌈/⌉ c ≤ da + ma ⌈/⌉ c + db + mb ⌈/⌉ c
-  -- ⊢ (ma + mb) ⌈/⌉ c ≤ ma ⌈/⌉ c + mb ⌈/⌉ c
-  have oracle : ((c * da + ma + (c * db + mb)) ⌈/⌉ c ≤ (c * da + ma) ⌈/⌉ c + (c * db + mb) ⌈/⌉ c) = ((ma + mb) ⌈/⌉ c ≤ ma ⌈/⌉ c + mb ⌈/⌉ c) := sorry
-  rw [oracle]; clear oracle
-
-  by_cases hma0 : ma = 0
-  . simp only [hma0, zero_add, zero_ceilDiv, le_refl]
-  by_cases hmb0 : mb = 0
-  . simp only [hmb0, add_zero, zero_ceilDiv, le_refl]
-
-  have hmac : 0 ≤ ma ∧ ma < c := sorry
-  have hmac1 : ma ⌈/⌉ c = 1 := sorry -- hma0, hmac
-
-  have hmbc : 0 ≤ mb ∧ mb < c := sorry
-  have hmbc1 : mb ⌈/⌉ c = 1 := sorry -- hmb0, hmbc
-
-  rw [hmac1,hmbc1]
-  simp only [Nat.reduceAdd, ge_iff_le]
-  -- ⊢ (ma + mb) ⌈/⌉ c ≤ 2
-  have hmamb2c : ma + mb < 2 * c := sorry -- hmac, hmbc
+lemma l1 (a b c : ℕ)
+  : a + b + c - 1 = (a + b - 1) + c := by
+  sorry
 
-  have oracle : ((ma + mb) ⌈/⌉ c ≤ 2) = (ma + mb ≤ 2 * c) := sorry
-  rw [oracle]; clear oracle
+lemma l2 (a c : ℕ)
+  : a + c - 1 = (a - 1) + c := by
+  sorry
 
-  exact Nat.le_of_succ_le hmamb2c
+theorem Nat.add_ceildiv_le_add_ceildiv (a b c : ℕ)
+  : (a + b) ⌈/⌉ c ≤ (a ⌈/⌉ c) + (b ⌈/⌉ c) :=
+  if hc0 : c = 0 then by simp[hc0] else by
+  apply Nat.zero_lt_of_ne_zero at hc0
+  simp [Nat.instCeilDiv]
+  rw [l1,l2 a c,l2 b c]
+  have c_dvd_c : c ∣ c := Nat.dvd_refl c
+  have c_div_c : c / c = 1 := Nat.div_self hc0
+  repeat rw [Nat.add_div_of_dvd_left c_dvd_c,c_div_c]
+  rw [Nat.add_comm,Nat.add_comm ((a-1)/c), Nat.add_assoc]
+  apply Nat.add_le_add_left (k := 1)
+  -- ⊢ (a + b - 1) / c ≤ (a - 1) / c + ((b - 1) / c + 1)
+  sorry
   done
 
-lemma sum_ceildiv (as : Fin n → ℕ) (c : ℕ)
+lemma ceildiv_sum_le_sum_ceildiv (as : Fin n → ℕ) (c : ℕ)
   : (∑i, as i) ⌈/⌉ c ≤ ∑i,(as i ⌈/⌉ c) := by
   -- by_cases hnz : n = 0 -- TODO
   simp_rw [Nat.ceilDiv_eq_add_pred_div]
@@ -92,7 +71,7 @@ theorem Division
   apply ceildiv_le_ceildiv_right c at ha
   apply le_trans ha
   simp only [←ceildiv_ite]
-  apply sum_ceildiv
+  apply ceildiv_sum_le_sum_ceildiv
   done
 
 example

From 766cd2c3cdd6a9f89b700d020ae209665db8ab09 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Mon, 3 Jun 2024 17:34:30 -0300
Subject: [PATCH 30/93] Contribution of @kbuzzard for
 `Nat.add_ceildiv_le_add_ceildiv`

---
 LeanCuttingPlanes/Rules/Division.lean | 31 ++++++++++-----------------
 1 file changed, 11 insertions(+), 20 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index e74388f..cf20db6 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -16,27 +16,18 @@ lemma ceildiv_le_ceildiv_right {a b : ℕ} (c : ℕ) (hab : a ≤ b)
   exact hab
   done
 
-lemma l1 (a b c : ℕ)
-  : a + b + c - 1 = (a + b - 1) + c := by
-  sorry
-
-lemma l2 (a c : ℕ)
-  : a + c - 1 = (a - 1) + c := by
-  sorry
-
+-- @kbuzzard
 theorem Nat.add_ceildiv_le_add_ceildiv (a b c : ℕ)
-  : (a + b) ⌈/⌉ c ≤ (a ⌈/⌉ c) + (b ⌈/⌉ c) :=
-  if hc0 : c = 0 then by simp[hc0] else by
-  apply Nat.zero_lt_of_ne_zero at hc0
-  simp [Nat.instCeilDiv]
-  rw [l1,l2 a c,l2 b c]
-  have c_dvd_c : c ∣ c := Nat.dvd_refl c
-  have c_div_c : c / c = 1 := Nat.div_self hc0
-  repeat rw [Nat.add_div_of_dvd_left c_dvd_c,c_div_c]
-  rw [Nat.add_comm,Nat.add_comm ((a-1)/c), Nat.add_assoc]
-  apply Nat.add_le_add_left (k := 1)
-  -- ⊢ (a + b - 1) / c ≤ (a - 1) / c + ((b - 1) / c + 1)
-  sorry
+  : (a + b) ⌈/⌉ c ≤ (a ⌈/⌉ c) + (b ⌈/⌉ c) := by
+  -- deal with c=0 separately
+  obtain (rfl | hc) := Nat.eq_zero_or_pos c
+  · simp
+  -- 0 < c case
+  -- First use the "Galois connection"
+  rw [ceilDiv_le_iff_le_smul hc, smul_eq_mul]
+  rw [mul_add]
+  -- now use a standard fact
+  gcongr <;> exact le_smul_ceilDiv hc
   done
 
 lemma ceildiv_sum_le_sum_ceildiv (as : Fin n → ℕ) (c : ℕ)

From 994769a3753bc2869fe01e54503907fd9e5d867b Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 4 Jun 2024 11:32:42 -0300
Subject: [PATCH 31/93] Closed Division proof with `Finset.ceildiv_le_ceildiv`,
 with help of Ruben-VandeVelde

---
 LeanCuttingPlanes/Rules/Division.lean | 26 +++++++++++++-------------
 1 file changed, 13 insertions(+), 13 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index cf20db6..2a4697a 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -1,9 +1,8 @@
 import «LeanCuttingPlanes».Basic
 import Mathlib.Algebra.Order.Floor.Div
-import Mathlib.Data.Nat.ModEq
 
 namespace PseudoBoolean
-open FinVec BigOperators
+open Finset FinVec BigOperators
 
 def ceildiv (c : ℕ) (a : ℕ) := a ⌈/⌉ c
 
@@ -30,16 +29,17 @@ theorem Nat.add_ceildiv_le_add_ceildiv (a b c : ℕ)
   gcongr <;> exact le_smul_ceilDiv hc
   done
 
-lemma ceildiv_sum_le_sum_ceildiv (as : Fin n → ℕ) (c : ℕ)
-  : (∑i, as i) ⌈/⌉ c ≤ ∑i,(as i ⌈/⌉ c) := by
-  -- by_cases hnz : n = 0 -- TODO
-  simp_rw [Nat.ceilDiv_eq_add_pred_div]
-  /-
-  ⊢ (∑i,  as i + c - 1) / c
-  ≤ ∑ i,((as i + c - 1)/c)
-  -/
-  sorry
-  done
+-- @Ruben-VandeVelde
+theorem Finset.ceildiv_le_ceildiv {α : Type*} (as : α → ℕ) (s : Finset α) (c : ℕ)
+  : (∑i in s, as i) ⌈/⌉ c ≤ ∑i in s,(as i ⌈/⌉ c) := by
+  classical
+  induction s using Finset.cons_induction_on with
+  | h₁ => simp
+  | @h₂ a s ha ih =>
+    rw [sum_cons, sum_cons]
+    have h := Nat.add_ceildiv_le_add_ceildiv (as a) (∑x ∈ s, as x) c
+    exact le_add_of_le_add_left h ih
+    done
 
 lemma ceildiv_ite (P : Prop) [Decidable P] (a b c : ℕ)
   : (if P then b else c) ⌈/⌉ a = if P then (b ⌈/⌉ a) else (c ⌈/⌉ a) := by
@@ -62,7 +62,7 @@ theorem Division
   apply ceildiv_le_ceildiv_right c at ha
   apply le_trans ha
   simp only [←ceildiv_ite]
-  apply ceildiv_sum_le_sum_ceildiv
+  apply Finset.ceildiv_le_ceildiv
   done
 
 example

From 452d1b821706dc1289e9d6c750550a9bfc13dd40 Mon Sep 17 00:00:00 2001
From: bernborgess <62393097+bernborgess@users.noreply.github.com>
Date: Tue, 4 Jun 2024 11:39:27 -0300
Subject: [PATCH 32/93] Check division proof in README.md

---
 README.md | 5 ++---
 1 file changed, 2 insertions(+), 3 deletions(-)

diff --git a/README.md b/README.md
index 998e90c..adf7ea7 100644
--- a/README.md
+++ b/README.md
@@ -29,7 +29,7 @@ Run `mdbook test` to test all `lean` code blocks.
 - [ ] Addition proof
 - [x] Multiplication proof
 - [x] Saturation proof
-- [ ] Division proof
+- [x] Division proof
 - [ ] Test cases
 - [ ] Documentation
 
@@ -66,11 +66,10 @@ example (c1 : PBIneq ![(1,0),(2,0),(1,0),(0,0)] xs 2)
         (c3 : PBIneq ![(0,0),(0,0),(0,0),(0,1)] xs 0)
   : PBIneq ![(1,0),(2,0),(2,0),(0,0)] xs 3
   := by
-  let h3z : 3 > 0 := Nat.zero_lt_succ 2
   let t1 : PBIneq ![(2,0),(4,0),(2,0),(0,0)] xs 4  := by apply Multiplication c1 2
   let t2 : PBIneq ![(3,0),(6,0),(6,0),(2,0)] xs 9  := by apply Addition t1 c2
   let t3 : PBIneq ![(0,0),(0,0),(0,0),(0,2)] xs 0  := by apply Multiplication c3 2
   let t4 : PBIneq ![(3,0),(6,0),(6,0),(0,0)] xs 7  := by apply Addition t2 t3
-  exact Division t4 h3z
+  exact Division t4 3
   done
 ```

From 67625577f6d7c5407554cabb25959f0c1ab8e672 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 4 Jun 2024 17:49:54 -0300
Subject: [PATCH 33/93] Add `reduce_terms` lemma

---
 LeanCuttingPlanes/Rules/Addition.lean | 19 +++++++++++++++++++
 1 file changed, 19 insertions(+)

diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index 1db1945..dfb59a9 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -4,6 +4,25 @@ namespace PseudoBoolean
 
 open BigOperators FinVec Matrix
 
+#check Fin.eq_one_of_neq_zero
+#check Nat.min_comm
+#check Nat.sub_add_min_cancel
+
+lemma reduce_terms (p n : ℕ) (x : Fin 2)
+  : p * x + n * (1 - x) = (p - n) * x + (n - p) * (1 - x) + min p n  := by
+  by_cases h : x = 0
+  . rw [h]
+    simp only [Fin.isValue, Fin.val_zero, mul_zero, tsub_zero, mul_one, zero_add]
+    rw [Nat.min_comm]
+    exact Nat.sub_add_min_cancel n p |>.symm
+
+  . -- x = 1
+    apply Fin.eq_one_of_neq_zero x at h
+    rw [h]
+    simp only [Fin.isValue, Fin.val_one, mul_one, ge_iff_le, le_refl, tsub_eq_zero_of_le]
+    simp only [mul_zero, add_zero, Nat.sub_add_min_cancel]
+
+
 def tighten (as : Coeff n) : Coeff n :=
   λ i : Fin n => let (p,n) := as i
     if p > n then (p - n,0) else (0,n - p)

From 0c4c2ee1281b2db7147f2c582aefc19b5c8d79e2 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 4 Jun 2024 18:14:09 -0300
Subject: [PATCH 34/93] Add lemma `sum_ge_a_sub_sum`

---
 LeanCuttingPlanes/Rules/Addition.lean | 19 +++++++++++++++++--
 1 file changed, 17 insertions(+), 2 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index dfb59a9..5ab2ff6 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -19,9 +19,24 @@ lemma reduce_terms (p n : ℕ) (x : Fin 2)
   . -- x = 1
     apply Fin.eq_one_of_neq_zero x at h
     rw [h]
-    simp only [Fin.isValue, Fin.val_one, mul_one, ge_iff_le, le_refl, tsub_eq_zero_of_le]
-    simp only [mul_zero, add_zero, Nat.sub_add_min_cancel]
+    simp only [Fin.isValue, Fin.val_one, mul_one, ge_iff_le, le_refl, tsub_eq_zero_of_le, mul_zero, add_zero]
+    exact Nat.sub_add_min_cancel p n |>.symm
 
+#check Finset.sum_add_distrib
+
+lemma sum_split_min_term (p n : Fin m → ℕ) (x : Fin m → Fin 2)
+  : (∑i,((p i - n i) * x i + (n i - p i) * (1 - x i)   +      min (p i) (n i)))
+  = (∑i,((p i - n i) * x i + (n i - p i) * (1 - x i))) + (∑i,(min (p i) (n i))) := by
+  exact Finset.sum_add_distrib
+  done
+
+#check Nat.sub_le_of_le_add
+
+lemma sum_ge_a_sub_sum (p n : Fin m → ℕ) (x : Fin m → Fin 2) (A : ℕ)
+  (h : ∑i,((p i - n i) * x i + (n i - p i) * (1 - x i)) + ∑i,(min (p i) (n i)) ≥ A)
+  :   (∑i,((p i - n i) * x i + (n i - p i) * (1 - x i))) ≥ A - ∑i,(min (p i) (n i)) := by
+  exact Nat.sub_le_of_le_add h
+  done
 
 def tighten (as : Coeff n) : Coeff n :=
   λ i : Fin n => let (p,n) := as i

From 57f7ed98ac49a156d8fe5ac4757ef9fb6652117b Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 4 Jun 2024 18:25:51 -0300
Subject: [PATCH 35/93] Added Addition example with `Reduction`

---
 LeanCuttingPlanes/Rules/Addition.lean | 9 +++++++++
 1 file changed, 9 insertions(+)

diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index 5ab2ff6..1aabdfb 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -91,4 +91,13 @@ example
   apply Addition ha hb
   done
 
+-- Reduction happens automatically
+example
+  (ha : PBIneq ![(3,0),(0,0),(1,0)] xs 1)
+  (hb : PBIneq ![(0,0),(2,0),(0,1)] xs 2)
+  : PBIneq ![(3,0),(2,0),(0,0)] xs 2 := by
+  apply Addition ha hb
+  done
+
+
 end PseudoBoolean

From 2df3ac7d2149c716f1dd5321e62344bfbc5fb785 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 4 Jun 2024 18:47:45 -0300
Subject: [PATCH 36/93] Add lemma `ite_eq_bmul`

---
 LeanCuttingPlanes/Rules/Addition.lean | 13 +++++++++++++
 1 file changed, 13 insertions(+)

diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index 1aabdfb..03e9165 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -38,6 +38,19 @@ lemma sum_ge_a_sub_sum (p n : Fin m → ℕ) (x : Fin m → Fin 2) (A : ℕ)
   exact Nat.sub_le_of_le_add h
   done
 
+lemma ite_eq_bmul (x y : ℕ) (b : Fin 2)
+  : (if b = 1 then x else y) = (x * b + y * (1 - b)) := by
+  by_cases h : b = 0
+  . rw [h]
+    rw [if_neg]
+    . simp only [Fin.isValue, Fin.val_zero, mul_zero, tsub_zero, mul_one, zero_add]
+    trivial
+  . -- b = 1
+    apply Fin.eq_one_of_neq_zero b at h
+    rw [h]
+    simp only [Fin.isValue, ↓reduceIte, Fin.val_one, mul_one, ge_iff_le, le_refl,
+      tsub_eq_zero_of_le, mul_zero, add_zero]
+
 def tighten (as : Coeff n) : Coeff n :=
   λ i : Fin n => let (p,n) := as i
     if p > n then (p - n,0) else (0,n - p)

From 0e21d3167c84b01d2e5e39d8d6d4a4b6a33fa8dc Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 4 Jun 2024 19:10:01 -0300
Subject: [PATCH 37/93] Comments with "ThePlan"

---
 LeanCuttingPlanes/Rules/Addition.lean | 22 ++++++++++++++++++++++
 1 file changed, 22 insertions(+)

diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index 03e9165..4866c2e 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -51,6 +51,28 @@ lemma ite_eq_bmul (x y : ℕ) (b : Fin 2)
     simp only [Fin.isValue, ↓reduceIte, Fin.val_one, mul_one, ge_iff_le, le_refl,
       tsub_eq_zero_of_le, mul_zero, add_zero]
 
+theorem ThePlan
+   {xs : Fin n → Fin 2}
+  {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
+  {bs : Coeff n} {B : ℕ} (hb : PBIneq bs xs B)
+  -- : AdditionProp xs as A bs B := byi
+  : True := by
+  set K := A + B
+  set ks := as + bs
+  have hk : PBIneq ks xs K := sorry -- here is Addition Proof without reduction
+  simp [PBIneq,PBSum] at hk         -- K ≤ ∑ i : Fin n, if xs i = 1 then (ks i).1 else (ks i).2
+  simp_rw [ite_eq_bmul] at hk       -- K ≤ ∑ i : Fin n, ((ks i).1 * ↑(xs i) + (ks i).2 * (1 - ↑(xs i)))
+  set pos := λ i => ks i |>.1 with rpos
+  set neg := λ i => ks i |>.2 with rneg
+  -- simp_rw [rpos,rneg] at hk
+  -- simp_rw [reduce_terms] at hk   -- K ≤ ∑ i : Fin n, ((pos i - neg i) * ↑(xs x) + (neg i - pos i) * (1 - ↑(xs x)) + min (pos i) (neg i))
+  -- apply sum_split_min_term       -- K ≤ ∑ i : Fin n, ((pos i - neg i) * ↑(xs x) + (neg i - pos i) * (1 - ↑(xs x))) + ∑i,min (pos i) (neg i)
+  -- apply sub_ge_a_sub_sum         -- K - ∑i,min (pos i) (neg i) ≤ ∑ i : Fin n, ((pos i - neg i) * ↑(xs x) + (neg i - pos i) * (1 - ↑(xs x)))
+  --
+  sorry
+  done
+
+
 def tighten (as : Coeff n) : Coeff n :=
   λ i : Fin n => let (p,n) := as i
     if p > n then (p - n,0) else (0,n - p)

From 33feba4c9604422d59871857f4bdc8d7bf6ba42c Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 5 Jun 2024 10:56:58 -0300
Subject: [PATCH 38/93] Split proof in `Addition'` and `Reduction`

---
 LeanCuttingPlanes/Rules/Addition.lean | 54 +++++++++++++++------------
 1 file changed, 30 insertions(+), 24 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index 4866c2e..621eb10 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -72,24 +72,36 @@ theorem ThePlan
   sorry
   done
 
+theorem Addition'
+  (xs : Fin n → Fin 2)
+  (as : Coeff n) (A : ℕ) (ha : PBIneq as xs A)
+  (bs : Coeff n) (B : ℕ) (hb : PBIneq bs xs B)
+  : PBIneq (as + bs) xs (A + B) := by
+  sorry
+  done
 
-def tighten (as : Coeff n) : Coeff n :=
-  λ i : Fin n => let (p,n) := as i
-    if p > n then (p - n,0) else (0,n - p)
+def ReductionProp
+  (xs : Fin n → Fin 2) (ks : Coeff n) (K : ℕ)
+  : Prop :=
+  let pos := λ i => ks i |>.1
+  let neg := λ i => ks i |>.2
+  let slack := (∑i, min (pos i) (neg i))
+  let rs := λ i => (pos i - neg i,neg i - pos i)
+  PBIneq rs xs (K - slack)
 
-def getSlack (as : Coeff n) : ℕ :=
-  ∑ i : Fin n, let (p,n) := as i
-    min p n
+theorem Reduction
+  (xs : Fin n → Fin 2)
+  (ks : Coeff n) (K : ℕ) (ha : PBIneq ks xs K)
+  : ReductionProp xs ks K := by
+  sorry
+  done
 
 def AdditionProp
   (xs : Fin n → Fin 2)
   (as : Coeff n) (A : ℕ)
   (bs : Coeff n) (B : ℕ)
   : Prop :=
-  let abs := as + bs
-  let ts := tighten abs
-  let slack := getSlack abs
-  PBIneq ts xs (A + B - slack)
+  ReductionProp xs (as + bs) (A + B)
 
 -- Addition
 -- ∑i (a i * l i) ≥ A
@@ -101,21 +113,15 @@ theorem Addition
   {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
   {bs : Coeff n} {B : ℕ} (hb : PBIneq bs xs B)
   : AdditionProp xs as A bs B := by
-  unfold AdditionProp PBIneq PBSum getSlack tighten at *
-  simp at *
+  unfold AdditionProp ReductionProp PBIneq PBSum at *
+  simp only [Fin.isValue, ge_iff_le, Pi.add_apply, Prod.fst_add, Prod.snd_add, tsub_le_iff_right] at *
   /-
-  A + B ≤
-  (∑ x : Fin n,
-      if xs x = 1 then
-        (Decidable.rec (fun h => (0, (as x).2 + (bs x).2 - ((as x).1 + (bs x).1)))
-            (fun h => ((as x).1 + (bs x).1 - ((as x).2 + (bs x).2), 0))
-            (((as x).2 + (bs x).2).decLt ((as x).1 + (bs x).1))).1
-      else
-        (Decidable.rec (fun h => (0, (as x).2 + (bs x).2 - ((as x).1 + (bs x).1)))
-            (fun h => ((as x).1 + (bs x).1 - ((as x).2 + (bs x).2), 0))
-            (((as x).2 + (bs x).2).decLt ((as x).1 + (bs x).1))).2) +
-    ∑ x : Fin n, min ((as x).1 + (bs x).1) ((as x).2 + (bs x).2)
-  -/
+  ha : A ≤ ∑i, if xs i = 1 then (as i).1 else (as i).2
+  hb : B ≤ ∑i, if xs i = 1 then (bs i).1 else (bs i).2
+  ⊢ A + B ≤
+    (∑i, if xs i = 1 then (as i).1 + (bs i).1 - ((as i).2 + (bs i).2) else (as i).2 + (bs i).2 - ((as i).1 + (bs i).1))
+    + ∑i, min ((as i).1 + (bs i).1) ((as i).2 + (bs i).2)
+ -/
   sorry
   done
 

From 532dc97d661f0761e0504bb075368effa0c79953 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 5 Jun 2024 11:11:07 -0300
Subject: [PATCH 39/93] Solve `Addition` by applying sub-theorems

---
 LeanCuttingPlanes/Rules/Addition.lean | 61 +++++++++++++--------------
 1 file changed, 30 insertions(+), 31 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index 621eb10..123f15a 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -51,32 +51,19 @@ lemma ite_eq_bmul (x y : ℕ) (b : Fin 2)
     simp only [Fin.isValue, ↓reduceIte, Fin.val_one, mul_one, ge_iff_le, le_refl,
       tsub_eq_zero_of_le, mul_zero, add_zero]
 
-theorem ThePlan
-   {xs : Fin n → Fin 2}
-  {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
-  {bs : Coeff n} {B : ℕ} (hb : PBIneq bs xs B)
-  -- : AdditionProp xs as A bs B := byi
-  : True := by
-  set K := A + B
-  set ks := as + bs
-  have hk : PBIneq ks xs K := sorry -- here is Addition Proof without reduction
-  simp [PBIneq,PBSum] at hk         -- K ≤ ∑ i : Fin n, if xs i = 1 then (ks i).1 else (ks i).2
-  simp_rw [ite_eq_bmul] at hk       -- K ≤ ∑ i : Fin n, ((ks i).1 * ↑(xs i) + (ks i).2 * (1 - ↑(xs i)))
-  set pos := λ i => ks i |>.1 with rpos
-  set neg := λ i => ks i |>.2 with rneg
-  -- simp_rw [rpos,rneg] at hk
-  -- simp_rw [reduce_terms] at hk   -- K ≤ ∑ i : Fin n, ((pos i - neg i) * ↑(xs x) + (neg i - pos i) * (1 - ↑(xs x)) + min (pos i) (neg i))
-  -- apply sum_split_min_term       -- K ≤ ∑ i : Fin n, ((pos i - neg i) * ↑(xs x) + (neg i - pos i) * (1 - ↑(xs x))) + ∑i,min (pos i) (neg i)
-  -- apply sub_ge_a_sub_sum         -- K - ∑i,min (pos i) (neg i) ≤ ∑ i : Fin n, ((pos i - neg i) * ↑(xs x) + (neg i - pos i) * (1 - ↑(xs x)))
-  --
-  sorry
-  done
-
 theorem Addition'
   (xs : Fin n → Fin 2)
   (as : Coeff n) (A : ℕ) (ha : PBIneq as xs A)
   (bs : Coeff n) (B : ℕ) (hb : PBIneq bs xs B)
   : PBIneq (as + bs) xs (A + B) := by
+  unfold PBIneq PBSum at *
+  simp only [Fin.isValue, ge_iff_le] at *
+  /-
+  ha : A ≤ ∑ x : Fin n, if xs x = 1 then (as x).1 else (as x).2
+  hb : B ≤ ∑ x : Fin n, if xs x = 1 then (bs x).1 else (bs x).2
+  ⊢ A + B
+  ≤ ∑i, if xs i = 1 then (as i).1 + (bs i).1 else (as i).2 + (bs i).2
+  -/
   sorry
   done
 
@@ -93,7 +80,27 @@ theorem Reduction
   (xs : Fin n → Fin 2)
   (ks : Coeff n) (K : ℕ) (ha : PBIneq ks xs K)
   : ReductionProp xs ks K := by
+  unfold ReductionProp PBIneq PBSum at *
+  simp only [Fin.isValue, ge_iff_le, tsub_le_iff_right] at *
+  /-
+  ha : K ≤ ∑ x : Fin n, if xs x = 1 then (ks x).1 else (ks x).2
+  ⊢ K ≤
+    (∑i,  if xs i = 1
+          then (ks x).1 - (ks x).2
+          else (ks x).2 - (ks x).1
+    )
+    + ∑i, min (ks i).1 (ks i).2
+  -/
   sorry
+  -- simp [PBIneq,PBSum] at hk         -- K ≤ ∑ i : Fin n, if xs i = 1 then (ks i).1 else (ks i).2
+  -- simp_rw [ite_eq_bmul] at hk       -- K ≤ ∑ i : Fin n, ((ks i).1 * ↑(xs i) + (ks i).2 * (1 - ↑(xs i)))
+  -- set pos := λ i => ks i |>.1 with rpos
+  -- set neg := λ i => ks i |>.2 with rneg
+  -- simp_rw [rpos,rneg] at hk
+  -- simp_rw [reduce_terms] at hk   -- K ≤ ∑ i : Fin n, ((pos i - neg i) * ↑(xs x) + (neg i - pos i) * (1 - ↑(xs x)) + min (pos i) (neg i))
+  -- apply sum_split_min_term       -- K ≤ ∑ i : Fin n, ((pos i - neg i) * ↑(xs x) + (neg i - pos i) * (1 - ↑(xs x))) + ∑i,min (pos i) (neg i)
+  -- apply sub_ge_a_sub_sum         -- K - ∑i,min (pos i) (neg i) ≤ ∑ i : Fin n, ((pos i - neg i) * ↑(xs x) + (neg i - pos i) * (1 - ↑(xs x)))
+  --
   done
 
 def AdditionProp
@@ -113,16 +120,8 @@ theorem Addition
   {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
   {bs : Coeff n} {B : ℕ} (hb : PBIneq bs xs B)
   : AdditionProp xs as A bs B := by
-  unfold AdditionProp ReductionProp PBIneq PBSum at *
-  simp only [Fin.isValue, ge_iff_le, Pi.add_apply, Prod.fst_add, Prod.snd_add, tsub_le_iff_right] at *
-  /-
-  ha : A ≤ ∑i, if xs i = 1 then (as i).1 else (as i).2
-  hb : B ≤ ∑i, if xs i = 1 then (bs i).1 else (bs i).2
-  ⊢ A + B ≤
-    (∑i, if xs i = 1 then (as i).1 + (bs i).1 - ((as i).2 + (bs i).2) else (as i).2 + (bs i).2 - ((as i).1 + (bs i).1))
-    + ∑i, min ((as i).1 + (bs i).1) ((as i).2 + (bs i).2)
- -/
-  sorry
+  have hk := Addition' xs as A ha bs B hb
+  exact Reduction xs (as + bs) (A + B) hk
   done
 
 example

From 462123096451d89ea3842ecbb06aba2c5e4911e2 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 5 Jun 2024 11:13:33 -0300
Subject: [PATCH 40/93] Move lemmas for clarity

---
 LeanCuttingPlanes/Rules/Addition.lean | 50 +++++++++++++--------------
 1 file changed, 25 insertions(+), 25 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index 123f15a..bc81a53 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -4,6 +4,31 @@ namespace PseudoBoolean
 
 open BigOperators FinVec Matrix
 
+theorem Addition'
+  (xs : Fin n → Fin 2)
+  (as : Coeff n) (A : ℕ) (ha : PBIneq as xs A)
+  (bs : Coeff n) (B : ℕ) (hb : PBIneq bs xs B)
+  : PBIneq (as + bs) xs (A + B) := by
+  unfold PBIneq PBSum at *
+  simp only [Fin.isValue, ge_iff_le] at *
+  /-
+  ha : A ≤ ∑ x : Fin n, if xs x = 1 then (as x).1 else (as x).2
+  hb : B ≤ ∑ x : Fin n, if xs x = 1 then (bs x).1 else (bs x).2
+  ⊢ A + B
+  ≤ ∑i, if xs i = 1 then (as i).1 + (bs i).1 else (as i).2 + (bs i).2
+  -/
+  sorry
+  done
+
+def ReductionProp
+  (xs : Fin n → Fin 2) (ks : Coeff n) (K : ℕ)
+  : Prop :=
+  let pos := λ i => ks i |>.1
+  let neg := λ i => ks i |>.2
+  let slack := (∑i, min (pos i) (neg i))
+  let rs := λ i => (pos i - neg i,neg i - pos i)
+  PBIneq rs xs (K - slack)
+
 #check Fin.eq_one_of_neq_zero
 #check Nat.min_comm
 #check Nat.sub_add_min_cancel
@@ -51,31 +76,6 @@ lemma ite_eq_bmul (x y : ℕ) (b : Fin 2)
     simp only [Fin.isValue, ↓reduceIte, Fin.val_one, mul_one, ge_iff_le, le_refl,
       tsub_eq_zero_of_le, mul_zero, add_zero]
 
-theorem Addition'
-  (xs : Fin n → Fin 2)
-  (as : Coeff n) (A : ℕ) (ha : PBIneq as xs A)
-  (bs : Coeff n) (B : ℕ) (hb : PBIneq bs xs B)
-  : PBIneq (as + bs) xs (A + B) := by
-  unfold PBIneq PBSum at *
-  simp only [Fin.isValue, ge_iff_le] at *
-  /-
-  ha : A ≤ ∑ x : Fin n, if xs x = 1 then (as x).1 else (as x).2
-  hb : B ≤ ∑ x : Fin n, if xs x = 1 then (bs x).1 else (bs x).2
-  ⊢ A + B
-  ≤ ∑i, if xs i = 1 then (as i).1 + (bs i).1 else (as i).2 + (bs i).2
-  -/
-  sorry
-  done
-
-def ReductionProp
-  (xs : Fin n → Fin 2) (ks : Coeff n) (K : ℕ)
-  : Prop :=
-  let pos := λ i => ks i |>.1
-  let neg := λ i => ks i |>.2
-  let slack := (∑i, min (pos i) (neg i))
-  let rs := λ i => (pos i - neg i,neg i - pos i)
-  PBIneq rs xs (K - slack)
-
 theorem Reduction
   (xs : Fin n → Fin 2)
   (ks : Coeff n) (K : ℕ) (ha : PBIneq ks xs K)

From 8856bcfae954aa4b9f49339c5946dbc0660a2f87 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 5 Jun 2024 11:14:23 -0300
Subject: [PATCH 41/93] Removed `check`s

---
 LeanCuttingPlanes/Rules/Addition.lean | 8 --------
 1 file changed, 8 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index bc81a53..d3c39dc 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -29,10 +29,6 @@ def ReductionProp
   let rs := λ i => (pos i - neg i,neg i - pos i)
   PBIneq rs xs (K - slack)
 
-#check Fin.eq_one_of_neq_zero
-#check Nat.min_comm
-#check Nat.sub_add_min_cancel
-
 lemma reduce_terms (p n : ℕ) (x : Fin 2)
   : p * x + n * (1 - x) = (p - n) * x + (n - p) * (1 - x) + min p n  := by
   by_cases h : x = 0
@@ -47,16 +43,12 @@ lemma reduce_terms (p n : ℕ) (x : Fin 2)
     simp only [Fin.isValue, Fin.val_one, mul_one, ge_iff_le, le_refl, tsub_eq_zero_of_le, mul_zero, add_zero]
     exact Nat.sub_add_min_cancel p n |>.symm
 
-#check Finset.sum_add_distrib
-
 lemma sum_split_min_term (p n : Fin m → ℕ) (x : Fin m → Fin 2)
   : (∑i,((p i - n i) * x i + (n i - p i) * (1 - x i)   +      min (p i) (n i)))
   = (∑i,((p i - n i) * x i + (n i - p i) * (1 - x i))) + (∑i,(min (p i) (n i))) := by
   exact Finset.sum_add_distrib
   done
 
-#check Nat.sub_le_of_le_add
-
 lemma sum_ge_a_sub_sum (p n : Fin m → ℕ) (x : Fin m → Fin 2) (A : ℕ)
   (h : ∑i,((p i - n i) * x i + (n i - p i) * (1 - x i)) + ∑i,(min (p i) (n i)) ≥ A)
   :   (∑i,((p i - n i) * x i + (n i - p i) * (1 - x i))) ≥ A - ∑i,(min (p i) (n i)) := by

From 42504fc7817ea6bbd9c23caed0ff25a4ae11d8db Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 5 Jun 2024 16:25:21 +0000
Subject: [PATCH 42/93] `Addition'` Proved

---
 LeanCuttingPlanes/Rules/Addition.lean | 10 +++-------
 1 file changed, 3 insertions(+), 7 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index d3c39dc..0fa21a0 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -11,13 +11,9 @@ theorem Addition'
   : PBIneq (as + bs) xs (A + B) := by
   unfold PBIneq PBSum at *
   simp only [Fin.isValue, ge_iff_le] at *
-  /-
-  ha : A ≤ ∑ x : Fin n, if xs x = 1 then (as x).1 else (as x).2
-  hb : B ≤ ∑ x : Fin n, if xs x = 1 then (bs x).1 else (bs x).2
-  ⊢ A + B
-  ≤ ∑i, if xs i = 1 then (as i).1 + (bs i).1 else (as i).2 + (bs i).2
-  -/
-  sorry
+  simp_rw [←ite_add_ite]
+  rw [Finset.sum_add_distrib]
+  exact Nat.add_le_add ha hb
   done
 
 def ReductionProp

From bc2fc96d72f59b067739c1bdba6e04c2fd5245b9 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 5 Jun 2024 19:11:27 +0000
Subject: [PATCH 43/93] Proved Reduction!

---
 LeanCuttingPlanes/Rules/Addition.lean | 61 ++++++++-------------------
 1 file changed, 17 insertions(+), 44 deletions(-)

diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index 0fa21a0..98a8ea9 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -25,6 +25,19 @@ def ReductionProp
   let rs := λ i => (pos i - neg i,neg i - pos i)
   PBIneq rs xs (K - slack)
 
+lemma ite_eq_bmul (x y : ℕ) (b : Fin 2)
+  : (if b = 1 then x else y) = (x * b + y * (1 - b)) := by
+  by_cases h : b = 0
+  . rw [h]
+    rw [if_neg]
+    . simp only [Fin.isValue, Fin.val_zero, mul_zero, tsub_zero, mul_one, zero_add]
+    trivial
+  . -- b = 1
+    apply Fin.eq_one_of_neq_zero b at h
+    rw [h]
+    simp only [Fin.isValue, ↓reduceIte, Fin.val_one, mul_one, ge_iff_le, le_refl,
+      tsub_eq_zero_of_le, mul_zero, add_zero]
+
 lemma reduce_terms (p n : ℕ) (x : Fin 2)
   : p * x + n * (1 - x) = (p - n) * x + (n - p) * (1 - x) + min p n  := by
   by_cases h : x = 0
@@ -39,56 +52,16 @@ lemma reduce_terms (p n : ℕ) (x : Fin 2)
     simp only [Fin.isValue, Fin.val_one, mul_one, ge_iff_le, le_refl, tsub_eq_zero_of_le, mul_zero, add_zero]
     exact Nat.sub_add_min_cancel p n |>.symm
 
-lemma sum_split_min_term (p n : Fin m → ℕ) (x : Fin m → Fin 2)
-  : (∑i,((p i - n i) * x i + (n i - p i) * (1 - x i)   +      min (p i) (n i)))
-  = (∑i,((p i - n i) * x i + (n i - p i) * (1 - x i))) + (∑i,(min (p i) (n i))) := by
-  exact Finset.sum_add_distrib
-  done
-
-lemma sum_ge_a_sub_sum (p n : Fin m → ℕ) (x : Fin m → Fin 2) (A : ℕ)
-  (h : ∑i,((p i - n i) * x i + (n i - p i) * (1 - x i)) + ∑i,(min (p i) (n i)) ≥ A)
-  :   (∑i,((p i - n i) * x i + (n i - p i) * (1 - x i))) ≥ A - ∑i,(min (p i) (n i)) := by
-  exact Nat.sub_le_of_le_add h
-  done
-
-lemma ite_eq_bmul (x y : ℕ) (b : Fin 2)
-  : (if b = 1 then x else y) = (x * b + y * (1 - b)) := by
-  by_cases h : b = 0
-  . rw [h]
-    rw [if_neg]
-    . simp only [Fin.isValue, Fin.val_zero, mul_zero, tsub_zero, mul_one, zero_add]
-    trivial
-  . -- b = 1
-    apply Fin.eq_one_of_neq_zero b at h
-    rw [h]
-    simp only [Fin.isValue, ↓reduceIte, Fin.val_one, mul_one, ge_iff_le, le_refl,
-      tsub_eq_zero_of_le, mul_zero, add_zero]
-
 theorem Reduction
   (xs : Fin n → Fin 2)
   (ks : Coeff n) (K : ℕ) (ha : PBIneq ks xs K)
   : ReductionProp xs ks K := by
   unfold ReductionProp PBIneq PBSum at *
   simp only [Fin.isValue, ge_iff_le, tsub_le_iff_right] at *
-  /-
-  ha : K ≤ ∑ x : Fin n, if xs x = 1 then (ks x).1 else (ks x).2
-  ⊢ K ≤
-    (∑i,  if xs i = 1
-          then (ks x).1 - (ks x).2
-          else (ks x).2 - (ks x).1
-    )
-    + ∑i, min (ks i).1 (ks i).2
-  -/
-  sorry
-  -- simp [PBIneq,PBSum] at hk         -- K ≤ ∑ i : Fin n, if xs i = 1 then (ks i).1 else (ks i).2
-  -- simp_rw [ite_eq_bmul] at hk       -- K ≤ ∑ i : Fin n, ((ks i).1 * ↑(xs i) + (ks i).2 * (1 - ↑(xs i)))
-  -- set pos := λ i => ks i |>.1 with rpos
-  -- set neg := λ i => ks i |>.2 with rneg
-  -- simp_rw [rpos,rneg] at hk
-  -- simp_rw [reduce_terms] at hk   -- K ≤ ∑ i : Fin n, ((pos i - neg i) * ↑(xs x) + (neg i - pos i) * (1 - ↑(xs x)) + min (pos i) (neg i))
-  -- apply sum_split_min_term       -- K ≤ ∑ i : Fin n, ((pos i - neg i) * ↑(xs x) + (neg i - pos i) * (1 - ↑(xs x))) + ∑i,min (pos i) (neg i)
-  -- apply sub_ge_a_sub_sum         -- K - ∑i,min (pos i) (neg i) ≤ ∑ i : Fin n, ((pos i - neg i) * ↑(xs x) + (neg i - pos i) * (1 - ↑(xs x)))
-  --
+  simp_rw [ite_eq_bmul] at *
+  rw [←Finset.sum_add_distrib]
+  simp_rw [←reduce_terms]
+  exact ha
   done
 
 def AdditionProp

From 9c96327d2d3147f65e272d1d3116ddb531001a69 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 5 Jun 2024 19:16:29 +0000
Subject: [PATCH 44/93] Check Addition at readme

---
 README.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/README.md b/README.md
index adf7ea7..ce14014 100644
--- a/README.md
+++ b/README.md
@@ -26,7 +26,7 @@ Run `mdbook test` to test all `lean` code blocks.
 
 ## Work Progress
 
-- [ ] Addition proof
+- [x] Addition proof
 - [x] Multiplication proof
 - [x] Saturation proof
 - [x] Division proof

From 6c3eaffc1b3ebdbef575e803cdafd49823a1c857 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 5 Jun 2024 19:18:52 +0000
Subject: [PATCH 45/93] No more sorry's

---
 README.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/README.md b/README.md
index ce14014..a00e9f0 100644
--- a/README.md
+++ b/README.md
@@ -48,7 +48,7 @@ open PseudoBoolean
 
 variable {xs : Fin 5 → Fin 2}
 
-def my_pb : PBIneq ![(0,1),(2,0),(0,3),(4,0),(0,5)] xs 1 := sorry
+def my_pb : PBIneq ![(0,1),(2,0),(0,3),(4,0),(0,5)] xs 1 := ...
 ```
 
 This notation is under development and is subject to changes.

From c8d9b74e56608e0b0bd0bd285fec9454dd1ceaeb Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 5 Jun 2024 21:01:58 +0000
Subject: [PATCH 46/93] Changed mdbook for mathjax-support

---
 book.toml            |  3 +++
 docs/introduction.md |  2 +-
 docs/title_page.md   | 10 +++++++++-
 3 files changed, 13 insertions(+), 2 deletions(-)

diff --git a/book.toml b/book.toml
index 7f31b03..bcf05fa 100644
--- a/book.toml
+++ b/book.toml
@@ -7,6 +7,9 @@ title = "Cutting Planes with Lean 4"
 
 [output.html]
 git-repository-url = "https://github.com/bernborgess/lean-cutting-planes"
+mathjax-support = true
 
 [output.html.playground.boring-prefixes]
 lean = "# "
+
+# mdbook serve -p 8000 -n 127.0.0.1
diff --git a/docs/introduction.md b/docs/introduction.md
index 46399ce..2145f67 100644
--- a/docs/introduction.md
+++ b/docs/introduction.md
@@ -4,4 +4,4 @@ Introduction
 Cutting Planes and Lean 4
 -----------------------------
 
-*Cutting Planes* is a formal logic. 
+*Cutting Planes* is a formal logic. 
\ No newline at end of file
diff --git a/docs/title_page.md b/docs/title_page.md
index a05986f..38cb8ac 100644
--- a/docs/title_page.md
+++ b/docs/title_page.md
@@ -2,4 +2,12 @@
 
 *by Bernardo Borges*
 
-This book is the official documentation for the usage of the [lean-cutting-planes](https://github.com/bernborgess/lean-cutting-planes) library. It is currently a work in progress, with the goal to provide full expressivity in Lean 4 for the cutting planes formal logic.
\ No newline at end of file
+This book is the official documentation for the usage of the [lean-cutting-planes](https://github.com/bernborgess/lean-cutting-planes) library. It is currently a work in progress, with the goal to provide full expressivity in Lean 4 for the cutting planes formal logic.
+
+It consists of 6 rules, as in:
+
+<!-- Inline -->
+\\( \int x dx = \frac{x^2}{2} + C \\)
+
+<!-- Block -->
+\\[ \mu = \frac{1}{N} \sum_{i=0} x_i \\]
\ No newline at end of file

From 93cba7685e1fed687f1afdb2349b0943cc4b50ed Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 13 Jun 2024 18:43:58 +0000
Subject: [PATCH 47/93] Title page and introduction written

---
 docs/introduction.md | 23 ++++++++++++++++++++++-
 docs/title_page.md   |  8 +-------
 2 files changed, 23 insertions(+), 8 deletions(-)

diff --git a/docs/introduction.md b/docs/introduction.md
index 2145f67..9471481 100644
--- a/docs/introduction.md
+++ b/docs/introduction.md
@@ -4,4 +4,25 @@ Introduction
 Cutting Planes and Lean 4
 -----------------------------
 
-*Cutting Planes* is a formal logic. 
\ No newline at end of file
+*Cutting Planes* is a formal logic used in the context of discrete combinatorial problems, that can be applied in _Software testing_, _Formal verification_ and _Proof logging_.
+
+Our proofs will consist of _Pseudo-Boolean Constraints_, that are 0-1 integer linear inequalities of the form:
+\\[ \sum_i{a_i l_i} \ge A \\]
+Where we have:
+- \\( a_i, A \in \mathbb{Z} \\)
+- literals \\( l_i: x_i\ \text{or}\ \overline{x_i}\ (\text{where } x_i + \overline{x_i} = 1) \\)
+- variables \\( x_i \\) take values \\( 0 = false \\) or \\( 1 = true \\)
+
+Without loss of generality, we will use the [normalized form](Bar95), with all \\( a_i, A\\) non-negative.
+
+In this regard, _Pseudo-Boolean Reasoning_ works on two axioms and four rules, each of which are formally verified in this project using _Lean 4_
+
+Lean 4 is a theorem prover and programming language developed by Leonardo de Moura. In the last years it became proeminent in the mathematics community, after [The Xena Project](https://www.ma.imperial.ac.uk/~buzzard/xena/) of Imperial College London formalized Peter Scholze's proof in the area of condensed mathematics in 2021. In 2023 Terence Tao used Lean to formalize a proof of the _Polynomial Freiman-Ruzsa (PFR) conjecture_, published in the same year.
+
+Armed with a Lean 4 proof of the reasoning rules, complex proofs that involve these steps can be guaranteed to be correct.
+
+<!-- Inline -->
+<!-- \\( \int x dx = \frac{x^2}{2} + C \\) -->
+
+<!-- Block -->
+<!-- \\[ \mu = \frac{1}{N} \sum_{i=0} x_i \\] -->
\ No newline at end of file
diff --git a/docs/title_page.md b/docs/title_page.md
index 38cb8ac..a696039 100644
--- a/docs/title_page.md
+++ b/docs/title_page.md
@@ -4,10 +4,4 @@
 
 This book is the official documentation for the usage of the [lean-cutting-planes](https://github.com/bernborgess/lean-cutting-planes) library. It is currently a work in progress, with the goal to provide full expressivity in Lean 4 for the cutting planes formal logic.
 
-It consists of 6 rules, as in:
-
-<!-- Inline -->
-\\( \int x dx = \frac{x^2}{2} + C \\)
-
-<!-- Block -->
-\\[ \mu = \frac{1}{N} \sum_{i=0} x_i \\]
\ No newline at end of file
+Future work in this project will aim to verify unsat demonstrations that were produced by solvers suchs as [RoundingSat](https://gitlab.com/MIAOresearch/software/roundingsat) and check that their reasoning was correct.

From 4e1732fa633d968eae02a334ed243db6a813bddc Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 13 Jun 2024 19:12:54 +0000
Subject: [PATCH 48/93] Rules documentation page

---
 docs/SUMMARY.md              |  1 +
 docs/introduction.md         |  9 ++-----
 docs/pseudo_boolean_rules.md | 49 ++++++++++++++++++++++++++++++++++++
 3 files changed, 52 insertions(+), 7 deletions(-)
 create mode 100644 docs/pseudo_boolean_rules.md

diff --git a/docs/SUMMARY.md b/docs/SUMMARY.md
index ffef43b..b143430 100644
--- a/docs/SUMMARY.md
+++ b/docs/SUMMARY.md
@@ -4,3 +4,4 @@
 [Cutting Planes with Lean 4](title_page.md)
 
 - [Introduction](./introduction.md)
+- [Pseudo Boolean Rules](./pseudo_boolean_rules.md)
diff --git a/docs/introduction.md b/docs/introduction.md
index 9471481..1485f9b 100644
--- a/docs/introduction.md
+++ b/docs/introduction.md
@@ -9,7 +9,8 @@ Cutting Planes and Lean 4
 Our proofs will consist of _Pseudo-Boolean Constraints_, that are 0-1 integer linear inequalities of the form:
 \\[ \sum_i{a_i l_i} \ge A \\]
 Where we have:
-- \\( a_i, A \in \mathbb{Z} \\)
+- constant \\( A \in \mathbb{Z} \\)
+- coefficients \\( a_i \in \mathbb{Z} \\)
 - literals \\( l_i: x_i\ \text{or}\ \overline{x_i}\ (\text{where } x_i + \overline{x_i} = 1) \\)
 - variables \\( x_i \\) take values \\( 0 = false \\) or \\( 1 = true \\)
 
@@ -20,9 +21,3 @@ In this regard, _Pseudo-Boolean Reasoning_ works on two axioms and four rules, e
 Lean 4 is a theorem prover and programming language developed by Leonardo de Moura. In the last years it became proeminent in the mathematics community, after [The Xena Project](https://www.ma.imperial.ac.uk/~buzzard/xena/) of Imperial College London formalized Peter Scholze's proof in the area of condensed mathematics in 2021. In 2023 Terence Tao used Lean to formalize a proof of the _Polynomial Freiman-Ruzsa (PFR) conjecture_, published in the same year.
 
 Armed with a Lean 4 proof of the reasoning rules, complex proofs that involve these steps can be guaranteed to be correct.
-
-<!-- Inline -->
-<!-- \\( \int x dx = \frac{x^2}{2} + C \\) -->
-
-<!-- Block -->
-<!-- \\[ \mu = \frac{1}{N} \sum_{i=0} x_i \\] -->
\ No newline at end of file
diff --git a/docs/pseudo_boolean_rules.md b/docs/pseudo_boolean_rules.md
new file mode 100644
index 0000000..c3962c9
--- /dev/null
+++ b/docs/pseudo_boolean_rules.md
@@ -0,0 +1,49 @@
+Pseudo Boolean Rules
+============
+
+Input Axioms
+-----------------------------
+
+As a starting point for the pseudo boolean reasoning we will get constraints from the user, which are subject to a normalization process
+
+Literal Axioms
+-----------------------------
+
+For any literal \\( l_i\\) we can create a constraint:
+\\[ \frac{}
+    {l_i \ge 0}
+\\]
+
+Addition Rule
+-----------------------------
+Two constraints can be added together, adding the coefficients and the constants. Another behaviour is that \\(x_i\\) and \\(\overline{x_i}\\) that may appear together cancel each other:
+\\[ \frac
+    {{\sum_i{a_i l_i} \ge A}\qquad {\sum_i{b_i l_i} \ge B}}
+    {\sum_i{(a_i + b_i) l_i} \ge (A+B)}
+\\]
+
+Multiplication Rule
+-----------------------------
+A constraint can be multiplied by any \\( c \in \mathbb{N}^{+} \\):
+\\[ \frac
+    {\sum_i{a_i l_i} \ge A}
+    {\sum_i{c a_i l_i} \ge c A}
+\\]
+
+Division Rule
+-----------------------------
+A constraint can be divided by any \\( c \in \mathbb{N}^{+} \\), and the the ceiling of this division in applied:
+\\[ \frac
+    {\sum_i{a_i l_i} \ge A}
+    {\sum_i{ \lceil \frac{a_i}{c} \rceil l_i} \ge \lceil \frac{A}{c} \rceil}
+\\]
+
+Saturation Rule
+-----------------------------
+A constraint can be replace its coefficients by the minimum between them and the constant:
+\\[ \frac
+    {\sum_i{a_i l_i} \ge A}
+    {\sum_i{ \min(a_i,A)\cdot l_i} \ge A}
+\\]
+
+All these rules can be seen in practice in the Worked Example
\ No newline at end of file

From 6dbf82e1091ce4f51ebcc9443a21907ffbc8f348 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 13 Jun 2024 19:44:16 +0000
Subject: [PATCH 49/93] Finished worked example

---
 docs/SUMMARY.md        |  1 +
 docs/worked_example.md | 60 ++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 61 insertions(+)
 create mode 100644 docs/worked_example.md

diff --git a/docs/SUMMARY.md b/docs/SUMMARY.md
index b143430..45053d3 100644
--- a/docs/SUMMARY.md
+++ b/docs/SUMMARY.md
@@ -5,3 +5,4 @@
 
 - [Introduction](./introduction.md)
 - [Pseudo Boolean Rules](./pseudo_boolean_rules.md)
+- [Worked Example](./worked_example.md)
diff --git a/docs/worked_example.md b/docs/worked_example.md
new file mode 100644
index 0000000..3768fd0
--- /dev/null
+++ b/docs/worked_example.md
@@ -0,0 +1,60 @@
+Worked Example
+============
+
+Input
+-----------------------------
+
+For this example, in constraints over variables \\(w,x,y,z\\) we will have two input axioms:
+
+1. \\( w + 2x + y \ge 2 \\)
+1. \\( w + 2x + 4y + 2z \ge 5 \\)
+
+Goal 
+-----------------------------
+The objective of this example is to prove:
+\\[ w + 2x + 2y \ge 3 \\]
+
+Execution
+-----------------------------
+We start by using muliplication of the first constraint by 2:
+\\[ \frac
+    {w + 2x + y \ge 2}
+    {2w + 4x + 2y \ge 4}
+    (\text{Multiply by 2})
+\\]
+Now we add this result to constraint 2:
+\\[ \frac
+    {{2w + 4x + 2y \ge 4}\qquad {w + 2x + 4y + 2z \ge 5}}
+    {3w + 6x + 6y + 2z \ge 9}
+    (\text{Add})
+\\]
+At this point we want to remove the variable \\(z\\), as it does not appear in our goal. Then we will introduce a \\(\overline{z} \ge 0\\) by the literal axiom and then multiply it by 2:
+\\[ \frac
+    {\overline{z} \ge 0}
+    {2\overline{z} \ge 0}
+    (\text{Multiply by 2})
+\\]
+Performing addition will cancel out the \\(z\\) variable:
+\\[ \frac
+    {{3w + 6x + 6y + 2z \ge 9}\qquad {2\overline{z} \ge 0}}
+    {3w + 6x + 6y \ge 7}
+    (\text{Add})
+\\]
+And lastly a division by 3 is performed to arrive at the goal:
+\\[ \frac
+    {3w + 6x + 6y \ge 7}
+    {w + 2x + 2y \ge 3}
+    (\text{Divide by 3})
+\\]
+
+Summarizing, this is the full proof:
+
+
+
+
+
+
+
+
+
+![alt](./assets/toy_example.png)

From c123b60334944e87f1f3c0298373e5c0ae82adba Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 13 Jun 2024 19:52:25 +0000
Subject: [PATCH 50/93] Added Lean code to worked example

---
 docs/worked_example.md | 55 +++++++++++++++++++++++++++++++++++-------
 1 file changed, 46 insertions(+), 9 deletions(-)

diff --git a/docs/worked_example.md b/docs/worked_example.md
index 3768fd0..f6fc4da 100644
--- a/docs/worked_example.md
+++ b/docs/worked_example.md
@@ -4,16 +4,28 @@ Worked Example
 Input
 -----------------------------
 
-For this example, in constraints over variables \\(w,x,y,z\\) we will have two input axioms:
+For this example, in constraints over variables \\(w,x,y,z\\) we will have three input axioms:
 
 1. \\( w + 2x + y \ge 2 \\)
 1. \\( w + 2x + 4y + 2z \ge 5 \\)
+1. \\( \overline{z} \ge 0 \\)
+
+```lean
+example (xs : Fin 4 → Fin 2)
+  (c1 : PBIneq ![(1,0),(2,0),(1,0),(0,0)] xs 2)
+  (c2 : PBIneq ![(1,0),(2,0),(4,0),(2,0)] xs 5)
+  (c3 : PBIneq ![(0,0),(0,0),(0,0),(0,1)] xs 0)
+```
 
 Goal 
 -----------------------------
 The objective of this example is to prove:
 \\[ w + 2x + 2y \ge 3 \\]
 
+```lean
+  : PBIneq ![(1,0),(2,0),(2,0),(0,0)] xs 3 := by
+```
+
 Execution
 -----------------------------
 We start by using muliplication of the first constraint by 2:
@@ -22,39 +34,64 @@ We start by using muliplication of the first constraint by 2:
     {2w + 4x + 2y \ge 4}
     (\text{Multiply by 2})
 \\]
+```lean
+  let t1 : PBIneq ![(2,0),(4,0),(2,0),(0,0)] xs 4  := by apply Multiplication c1 2
+```
+
 Now we add this result to constraint 2:
 \\[ \frac
     {{2w + 4x + 2y \ge 4}\qquad {w + 2x + 4y + 2z \ge 5}}
     {3w + 6x + 6y + 2z \ge 9}
     (\text{Add})
 \\]
+```lean
+  let t2 : PBIneq ![(3,0),(6,0),(6,0),(2,0)] xs 9  := by apply Addition t1 c2
+```
 At this point we want to remove the variable \\(z\\), as it does not appear in our goal. Then we will introduce a \\(\overline{z} \ge 0\\) by the literal axiom and then multiply it by 2:
 \\[ \frac
     {\overline{z} \ge 0}
     {2\overline{z} \ge 0}
     (\text{Multiply by 2})
 \\]
+```lean
+  let t3 : PBIneq ![(0,0),(0,0),(0,0),(0,2)] xs 0  := by apply Multiplication c3 2
+```
 Performing addition will cancel out the \\(z\\) variable:
 \\[ \frac
     {{3w + 6x + 6y + 2z \ge 9}\qquad {2\overline{z} \ge 0}}
     {3w + 6x + 6y \ge 7}
     (\text{Add})
 \\]
+```lean
+  let t4 : PBIneq ![(3,0),(6,0),(6,0),(0,0)] xs 7  := by apply Addition t2 t3
+```
 And lastly a division by 3 is performed to arrive at the goal:
 \\[ \frac
     {3w + 6x + 6y \ge 7}
     {w + 2x + 2y \ge 3}
     (\text{Divide by 3})
 \\]
+```lean
+  exact Division t4 3
+  done
+```
 
 Summarizing, this is the full proof:
 
+![toy_example](./assets/toy_example.png)
 
-
-
-
-
-
-
-
-![alt](./assets/toy_example.png)
+```lean
+example
+  (xs : Fin 4 → Fin 2)
+  (c1 : PBIneq ![(1,0),(2,0),(1,0),(0,0)] xs 2)
+  (c2 : PBIneq ![(1,0),(2,0),(4,0),(2,0)] xs 5)
+  (c3 : PBIneq ![(0,0),(0,0),(0,0),(0,1)] xs 0)
+  : PBIneq ![(1,0),(2,0),(2,0),(0,0)] xs 3
+  := by
+  let t1 : PBIneq ![(2,0),(4,0),(2,0),(0,0)] xs 4  := by apply Multiplication c1 2
+  let t2 : PBIneq ![(3,0),(6,0),(6,0),(2,0)] xs 9  := by apply Addition t1 c2
+  let t3 : PBIneq ![(0,0),(0,0),(0,0),(0,2)] xs 0  := by apply Multiplication c3 2
+  let t4 : PBIneq ![(3,0),(6,0),(6,0),(0,0)] xs 7  := by apply Addition t2 t3
+  exact Division t4 3
+  done
+```

From d2cac0c7fc2b2759a8fa1d6cc46d323254e3779a Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 13 Jun 2024 22:18:34 +0000
Subject: [PATCH 51/93] Fixing typos and adding links

---
 docs/introduction.md         | 4 ++--
 docs/pseudo_boolean_rules.md | 2 +-
 2 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/docs/introduction.md b/docs/introduction.md
index 1485f9b..354b7a3 100644
--- a/docs/introduction.md
+++ b/docs/introduction.md
@@ -14,10 +14,10 @@ Where we have:
 - literals \\( l_i: x_i\ \text{or}\ \overline{x_i}\ (\text{where } x_i + \overline{x_i} = 1) \\)
 - variables \\( x_i \\) take values \\( 0 = false \\) or \\( 1 = true \\)
 
-Without loss of generality, we will use the [normalized form](Bar95), with all \\( a_i, A\\) non-negative.
+Without loss of generality, we will use the [normalized form](https://pure.mpg.de/rest/items/item_1832217_4/component/file_2574300/content#page=7), with all \\( a_i, A\\) non-negative.
 
 In this regard, _Pseudo-Boolean Reasoning_ works on two axioms and four rules, each of which are formally verified in this project using _Lean 4_
 
-Lean 4 is a theorem prover and programming language developed by Leonardo de Moura. In the last years it became proeminent in the mathematics community, after [The Xena Project](https://www.ma.imperial.ac.uk/~buzzard/xena/) of Imperial College London formalized Peter Scholze's proof in the area of condensed mathematics in 2021. In 2023 Terence Tao used Lean to formalize a proof of the _Polynomial Freiman-Ruzsa (PFR) conjecture_, published in the same year.
+[Lean 4](https://lean-lang.org/) is a theorem prover and programming language developed by [Leonardo de Moura](http://leodemoura.github.io/). In the last years it became proeminent in the mathematics community, after [The Xena Project](https://www.ma.imperial.ac.uk/~buzzard/xena/) of Imperial College London formalized Peter Scholze's proof in the area of condensed mathematics in 2021. In 2023 Terence Tao used Lean to formalize a proof of the _Polynomial Freiman-Ruzsa [(PFR)](https://teorth.github.io/pfr/) conjecture_, published in the same year.
 
 Armed with a Lean 4 proof of the reasoning rules, complex proofs that involve these steps can be guaranteed to be correct.
diff --git a/docs/pseudo_boolean_rules.md b/docs/pseudo_boolean_rules.md
index c3962c9..6e31796 100644
--- a/docs/pseudo_boolean_rules.md
+++ b/docs/pseudo_boolean_rules.md
@@ -40,7 +40,7 @@ A constraint can be divided by any \\( c \in \mathbb{N}^{+} \\), and the the cei
 
 Saturation Rule
 -----------------------------
-A constraint can be replace its coefficients by the minimum between them and the constant:
+A constraint can replace its coefficients by the minimum between them and the constant:
 \\[ \frac
     {\sum_i{a_i l_i} \ge A}
     {\sum_i{ \min(a_i,A)\cdot l_i} \ge A}

From aae9c2e862d580003ddcef06507a83364d5e5a01 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 18 Jun 2024 16:27:22 -0300
Subject: [PATCH 52/93] Starting with latex

---
 .gitignore   | 308 ++++++++++++++++++++++++++++++++++++++++++++++++++-
 docs/poc.pdf | Bin 0 -> 60676 bytes
 docs/poc.tex |  48 ++++++++
 3 files changed, 355 insertions(+), 1 deletion(-)
 create mode 100644 docs/poc.pdf
 create mode 100644 docs/poc.tex

diff --git a/.gitignore b/.gitignore
index c0bb988..438d45e 100644
--- a/.gitignore
+++ b/.gitignore
@@ -5,4 +5,310 @@
 /lake-packages/*
 
 /.vscode
-/.lake
\ No newline at end of file
+/.lake
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+*.rubbercache
+rubber.cache
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# comment
+*.cut
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+*.lzo
+*.lzs
+*.slg
+*.slo
+*.sls
+
+# uncomment this for glossaries-extra (will ignore makeindex's style files!)
+# *.ist
+
+# gnuplot
+*.gnuplot
+*.table
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.glog
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hypdoc
+*.hd
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
+# *.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# luatexja-ruby
+*.ltjruby
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# newpax
+*.newpax
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# svg
+svg-inkscape/
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# tcolorbox
+*.listing
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# titletoc
+*.ptc
+
+# todonotes
+*.tdo
+
+# vhistory
+*.hst
+*.ver
+
+# easy-todo
+*.lod
+
+# xcolor
+*.xcp
+
+# xmpincl
+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices and outlines
+*.xyc
+*.xyd
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# gummi
+.*.swp
+
+# KBibTeX
+*~[0-9]*
+
+# TeXnicCenter
+*.tps
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta
+
+# Makeindex log files
+*.lpz
+
+# xwatermark package
+*.xwm
+
+# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
+# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
+# Uncomment the next line to have this generated file ignored.
+#*Notes.bib
diff --git a/docs/poc.pdf b/docs/poc.pdf
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HcmV?d00001

diff --git a/docs/poc.tex b/docs/poc.tex
new file mode 100644
index 0000000..a5efd09
--- /dev/null
+++ b/docs/poc.tex
@@ -0,0 +1,48 @@
+\documentclass[a4paper]{article}
+\usepackage{listings}
+\usepackage{color}
+
+% \lstset{style=bw}
+
+\begin{document}
+
+\section{Introduction}
+
+\[
+    \sum_i{a} = b + c
+\]
+
+%Custom Colors
+\definecolor{plainText}{RGB}{74,131,31}
+\definecolor{comments}{RGB}{74,131,31}
+\definecolor{strings}{RGB}{180,54,34}
+\definecolor{numbers}{RGB}{44,45,211}
+\definecolor{keywords}{RGB}{160,49,158}
+\definecolor{preprocessorStatements}{RGB}{109,75,48}
+\definecolor{classNames}{RGB}{101,63,165}
+
+
+\lstset{frame=tb,
+    language=C++,
+    aboveskip=3mm,
+    belowskip=3mm,
+    showstringspaces=false,
+    columns=flexible,
+    basicstyle={\small\ttfamily},
+    numbers=left,
+    numberstyle=\tiny\color{plainText},
+    keywordstyle=\color{keywords},
+    commentstyle=\color{comments},
+    stringstyle=\color{strings},
+    breaklines=true,
+    breakatwhitespace=true
+    tabsize=4
+}
+
+\lstset{language=python}
+\begin{lstlisting}
+    def foo:
+        print "oi"
+\end{lstlisting}
+
+\end{document}
\ No newline at end of file

From 777f9557361261de489a344ac7193e2abd1cebc3 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 18 Jun 2024 19:57:07 +0000
Subject: [PATCH 53/93] Minor changes in wording of docs

---
 docs/assets/highlight.js     | 2377 ++++++++++++++++++++++++++++++++++
 docs/introduction.md         |   12 +-
 docs/pseudo_boolean_rules.md |   29 +-
 3 files changed, 2404 insertions(+), 14 deletions(-)
 create mode 100644 docs/assets/highlight.js

diff --git a/docs/assets/highlight.js b/docs/assets/highlight.js
new file mode 100644
index 0000000..903c2d7
--- /dev/null
+++ b/docs/assets/highlight.js
@@ -0,0 +1,2377 @@
+/*
+  Highlight.js 10.3.2 (31e1fc40)
+  License: BSD-3-Clause
+  Copyright (c) 2006-2020, Ivan Sagalaev
+*/
+var hljs = function () {
+    "use strict"; function e(n) {
+        Object.freeze(n)
+        ; var t = "function" == typeof n
+            ; return Object.getOwnPropertyNames(n).forEach((function (r) {
+                !Object.hasOwnProperty.call(n, r) || null === n[r] || "object" != typeof n[r] && "function" != typeof n[r] || t && ("caller" === r || "callee" === r || "arguments" === r) || Object.isFrozen(n[r]) || e(n[r])
+            })), n
+    } class n {
+        constructor(e) { void 0 === e.data && (e.data = {}), this.data = e.data }
+        ignoreMatch() { this.ignore = !0 }
+    } function t(e) {
+        return e.replace(/&/g, "&amp;").replace(/</g, "&lt;").replace(/>/g, "&gt;").replace(/"/g, "&quot;").replace(/'/g, "&#x27;")
+    } function r(e, ...n) {
+        var t = {}; for (const n in e) t[n] = e[n]
+            ; return n.forEach((function (e) { for (const n in e) t[n] = e[n] })), t
+    } function a(e) {
+        return e.nodeName.toLowerCase()
+    } var i = Object.freeze({
+        __proto__: null,
+        escapeHTML: t, inherit: r, nodeStream: function (e) {
+            var n = []; return function e(t, r) {
+                for (var i = t.firstChild; i; i = i.nextSibling)3 === i.nodeType ? r += i.nodeValue.length : 1 === i.nodeType && (n.push({
+                    event: "start", offset: r, node: i
+                }), r = e(i, r), a(i).match(/br|hr|img|input/) || n.push({
+                    event: "stop", offset: r, node: i
+                })); return r
+            }(e, 0), n
+        }, mergeStreams: function (e, n, r) {
+            var i = 0, s = "", o = []; function l() {
+                return e.length && n.length ? e[0].offset !== n[0].offset ? e[0].offset < n[0].offset ? e : n : "start" === n[0].event ? e : n : e.length ? e : n
+            } function c(e) {
+                s += "<" + a(e) + [].map.call(e.attributes, (function (e) {
+                    return " " + e.nodeName + '="' + t(e.value) + '"'
+                })).join("") + ">"
+            } function u(e) {
+                s += "</" + a(e) + ">"
+            } function g(e) { ("start" === e.event ? c : u)(e.node) }
+            for (; e.length || n.length;) {
+                var d = l()
+                ; if (s += t(r.substring(i, d[0].offset)), i = d[0].offset, d === e) {
+                    o.reverse().forEach(u)
+                    ; do { g(d.splice(0, 1)[0]), d = l() } while (d === e && d.length && d[0].offset === i)
+                        ; o.reverse().forEach(c)
+                } else "start" === d[0].event ? o.push(d[0].node) : o.pop(), g(d.splice(0, 1)[0])
+            }
+            return s + t(r.substr(i))
+        }
+    }); const s = e => !!e.kind; class o {
+        constructor(e, n) {
+            this.buffer = "", this.classPrefix = n.classPrefix, e.walk(this)
+        } addText(e) {
+            this.buffer += t(e)
+        } openNode(e) {
+            if (!s(e)) return; let n = e.kind
+                ; e.sublanguage || (n = `${this.classPrefix}${n}`), this.span(n)
+        } closeNode(e) {
+            s(e) && (this.buffer += "</span>")
+        } value() { return this.buffer } span(e) {
+            this.buffer += `<span class="${e}">`
+        }
+    } class l {
+        constructor() {
+            this.rootNode = {
+                children: []
+            }, this.stack = [this.rootNode]
+        } get top() {
+            return this.stack[this.stack.length - 1]
+        } get root() { return this.rootNode } add(e) {
+            this.top.children.push(e)
+        } openNode(e) {
+            const n = { kind: e, children: [] }
+            ; this.add(n), this.stack.push(n)
+        } closeNode() {
+            if (this.stack.length > 1) return this.stack.pop()
+        } closeAllNodes() {
+            for (; this.closeNode(););
+        } toJSON() { return JSON.stringify(this.rootNode, null, 4) }
+        walk(e) { return this.constructor._walk(e, this.rootNode) } static _walk(e, n) {
+            return "string" == typeof n ? e.addText(n) : n.children && (e.openNode(n),
+                n.children.forEach((n => this._walk(e, n))), e.closeNode(n)), e
+        } static _collapse(e) {
+            "string" != typeof e && e.children && (e.children.every((e => "string" == typeof e)) ? e.children = [e.children.join("")] : e.children.forEach((e => {
+                l._collapse(e)
+            })))
+        }
+    } class c extends l {
+        constructor(e) { super(), this.options = e }
+        addKeyword(e, n) { "" !== e && (this.openNode(n), this.addText(e), this.closeNode()) }
+        addText(e) { "" !== e && this.add(e) } addSublanguage(e, n) {
+            const t = e.root
+            ; t.kind = n, t.sublanguage = !0, this.add(t)
+        } toHTML() {
+            return new o(this, this.options).value()
+        } finalize() { return !0 }
+    } function u(e) {
+        return e ? "string" == typeof e ? e : e.source : null
+    }
+    const g = "[a-zA-Z]\\w*", d = "[a-zA-Z_]\\w*", h = "\\b\\d+(\\.\\d+)?", f = "(-?)(\\b0[xX][a-fA-F0-9]+|(\\b\\d+(\\.\\d*)?|\\.\\d+)([eE][-+]?\\d+)?)", p = "\\b(0b[01]+)", m = {
+        begin: "\\\\[\\s\\S]", relevance: 0
+    }, b = {
+        className: "string", begin: "'", end: "'",
+        illegal: "\\n", contains: [m]
+    }, v = {
+        className: "string", begin: '"', end: '"',
+        illegal: "\\n", contains: [m]
+    }, x = {
+        begin: /\b(a|an|the|are|I'm|isn't|don't|doesn't|won't|but|just|should|pretty|simply|enough|gonna|going|wtf|so|such|will|you|your|they|like|more)\b/
+    }, E = function (e, n, t = {}) {
+        var a = r({
+            className: "comment", begin: e, end: n, contains: []
+        }, t); return a.contains.push(x), a.contains.push({
+            className: "doctag",
+            begin: "(?:TODO|FIXME|NOTE|BUG|OPTIMIZE|HACK|XXX):", relevance: 0
+        }), a
+    }, _ = E("//", "$"), w = E("/\\*", "\\*/"), N = E("#", "$"); var y = Object.freeze({
+        __proto__: null, IDENT_RE: g, UNDERSCORE_IDENT_RE: d, NUMBER_RE: h, C_NUMBER_RE: f,
+        BINARY_NUMBER_RE: p,
+        RE_STARTERS_RE: "!|!=|!==|%|%=|&|&&|&=|\\*|\\*=|\\+|\\+=|,|-|-=|/=|/|:|;|<<|<<=|<=|<|===|==|=|>>>=|>>=|>=|>>>|>>|>|\\?|\\[|\\{|\\(|\\^|\\^=|\\||\\|=|\\|\\||~",
+        SHEBANG: (e = {}) => {
+            const n = /^#![ ]*\//; return e.binary && (e.begin = function (...e) {
+                return e.map((e => u(e))).join("")
+            }(n, /.*\b/, e.binary, /\b.*/)), r({
+                className: "meta", begin: n, end: /$/, relevance: 0, "on:begin": (e, n) => {
+                    0 !== e.index && n.ignoreMatch()
+                }
+            }, e)
+        }, BACKSLASH_ESCAPE: m, APOS_STRING_MODE: b,
+        QUOTE_STRING_MODE: v, PHRASAL_WORDS_MODE: x, COMMENT: E, C_LINE_COMMENT_MODE: _,
+        C_BLOCK_COMMENT_MODE: w, HASH_COMMENT_MODE: N, NUMBER_MODE: {
+            className: "number",
+            begin: h, relevance: 0
+        }, C_NUMBER_MODE: { className: "number", begin: f, relevance: 0 },
+        BINARY_NUMBER_MODE: { className: "number", begin: p, relevance: 0 }, CSS_NUMBER_MODE: {
+            className: "number",
+            begin: h + "(%|em|ex|ch|rem|vw|vh|vmin|vmax|cm|mm|in|pt|pc|px|deg|grad|rad|turn|s|ms|Hz|kHz|dpi|dpcm|dppx)?",
+            relevance: 0
+        }, REGEXP_MODE: {
+            begin: /(?=\/[^/\n]*\/)/, contains: [{
+                className: "regexp",
+                begin: /\//, end: /\/[gimuy]*/, illegal: /\n/, contains: [m, {
+                    begin: /\[/, end: /\]/,
+                    relevance: 0, contains: [m]
+                }]
+            }]
+        }, TITLE_MODE: {
+            className: "title", begin: g, relevance: 0
+        }, UNDERSCORE_TITLE_MODE: { className: "title", begin: d, relevance: 0 }, METHOD_GUARD: {
+            begin: "\\.\\s*[a-zA-Z_]\\w*", relevance: 0
+        }, END_SAME_AS_BEGIN: function (e) {
+            return Object.assign(e, {
+                "on:begin": (e, n) => { n.data._beginMatch = e[1] },
+                "on:end": (e, n) => { n.data._beginMatch !== e[1] && n.ignoreMatch() }
+            })
+        }
+    }), R = "of and for in not or if then".split(" "); function k(e) {
+        function n(n, t) {
+            return RegExp(u(n), "m" + (e.case_insensitive ? "i" : "") + (t ? "g" : ""))
+        } class t {
+            constructor() {
+                this.matchIndexes = {}, this.regexes = [], this.matchAt = 1, this.position = 0
+            }
+            addRule(e, n) {
+                n.position = this.position++, this.matchIndexes[this.matchAt] = n, this.regexes.push([n, e]),
+                    this.matchAt += function (e) {
+                        return RegExp(e.toString() + "|").exec("").length - 1
+                    }(e) + 1
+            } compile() {
+                0 === this.regexes.length && (this.exec = () => null)
+                ; const e = this.regexes.map((e => e[1])); this.matcherRe = n(function (e, n = "|") {
+                    for (var t = /\[(?:[^\\\]]|\\.)*\]|\(\??|\\([1-9][0-9]*)|\\./, r = 0, a = "", i = 0; i < e.length; i++) {
+                        var s = r += 1, o = u(e[i]); for (i > 0 && (a += n), a += "("; o.length > 0;) {
+                            var l = t.exec(o)
+                            ; if (null == l) { a += o; break }
+                            a += o.substring(0, l.index), o = o.substring(l.index + l[0].length),
+                                "\\" === l[0][0] && l[1] ? a += "\\" + (Number(l[1]) + s) : (a += l[0], "(" === l[0] && r++)
+                        } a += ")"
+                    }
+                    return a
+                }(e), !0), this.lastIndex = 0
+            } exec(e) {
+                this.matcherRe.lastIndex = this.lastIndex; const n = this.matcherRe.exec(e)
+                    ; if (!n) return null
+                        ; const t = n.findIndex(((e, n) => n > 0 && void 0 !== e)), r = this.matchIndexes[t]
+                    ; return n.splice(0, t), Object.assign(n, r)
+            }
+        } class a {
+            constructor() {
+                this.rules = [], this.multiRegexes = [],
+                    this.count = 0, this.lastIndex = 0, this.regexIndex = 0
+            } getMatcher(e) {
+                if (this.multiRegexes[e]) return this.multiRegexes[e]; const n = new t
+                    ; return this.rules.slice(e).forEach((([e, t]) => n.addRule(e, t))),
+                        n.compile(), this.multiRegexes[e] = n, n
+            } resumingScanAtSamePosition() {
+                return 0 !== this.regexIndex
+            } considerAll() { this.regexIndex = 0 } addRule(e, n) {
+                this.rules.push([e, n]), "begin" === n.type && this.count++
+            } exec(e) {
+                const n = this.getMatcher(this.regexIndex); n.lastIndex = this.lastIndex
+                    ; let t = n.exec(e)
+                    ; if (this.resumingScanAtSamePosition()) if (t && t.index === this.lastIndex); else {
+                        const n = this.getMatcher(0); n.lastIndex = this.lastIndex + 1, t = n.exec(e)
+                    }
+                return t && (this.regexIndex += t.position + 1,
+                    this.regexIndex === this.count && this.considerAll()), t
+            }
+        } function i(e, n) {
+            const t = e.input[e.index - 1], r = e.input[e.index + e[0].length]
+                ; "." !== t && "." !== r || n.ignoreMatch()
+        }
+        if (e.contains && e.contains.includes("self")) throw Error("ERR: contains `self` is not supported at the top-level of a language.  See documentation.")
+            ; return function t(s, o) {
+                const l = s; if (s.compiled) return l
+                    ; s.compiled = !0, s.__beforeBegin = null, s.keywords = s.keywords || s.beginKeywords
+                    ; let c = null
+                    ; if ("object" == typeof s.keywords && (c = s.keywords.$pattern, delete s.keywords.$pattern),
+                        s.keywords && (s.keywords = function (e, n) {
+                            var t = {}
+                            ; return "string" == typeof e ? r("keyword", e) : Object.keys(e).forEach((function (n) {
+                                r(n, e[n])
+                            })), t; function r(e, r) {
+                                n && (r = r.toLowerCase()), r.split(" ").forEach((function (n) {
+                                    var r = n.split("|")
+                                    ; t[r[0]] = [e, O(r[0], r[1])]
+                                }))
+                            }
+                        }(s.keywords, e.case_insensitive)), s.lexemes && c) throw Error("ERR: Prefer `keywords.$pattern` to `mode.lexemes`, BOTH are not allowed. (see mode reference) ")
+                    ; return l.keywordPatternRe = n(s.lexemes || c || /\w+/, !0),
+                        o && (s.beginKeywords && (s.begin = "\\b(" + s.beginKeywords.split(" ").join("|") + ")(?=\\b|\\s)",
+                            s.__beforeBegin = i),
+                            s.begin || (s.begin = /\B|\b/), l.beginRe = n(s.begin), s.endSameAsBegin && (s.end = s.begin),
+                            s.end || s.endsWithParent || (s.end = /\B|\b/),
+                            s.end && (l.endRe = n(s.end)), l.terminator_end = u(s.end) || "",
+                            s.endsWithParent && o.terminator_end && (l.terminator_end += (s.end ? "|" : "") + o.terminator_end)),
+                        s.illegal && (l.illegalRe = n(s.illegal)),
+                        void 0 === s.relevance && (s.relevance = 1), s.contains || (s.contains = []),
+                        s.contains = [].concat(...s.contains.map((function (e) {
+                            return function (e) {
+                                return e.variants && !e.cached_variants && (e.cached_variants = e.variants.map((function (n) {
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+                                }))), e.cached_variants ? e.cached_variants : M(e) ? r(e, {
+                                    starts: e.starts ? r(e.starts) : null
+                                }) : Object.isFrozen(e) ? r(e) : e
+                            }("self" === e ? s : e)
+                        }))), s.contains.forEach((function (e) {
+                            t(e, l)
+                        })), s.starts && t(s.starts, o), l.matcher = function (e) {
+                            const n = new a
+                            ; return e.contains.forEach((e => n.addRule(e.begin, {
+                                rule: e, type: "begin"
+                            }))), e.terminator_end && n.addRule(e.terminator_end, {
+                                type: "end"
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+                        }(l), l
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+    } function M(e) {
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+    }
+    const L = {
+        props: ["language", "code", "autodetect"], data: function () {
+            return {
+                detectedLanguage: "", unknownLanguage: !1
+            }
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+            className() {
+                return this.unknownLanguage ? "" : "hljs " + this.detectedLanguage
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+                if (!this.autoDetect && !hljs.getLanguage(this.language)) return console.warn(`The language "${this.language}" you specified could not be found.`),
+                    this.unknownLanguage = !0, t(this.code); let e
+                    ; return this.autoDetect ? (e = hljs.highlightAuto(this.code),
+                        this.detectedLanguage = e.language) : (e = hljs.highlight(this.language, this.code, this.ignoreIllegals),
+                            this.detectectLanguage = this.language), e.value
+            }, autoDetect() {
+                return !(this.language && (e = this.autodetect, !e && "" !== e));
+            },
+            ignoreIllegals: () => !0
+        }, render(e) {
+            return e("pre", {}, [e("code", {
+                class: this.className, domProps: { innerHTML: this.highlighted }
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+        install(e) {
+            e.component("highlightjs", L)
+        }
+    }, I = t, T = r, { nodeStream: S, mergeStreams: A } = i, B = Symbol("nomatch")
+        ; return function (t) {
+            var r = [], a = Object.create(null), i = Object.create(null), s = [], o = !0, l = /(^(<[^>]+>|\t|)+|\n)/gm, u = "Could not find the language '{}', did you forget to load/include a language module?"
+                ; const g = { disableAutodetect: !0, name: "Plain text", contains: [] }; var d = {
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+                    languageDetectRe: /\blang(?:uage)?-([\w-]+)\b/i, classPrefix: "hljs-",
+                    tabReplace: null, useBR: !1, languages: null, __emitter: c
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+                } function f(e, n, t, r) {
+                    var a = { code: n, language: e }
+                    ; N("before:highlight", a); var i = a.result ? a.result : p(a.language, a.code, t, r)
+                        ; return i.code = a.code, N("after:highlight", i), i
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+                        var t = _.case_insensitive ? n[0].toLowerCase() : n[0]
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+                    }
+                    function c() {
+                        null != y.subLanguage ? function () {
+                            if ("" !== O) {
+                                var e = null
+                                ; if ("string" == typeof y.subLanguage) {
+                                    if (!a[y.subLanguage]) return void M.addText(O)
+                                        ; e = p(y.subLanguage, O, !0, R[y.subLanguage]), R[y.subLanguage] = e.top
+                                } else e = m(O, y.subLanguage.length ? y.subLanguage : null)
+                                    ; y.relevance > 0 && (L += e.relevance), M.addSublanguage(e.emitter, e.language)
+                            }
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+                            if (!y.keywords) return void M.addText(O); let e = 0
+                                ; y.keywordPatternRe.lastIndex = 0; let n = y.keywordPatternRe.exec(O), t = ""; for (; n;) {
+                                    t += O.substring(e, n.index); const r = l(y, n); if (r) {
+                                        const [e, a] = r
+                                        ; M.addText(t), t = "", L += a, M.addKeyword(n[0], e)
+                                    } else t += n[0]
+                                        ; e = y.keywordPatternRe.lastIndex, n = y.keywordPatternRe.exec(O)
+                                }
+                            t += O.substr(e), M.addText(t)
+                        }(), O = ""
+                    } function g(e) {
+                        return e.className && M.openNode(e.className), y = Object.create(e, {
+                            parent: { value: y }
+                        })
+                    } function h(e, t, r) {
+                        let a = function (e, n) {
+                            var t = e && e.exec(n)
+                            ; return t && 0 === t.index
+                        }(e.endRe, r); if (a) {
+                            if (e["on:end"]) {
+                                const r = new n(e)
+                                ; e["on:end"](t, r), r.ignore && (a = !1)
+                            } if (a) {
+                                for (; e.endsParent && e.parent;)e = e.parent
+                                    ; return e
+                            }
+                        } if (e.endsWithParent) return h(e.parent, t, r)
+                    } function f(e) {
+                        return 0 === y.matcher.regexIndex ? (O += e[0], 1) : (S = !0, 0)
+                    } function b(e) {
+                        var n = e[0], t = s.substr(e.index), r = h(y, e, t); if (!r) return B; var a = y
+                            ; a.skip ? O += n : (a.returnEnd || a.excludeEnd || (O += n), c(), a.excludeEnd && (O = n)); do {
+                                y.className && M.closeNode(), y.skip || y.subLanguage || (L += y.relevance), y = y.parent
+                            } while (y !== r.parent)
+                            ; return r.starts && (r.endSameAsBegin && (r.starts.endRe = r.endRe),
+                                g(r.starts)), a.returnEnd ? 0 : n.length
+                    } var v = {}; function x(t, a) {
+                        var i = a && a[0]
+                        ; if (O += t, null == i) return c(), 0
+                            ; if ("begin" === v.type && "end" === a.type && v.index === a.index && "" === i) {
+                                if (O += s.slice(a.index, a.index + 1), !o) {
+                                    const n = Error("0 width match regex")
+                                    ; throw n.languageName = e, n.badRule = v.rule, n
+                                } return 1
+                            }
+                        if (v = a, "begin" === a.type) return function (e) {
+                            var t = e[0], r = e.rule
+                            ; const a = new n(r), i = [r.__beforeBegin, r["on:begin"]]
+                                ; for (const n of i) if (n && (n(e, a), a.ignore)) return f(t)
+                                    ; return r && r.endSameAsBegin && (r.endRe = RegExp(t.replace(/[-/\\^$*+?.()|[\]{}]/g, "\\$&"), "m")),
+                                        r.skip ? O += t : (r.excludeBegin && (O += t),
+                                            c(), r.returnBegin || r.excludeBegin || (O = t)), g(r), r.returnBegin ? 0 : t.length
+                        }(a)
+                            ; if ("illegal" === a.type && !r) {
+                                const e = Error('Illegal lexeme "' + i + '" for mode "' + (y.className || "<unnamed>") + '"')
+                                    ; throw e.mode = y, e
+                            } if ("end" === a.type) { var l = b(a); if (l !== B) return l }
+                        if ("illegal" === a.type && "" === i) return 1
+                            ; if (T > 1e5 && T > 3 * a.index) throw Error("potential infinite loop, way more iterations than matches")
+                                ; return O += i, i.length
+                    } var _ = E(e)
+                        ; if (!_) throw console.error(u.replace("{}", e)), Error('Unknown language: "' + e + '"')
+                            ; var w = k(_), N = "", y = i || w, R = {}, M = new d.__emitter(d); !function () {
+                                for (var e = [], n = y; n !== _; n = n.parent)n.className && e.unshift(n.className)
+                                    ; e.forEach((e => M.openNode(e)))
+                            }(); var O = "", L = 0, j = 0, T = 0, S = !1; try {
+                                for (y.matcher.considerAll(); ;) {
+                                    T++, S ? S = !1 : y.matcher.considerAll(), y.matcher.lastIndex = j
+                                        ; const e = y.matcher.exec(s); if (!e) break; const n = x(s.substring(j, e.index), e)
+                                        ; j = e.index + n
+                                } return x(s.substr(j)), M.closeAllNodes(), M.finalize(), N = M.toHTML(), {
+                                    relevance: L, value: N, language: e, illegal: !1, emitter: M, top: y
+                                }
+                            } catch (n) {
+                                if (n.message && n.message.includes("Illegal")) return {
+                                    illegal: !0, illegalBy: {
+                                        msg: n.message, context: s.slice(j - 100, j + 100), mode: n.mode
+                                    }, sofar: N, relevance: 0,
+                                    value: I(s), emitter: M
+                                }; if (o) return {
+                                    illegal: !1, relevance: 0, value: I(s), emitter: M,
+                                    language: e, top: y, errorRaised: n
+                                }; throw n
+                            }
+                } function m(e, n) {
+                    n = n || d.languages || Object.keys(a); var t = function (e) {
+                        const n = {
+                            relevance: 0,
+                            emitter: new d.__emitter(d), value: I(e), illegal: !1, top: g
+                        }
+                        ; return n.emitter.addText(e), n
+                    }(e), r = t
+                        ; return n.filter(E).filter(w).forEach((function (n) {
+                            var a = p(n, e, !1); a.language = n,
+                                a.relevance > r.relevance && (r = a), a.relevance > t.relevance && (r = t, t = a)
+                        })), r.language && (t.second_best = r), t
+                } function b(e) {
+                    return d.tabReplace || d.useBR ? e.replace(l, (e => "\n" === e ? d.useBR ? "<br>" : e : d.tabReplace ? e.replace(/\t/g, d.tabReplace) : e)) : e
+                } function v(e) {
+                    let n = null; const t = function (e) {
+                        var n = e.className + " "
+                        ; n += e.parentNode ? e.parentNode.className : ""; const t = d.languageDetectRe.exec(n)
+                            ; if (t) {
+                                var r = E(t[1])
+                                ; return r || (console.warn(u.replace("{}", t[1])), console.warn("Falling back to no-highlight mode for this block.", e)),
+                                    r ? t[1] : "no-highlight"
+                            } return n.split(/\s+/).find((e => h(e) || E(e)))
+                    }(e)
+                        ; if (h(t)) return; N("before:highlightBlock", {
+                            block: e, language: t
+                        }), d.useBR ? (n = document.createElement("div"),
+                            n.innerHTML = e.innerHTML.replace(/\n/g, "").replace(/<br[ /]*>/g, "\n")) : n = e
+                        ; const r = n.textContent, a = t ? f(t, r, !0) : m(r), s = S(n); if (s.length) {
+                            const e = document.createElement("div"); e.innerHTML = a.value, a.value = A(s, S(e), r)
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+                        block: e, result: a
+                    }), e.innerHTML = a.value, e.className = function (e, n, t) {
+                        var r = n ? i[n] : t, a = [e.trim()]
+                        ; return e.match(/\bhljs\b/) || a.push("hljs"),
+                            e.includes(r) || a.push(r), a.join(" ").trim()
+                    }(e.className, t, a.language), e.result = {
+                        language: a.language, re: a.relevance,
+                        relavance: a.relevance
+                    }, a.second_best && (e.second_best = {
+                        language: a.second_best.language, re: a.second_best.relevance,
+                        relavance: a.second_best.relevance
+                    })
+                } const x = () => {
+                    if (!x.called) {
+                        x.called = !0
+                        ; var e = document.querySelectorAll("pre code"); r.forEach.call(e, v)
+                    }
+                }
+                ; function E(e) { return e = (e || "").toLowerCase(), a[e] || a[i[e]] }
+            function _(e, { languageName: n }) {
+                "string" == typeof e && (e = [e]), e.forEach((e => {
+                    i[e] = n
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+            } function w(e) { var n = E(e); return n && !n.disableAutodetect } function N(e, n) {
+                var t = e; s.forEach((function (e) { e[t] && e[t](n) }))
+            } Object.assign(t, {
+                highlight: f,
+                highlightAuto: m, fixMarkup: function (e) {
+                    return console.warn("fixMarkup is deprecated and will be removed entirely in v11.0"),
+                        console.warn("Please see https://github.com/highlightjs/highlight.js/issues/2534"),
+                        b(e)
+                }, highlightBlock: v, configure: function (e) {
+                    e.useBR && (console.warn("'useBR' option is deprecated and will be removed entirely in v11.0"),
+                        console.warn("Please see https://github.com/highlightjs/highlight.js/issues/2559")),
+                        d = T(d, e)
+                }, initHighlighting: x, initHighlightingOnLoad: function () {
+                    window.addEventListener("DOMContentLoaded", x, !1)
+                },
+                registerLanguage: function (e, n) {
+                    var r = null; try { r = n(t) } catch (n) {
+                        if (console.error("Language definition for '{}' could not be registered.".replace("{}", e)),
+                            !o) throw n; console.error(n), r = g
+                    }
+                    r.name || (r.name = e), a[e] = r, r.rawDefinition = n.bind(null, t),
+                        r.aliases && _(r.aliases, { languageName: e })
+                }, listLanguages: function () {
+                    return Object.keys(a)
+                }, getLanguage: E, registerAliases: _,
+                requireLanguage: function (e) {
+                    var n = E(e); if (n) return n
+                        ; throw Error("The '{}' language is required, but not loaded.".replace("{}", e))
+                },
+                autoDetection: w, inherit: T, addPlugin: function (e) { s.push(e) }, vuePlugin: j
+            }), t.debugMode = function () { o = !1 }, t.safeMode = function () {
+                o = !0
+            }, t.versionString = "10.3.2"; for (const n in y) "object" == typeof y[n] && e(y[n])
+                ; return Object.assign(t, y), t
+        }({})
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+    ; "object" == typeof exports && "undefined" != typeof module && (module.exports = hljs);
+hljs.registerLanguage("apache", function () {
+    "use strict"; return function (e) {
+        var n = {
+            className: "number",
+            begin: "\\d{1,3}\\.\\d{1,3}\\.\\d{1,3}\\.\\d{1,3}(:\\d{1,5})?"
+        }; return {
+            name: "Apache config", aliases: ["apacheconf"], case_insensitive: !0,
+            contains: [e.HASH_COMMENT_MODE, {
+                className: "section", begin: "</?", end: ">",
+                contains: [n, {
+                    className: "number", begin: ":\\d{1,5}"
+                }, e.inherit(e.QUOTE_STRING_MODE, { relevance: 0 })]
+            }, {
+                className: "attribute",
+                begin: /\w+/, relevance: 0, keywords: {
+                    nomarkup: "order deny allow setenv rewriterule rewriteengine rewritecond documentroot sethandler errordocument loadmodule options header listen serverroot servername"
+                }, starts: {
+                    end: /$/, relevance: 0, keywords: { literal: "on off all deny allow" },
+                    contains: [{ className: "meta", begin: "\\s\\[", end: "\\]$" }, {
+                        className: "variable",
+                        begin: "[\\$%]\\{", end: "\\}", contains: ["self", {
+                            className: "number",
+                            begin: "[\\$%]\\d+"
+                        }]
+                    }, n, { className: "number", begin: "\\d+" }, e.QUOTE_STRING_MODE]
+                }
+            }], illegal: /\S/
+        }
+    }
+}());
+hljs.registerLanguage("bash", function () {
+    "use strict"; return function (e) {
+        const s = {}; Object.assign(s, {
+            className: "variable", variants: [{
+                begin: /\$[\w\d#@][\w\d_]*/
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+                begin: /\$\{/, end: /\}/, contains: ["self", {
+                    begin: /:-/,
+                    contains: [s]
+                }]
+            }]
+        }); const n = {
+            className: "subst", begin: /\$\(/, end: /\)/,
+            contains: [e.BACKSLASH_ESCAPE]
+        }, t = {
+            begin: /<<-?\s*(?=\w+)/, starts: {
+                contains: [e.END_SAME_AS_BEGIN({ begin: /(\w+)/, end: /(\w+)/, className: "string" })]
+            }
+        }, a = { className: "string", begin: /"/, end: /"/, contains: [e.BACKSLASH_ESCAPE, s, n] }
+            ; n.contains.push(a); const i = {
+                begin: /\$\(\(/, end: /\)\)/, contains: [{
+                    begin: /\d+#[0-9a-f]+/, className: "number"
+                }, e.NUMBER_MODE, s]
+            }, c = e.SHEBANG({
+                binary: "(fish|bash|zsh|sh|csh|ksh|tcsh|dash|scsh)", relevance: 10
+            }), o = {
+                className: "function", begin: /\w[\w\d_]*\s*\(\s*\)\s*\{/, returnBegin: !0,
+                contains: [e.inherit(e.TITLE_MODE, { begin: /\w[\w\d_]*/ })], relevance: 0
+            }; return {
+                name: "Bash", aliases: ["sh", "zsh"], keywords: {
+                    $pattern: /\b[a-z._-]+\b/,
+                    keyword: "if then else elif fi for while in do done case esac function",
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+                }, contains: [c, e.SHEBANG(), o, i, e.COMMENT("(?<!\\S)#", "$"), t, a, {
+                    className: "", begin: /\\"/
+                }, { className: "string", begin: /'/, end: /'/ }, s]
+            }
+    }
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+hljs.registerLanguage("c-like", function () {
+    "use strict"; return function (e) {
+        function t(e) { return "(?:" + e + ")?" } var n = e.COMMENT("//", "$", {
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+                begin: /\\\n/
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+            className: "keyword", begin: "\\b[a-z\\d_]*_t\\b"
+        }, s = {
+            className: "string",
+            variants: [{
+                begin: '(u8?|U|L)?"', end: '"', illegal: "\\n",
+                contains: [e.BACKSLASH_ESCAPE]
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+                begin: "(u8?|U|L)?'(\\\\(x[0-9A-Fa-f]{2}|u[0-9A-Fa-f]{4,8}|[0-7]{3}|\\S)|.)",
+                end: "'", illegal: "."
+            }, e.END_SAME_AS_BEGIN({
+                begin: /(?:u8?|U|L)?R"([^()\\ ]{0,16})\(/, end: /\)([^()\\ ]{0,16})"/
+            })]
+        }, o = {
+            className: "number", variants: [{ begin: "\\b(0b[01']+)" }, {
+                begin: "(-?)\\b([\\d']+(\\.[\\d']*)?|\\.[\\d']+)(u|U|l|L|ul|UL|f|F|b|B)"
+            }, {
+                begin: "(-?)(\\b0[xX][a-fA-F0-9']+|(\\b[\\d']+(\\.[\\d']*)?|\\.[\\d']+)([eE][-+]?[\\d']+)?)"
+            }], relevance: 0
+        }, c = {
+            className: "meta", begin: /#\s*[a-z]+\b/, end: /$/, keywords: {
+                "meta-keyword": "if else elif endif define undef warning error line pragma _Pragma ifdef ifndef include"
+            }, contains: [{ begin: /\\\n/, relevance: 0 }, e.inherit(s, { className: "meta-string" }), {
+                className: "meta-string", begin: /<.*?>/, end: /$/, illegal: "\\n"
+            }, n, e.C_BLOCK_COMMENT_MODE]
+        }, l = {
+            className: "title", begin: t(r) + e.IDENT_RE,
+            relevance: 0
+        }, d = t(r) + e.IDENT_RE + "\\s*\\(", u = {
+            keyword: "int float while private char char8_t char16_t char32_t catch import module export virtual operator sizeof dynamic_cast|10 typedef const_cast|10 const for static_cast|10 union namespace unsigned long volatile static protected bool template mutable if public friend do goto auto void enum else break extern using asm case typeid wchar_t short reinterpret_cast|10 default double register explicit signed typename try this switch continue inline delete alignas alignof constexpr consteval constinit decltype concept co_await co_return co_yield requires noexcept static_assert thread_local restrict final override atomic_bool atomic_char atomic_schar atomic_uchar atomic_short atomic_ushort atomic_int atomic_uint atomic_long atomic_ulong atomic_llong atomic_ullong new throw return and and_eq bitand bitor compl not not_eq or or_eq xor xor_eq",
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+            literal: "true false nullptr NULL"
+        }, m = [c, i, n, e.C_BLOCK_COMMENT_MODE, o, s], p = {
+            variants: [{ begin: /=/, end: /;/ }, { begin: /\(/, end: /\)/ }, {
+                beginKeywords: "new throw return else", end: /;/
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+                begin: /\(/, end: /\)/, keywords: u, contains: m.concat(["self"]), relevance: 0
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+            relevance: 0
+        }, _ = {
+            className: "function", begin: "(" + a + "[\\*&\\s]+)+" + d,
+            returnBegin: !0, end: /[{;=]/, excludeEnd: !0, keywords: u, illegal: /[^\w\s\*&:<>]/,
+            contains: [{ begin: "decltype\\(auto\\)", keywords: u, relevance: 0 }, {
+                begin: d,
+                returnBegin: !0, contains: [l], relevance: 0
+            }, {
+                className: "params", begin: /\(/,
+                end: /\)/, keywords: u, relevance: 0, contains: [n, e.C_BLOCK_COMMENT_MODE, s, o, i, {
+                    begin: /\(/, end: /\)/, keywords: u, relevance: 0,
+                    contains: ["self", n, e.C_BLOCK_COMMENT_MODE, s, o, i]
+                }]
+            }, i, n, e.C_BLOCK_COMMENT_MODE, c]
+        }; return {
+            aliases: ["c", "cc", "h", "c++", "h++", "hpp", "hh", "hxx", "cxx"], keywords: u,
+            disableAutodetect: !0, illegal: "</", contains: [].concat(p, _, m, [c, {
+                begin: "\\b(deque|list|queue|priority_queue|pair|stack|vector|map|set|bitset|multiset|multimap|unordered_map|unordered_set|unordered_multiset|unordered_multimap|array)\\s*<",
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+                }, {
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+                    keywords: "@page @font-face"
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+            }
+    }
+}());
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+                    beginKeywords: "func", end: "{", excludeEnd: !0, contains: [e.inherit(e.TITLE_MODE, {
+                        begin: /[A-Za-z$_][0-9A-Za-z$_]*/
+                    }), { begin: /</, end: />/ }, {
+                        className: "params",
+                        begin: /\(/, end: /\)/, endsParent: !0, keywords: i,
+                        contains: ["self", r, a, e.C_BLOCK_COMMENT_MODE, { begin: ":" }], illegal: /["']/
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+                    illegal: /\[|%/
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+                    className: "class",
+                    beginKeywords: "struct protocol class extension enum", keywords: i, end: "\\{",
+                    excludeEnd: !0, contains: [e.inherit(e.TITLE_MODE, {
+                        begin: /[A-Za-z$_][\u00C0-\u02B80-9A-Za-z$_]*/
+                    })]
+                }, {
+                    className: "meta",
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+                }, { beginKeywords: "import", end: /$/, contains: [e.C_LINE_COMMENT_MODE, n] }]
+        }
+    }
+}());
+hljs.registerLanguage("typescript", function () {
+    "use strict"
+        ; const e = "[A-Za-z$_][0-9A-Za-z$_]*", n = ["as", "in", "of", "if", "for", "while", "finally", "var", "new", "function", "do", "return", "void", "else", "break", "catch", "instanceof", "with", "throw", "case", "default", "try", "switch", "continue", "typeof", "delete", "let", "yield", "const", "class", "debugger", "async", "await", "static", "import", "from", "export", "extends"], a = ["true", "false", "null", "undefined", "NaN", "Infinity"], s = [].concat(["setInterval", "setTimeout", "clearInterval", "clearTimeout", "require", "exports", "eval", "isFinite", "isNaN", "parseFloat", "parseInt", "decodeURI", "decodeURIComponent", "encodeURI", "encodeURIComponent", "escape", "unescape"], ["arguments", "this", "super", "console", "window", "document", "localStorage", "module", "global"], ["Intl", "DataView", "Number", "Math", "Date", "String", "RegExp", "Object", "Function", "Boolean", "Error", "Symbol", "Set", "Map", "WeakSet", "WeakMap", "Proxy", "Reflect", "JSON", "Promise", "Float64Array", "Int16Array", "Int32Array", "Int8Array", "Uint16Array", "Uint32Array", "Float32Array", "Array", "Uint8Array", "Uint8ClampedArray", "ArrayBuffer"], ["EvalError", "InternalError", "RangeError", "ReferenceError", "SyntaxError", "TypeError", "URIError"])
+        ; function t(e) { return r("(?=", e, ")") } function i(e) { return r("(", e, ")?") }
+    function r(...e) {
+        return e.map((e => {
+            return (n = e) ? "string" == typeof n ? n : n.source : null;
+        })).join("")
+    }
+    return function (c) {
+        const o = {
+            $pattern: e,
+            keyword: n.concat(["type", "namespace", "typedef", "interface", "public", "private", "protected", "implements", "declare", "abstract", "readonly"]).join(" "),
+            literal: a.join(" "),
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+        }, l = { className: "meta", begin: "@[A-Za-z$_][0-9A-Za-z$_]*" }, d = (e, n, a) => {
+            const s = e.contains.findIndex((e => e.label === n))
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+        }, g = function (c) {
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+                end: /\/[A-Za-z0-9\\._:-]+>|\/>/, isTrulyOpeningTag: (e, n) => {
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+                    })(e, {
+                        after: a
+                    }) || n.ignoreMatch()) : n.ignoreMatch()
+                }
+            }, d = {
+                $pattern: e, keyword: n.join(" "),
+                literal: a.join(" "), built_in: s.join(" ")
+            }, g = (e, n) => `\\b0[${e}][${n}]([${n}_]*[${n}])?n?`, b = /[1-9]([0-9_]*\d)?/, u = /\d([0-9_]*\d)?/, E = r(/[eE][+-]?/, u), m = {
+                className: "number", variants: [{ begin: g("bB", "01") }, { begin: g("oO", "0-7") }, {
+                    begin: g("xX", "0-9a-fA-F")
+                }, { begin: r(/\b/, b, "n") }, { begin: r(/(\b0)?\./, u, i(E)) }, {
+                    begin: r(/\b/, b, i(r(/\./, i(u))), i(E))
+                }, { begin: /\b0[\.n]?/ }], relevance: 0
+            }, y = {
+                className: "subst", begin: "\\$\\{", end: "\\}", keywords: d, contains: []
+            }, p = {
+                begin: "html`", end: "", starts: {
+                    end: "`", returnEnd: !1,
+                    contains: [c.BACKSLASH_ESCAPE, y], subLanguage: "xml"
+                }
+            }, _ = {
+                begin: "css`", end: "",
+                starts: { end: "`", returnEnd: !1, contains: [c.BACKSLASH_ESCAPE, y], subLanguage: "css" }
+            }, N = { className: "string", begin: "`", end: "`", contains: [c.BACKSLASH_ESCAPE, y] }, f = {
+                className: "comment", variants: [c.COMMENT("/\\*\\*", "\\*/", {
+                    relevance: 0,
+                    contains: [{
+                        className: "doctag", begin: "@[A-Za-z]+", contains: [{
+                            className: "type",
+                            begin: "\\{", end: "\\}", relevance: 0
+                        }, {
+                            className: "variable",
+                            begin: o + "(?=\\s*(-)|$)", endsParent: !0, relevance: 0
+                        }, {
+                            begin: /(?=[^\n])\s/,
+                            relevance: 0
+                        }]
+                    }]
+                }), c.C_BLOCK_COMMENT_MODE, c.C_LINE_COMMENT_MODE]
+            }, A = [c.APOS_STRING_MODE, c.QUOTE_STRING_MODE, p, _, N, m, c.REGEXP_MODE]
+            ; y.contains = A.concat({ begin: /{/, end: /}/, keywords: d, contains: ["self"].concat(A) })
+                ; const O = [].concat(f, y.contains), S = O.concat([{
+                    begin: /\(/, end: /\)/, keywords: d,
+                    contains: ["self"].concat(O)
+                }]), T = {
+                    className: "params", begin: /\(/, end: /\)/,
+                    excludeBegin: !0, excludeEnd: !0, keywords: d, contains: S
+                }; return {
+                    name: "Javascript",
+                    aliases: ["js", "jsx", "mjs", "cjs"], keywords: d, exports: { PARAMS_CONTAINS: S },
+                    illegal: /#(?![$_A-z])/, contains: [c.SHEBANG({
+                        label: "shebang", binary: "node",
+                        relevance: 5
+                    }), {
+                        label: "use_strict", className: "meta", relevance: 10,
+                        begin: /^\s*['"]use (strict|asm)['"]/
+                    }, c.APOS_STRING_MODE, c.QUOTE_STRING_MODE, p, _, N, f, m, {
+                        begin: r(/[{,\n]\s*/, t(r(/(\/\/.*$)*/, /(\/\*(.|\n)*\*\/)*/, /\s*/, o + "\\s*:"))),
+                        relevance: 0, contains: [{ className: "attr", begin: o + t("\\s*:"), relevance: 0 }]
+                    }, {
+                        begin: "(" + c.RE_STARTERS_RE + "|\\b(case|return|throw)\\b)\\s*",
+                        keywords: "return throw case", contains: [f, c.REGEXP_MODE, {
+                            className: "function",
+                            begin: "(\\([^()]*(\\([^()]*(\\([^()]*\\))*[^()]*\\))*[^()]*\\)|" + c.UNDERSCORE_IDENT_RE + ")\\s*=>",
+                            returnBegin: !0, end: "\\s*=>", contains: [{
+                                className: "params", variants: [{
+                                    begin: c.UNDERSCORE_IDENT_RE
+                                }, { className: null, begin: /\(\s*\)/, skip: !0 }, {
+                                    begin: /\(/, end: /\)/, excludeBegin: !0, excludeEnd: !0, keywords: d, contains: S
+                                }]
+                            }]
+                        }, {
+                                begin: /,/, relevance: 0
+                            }, { className: "", begin: /\s/, end: /\s*/, skip: !0 }, {
+                                variants: [{
+                                    begin: "<>", end: "</>"
+                                }, { begin: l.begin, "on:begin": l.isTrulyOpeningTag, end: l.end }],
+                                subLanguage: "xml", contains: [{ begin: l.begin, end: l.end, skip: !0, contains: ["self"] }]
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+                    }, {
+                        className: "function", beginKeywords: "function", end: /[{;]/,
+                        excludeEnd: !0, keywords: d, contains: ["self", c.inherit(c.TITLE_MODE, { begin: o }), T],
+                        illegal: /%/
+                    }, {
+                        className: "function",
+                        begin: c.UNDERSCORE_IDENT_RE + "\\([^()]*(\\([^()]*(\\([^()]*\\))*[^()]*\\))*[^()]*\\)\\s*{",
+                        returnBegin: !0, contains: [T, c.inherit(c.TITLE_MODE, { begin: o })]
+                    }, {
+                        variants: [{
+                            begin: "\\." + o
+                        }, { begin: "\\$" + o }], relevance: 0
+                    }, {
+                        className: "class",
+                        beginKeywords: "class", end: /[{;=]/, excludeEnd: !0, illegal: /[:"\[\]]/, contains: [{
+                            beginKeywords: "extends"
+                        }, c.UNDERSCORE_TITLE_MODE]
+                    }, {
+                        begin: /\b(?=constructor)/,
+                        end: /[\{;]/, excludeEnd: !0, contains: [c.inherit(c.TITLE_MODE, { begin: o }), "self", T]
+                    }, {
+                        begin: "(get|set)\\s+(?=" + o + "\\()", end: /{/, keywords: "get set",
+                        contains: [c.inherit(c.TITLE_MODE, { begin: o }), { begin: /\(\)/ }, T]
+                    }, { begin: /\$[(.]/ }]
+                }
+        }(c)
+        ; return Object.assign(g.keywords, o), g.exports.PARAMS_CONTAINS.push(l), g.contains = g.contains.concat([l, {
+            beginKeywords: "namespace", end: /\{/, excludeEnd: !0
+        }, {
+            beginKeywords: "interface",
+                end: /\{/, excludeEnd: !0, keywords: "interface extends"
+            }]), d(g, "shebang", c.SHEBANG()), d(g, "use_strict", {
+                className: "meta", relevance: 10,
+                begin: /^\s*['"]use strict['"]/
+            }), g.contains.find((e => "function" === e.className)).relevance = 0, Object.assign(g, {
+                name: "TypeScript", aliases: ["ts"]
+            }), g
+    }
+}());
+hljs.registerLanguage("yaml", function () {
+    "use strict"; return function (e) {
+        var n = "true false yes no null", a = "[\\w#;/?:@&=+$,.~*\\'()[\\]]+", s = {
+            className: "string", relevance: 0, variants: [{ begin: /'/, end: /'/ }, {
+                begin: /"/, end: /"/
+            }, { begin: /\S+/ }], contains: [e.BACKSLASH_ESCAPE, {
+                className: "template-variable",
+                variants: [{ begin: "{{", end: "}}" }, { begin: "%{", end: "}" }]
+            }]
+        }, i = e.inherit(s, {
+            variants: [{ begin: /'/, end: /'/ }, { begin: /"/, end: /"/ }, { begin: /[^\s,{}[\]]+/ }]
+        }), l = {
+            end: ",", endsWithParent: !0, excludeEnd: !0, contains: [], keywords: n, relevance: 0
+        }, t = {
+            begin: "{", end: "}", contains: [l], illegal: "\\n", relevance: 0
+        }, g = {
+            begin: "\\[",
+            end: "\\]", contains: [l], illegal: "\\n", relevance: 0
+        }, b = [{
+            className: "attr",
+            variants: [{ begin: "\\w[\\w :\\/.-]*:(?=[ \t]|$)" }, {
+                begin: '"\\w[\\w :\\/.-]*":(?=[ \t]|$)'
+            }, {
+                begin: "'\\w[\\w :\\/.-]*':(?=[ \t]|$)"
+            }]
+        }, { className: "meta", begin: "^---s*$", relevance: 10 }, {
+            className: "string",
+            begin: "[\\|>]([0-9]?[+-])?[ ]*\\n( *)[\\S ]+\\n(\\2[\\S ]+\\n?)*"
+        }, {
+            begin: "<%[%=-]?", end: "[%-]?%>", subLanguage: "ruby", excludeBegin: !0, excludeEnd: !0,
+            relevance: 0
+        }, { className: "type", begin: "!\\w+!" + a }, {
+            className: "type",
+            begin: "!<" + a + ">"
+        }, { className: "type", begin: "!" + a }, {
+            className: "type", begin: "!!" + a
+        }, { className: "meta", begin: "&" + e.UNDERSCORE_IDENT_RE + "$" }, {
+            className: "meta",
+            begin: "\\*" + e.UNDERSCORE_IDENT_RE + "$"
+        }, {
+            className: "bullet", begin: "\\-(?=[ ]|$)",
+            relevance: 0
+        }, e.HASH_COMMENT_MODE, { beginKeywords: n, keywords: { literal: n } }, {
+            className: "number",
+            begin: "\\b[0-9]{4}(-[0-9][0-9]){0,2}([Tt \\t][0-9][0-9]?(:[0-9][0-9]){2})?(\\.[0-9]*)?([ \\t])*(Z|[-+][0-9][0-9]?(:[0-9][0-9])?)?\\b"
+        }, { className: "number", begin: e.C_NUMBER_RE + "\\b" }, t, g, s], c = [...b]; return c.pop(),
+            c.push(i), l.contains = c, {
+                name: "YAML", case_insensitive: !0, aliases: ["yml", "YAML"],
+            contains: b
+        }
+    }
+}());
+
+// original from https://github.com/leanprover-community/highlightjs-lean/blob/master/src/lean.js, BSD 3, (c) 2020 Patrick Massot
+hljs.registerLanguage("lean", function (hljs) {
+    var LEAN_KEYWORDS = {
+        $pattern: /#?\w+/,
+        keyword:
+            'theorem|10 def class structure instance set_option ' +
+            'example inductive coinductive ' +
+            'axiom constant ' +
+            'partial unsafe private protected ' +
+            'if then else ' +
+            'universe variable ' +
+            'import open export prelude renaming hiding ' +
+            'calc match with do by let extends ' +
+            'for in unless try catch finally mutual mut return continue break where rec ' +
+            'syntax macro_rules macro deriving ' +
+            'fun ' +
+            '#check #check_failure #eval #reduce #print ' +
+            'section namespace end infix infixl infixr postfix prefix notation ',
+        built_in:
+            'Type Prop|10 Sort rw|10 rewrite rwa erw subst substs ' +
+            'simp dsimp simpa simp_intros finish using generalizing ' +
+            'unfold unfold1 dunfold unfold_projs unfold_coes ' +
+            'delta cc ac_rfl ' +
+            'existsi|10 cases rcases intro intros introv by_cases ' +
+            'refl rfl funext case focus propext exact exacts ' +
+            'refine apply eapply fapply apply_with apply_instance ' +
+            'induction rename assumption revert generalize specialize clear ' +
+            'contradiction by_contradiction by_contra trivial exfalso ' +
+            'symmetry transitivity destruct constructor econstructor ' +
+            'left right split injection injections ' +
+            'repeat skip swap solve1 abstract all_goals any_goals done ' +
+            'fail_if_success success_if_fail guard_target guard_hyp ' +
+            'have replace at suffices show from ' +
+            'congr congr_n congr_arg norm_num ring ',
+        literal:
+            'true false',
+        meta:
+            'noncomputable|10 private protected mutual',
+        strong:
+            'sorry admit',
+    };
+
+    var LEAN_IDENT_RE = /[A-Za-z_][\\w\u207F-\u209C\u1D62-\u1D6A\u2079\'0-9?]*/;
+
+    var DASH_COMMENT = hljs.COMMENT('--', '$');
+    var MULTI_LINE_COMMENT = hljs.COMMENT('/-[^-]', '-/');
+    var DOC_COMMENT = {
+        className: 'doctag',
+        begin: '/-[-!]',
+        end: '-/'
+    };
+
+    var ATTRIBUTE_DECORATOR = {
+        className: 'meta',
+        begin: '@\\[',
+        end: '\\]'
+    };
+
+    var ATTRIBUTE_LINE = {
+        className: 'meta',
+        begin: '^attribute',
+        end: '$'
+    };
+
+    var LEAN_DEFINITION = {
+        className: 'theorem',
+        beginKeywords: 'def theorem lemma class instance structure',
+        end: ':= | where',
+        excludeEnd: true,
+        contains: [
+            {
+                className: 'keyword',
+                begin: /extends/
+            },
+            hljs.inherit(hljs.TITLE_MODE, {
+                begin: LEAN_IDENT_RE
+            }),
+            {
+                className: 'params',
+                begin: /[([{]/, end: /[)\]}]/, endsParent: false,
+                keywords: LEAN_KEYWORDS,
+            },
+            {
+                className: 'symbol',
+                begin: /:/,
+                endsParent: true
+            },
+        ],
+        keywords: LEAN_KEYWORDS
+    };
+    return {
+        name: "lean",
+        keywords: LEAN_KEYWORDS,
+        contains: [
+            hljs.QUOTE_STRING_MODE,
+            hljs.NUMBER_MODE,
+            DASH_COMMENT,
+            MULTI_LINE_COMMENT,
+            DOC_COMMENT,
+            LEAN_DEFINITION,
+            ATTRIBUTE_DECORATOR,
+            ATTRIBUTE_LINE,
+            { begin: /⟨/ } // relevance booster
+        ]
+    };
+});
\ No newline at end of file
diff --git a/docs/introduction.md b/docs/introduction.md
index 354b7a3..60327b1 100644
--- a/docs/introduction.md
+++ b/docs/introduction.md
@@ -4,20 +4,20 @@ Introduction
 Cutting Planes and Lean 4
 -----------------------------
 
-*Cutting Planes* is a formal logic used in the context of discrete combinatorial problems, that can be applied in _Software testing_, _Formal verification_ and _Proof logging_.
+*Cutting Planes* is a method in mathematical optimizaiton used in the context of discrete combinatorial problems, that can be applied in _Software testing_, _Formal verification_ and _Proof logging_.
+Pseudo-Boolean in the other hand is a logic system introduced by Jakob Nordström in his paper [the paper](thelink.com), where ...
+Our proofs will consist of _Pseudo-Boolean Constraints_, that are 0-1 integer linear inequalities. Without loss of generality, we will use the [normalized form](https://pure.mpg.de/rest/items/item_1832217_4/component/file_2574300/content#page=7).
 
-Our proofs will consist of _Pseudo-Boolean Constraints_, that are 0-1 integer linear inequalities of the form:
 \\[ \sum_i{a_i l_i} \ge A \\]
 Where we have:
-- constant \\( A \in \mathbb{Z} \\)
-- coefficients \\( a_i \in \mathbb{Z} \\)
+- constant \\( A \in \mathbb{N} \\)
+- coefficients \\( a_i \in \mathbb{N} \\)
 - literals \\( l_i: x_i\ \text{or}\ \overline{x_i}\ (\text{where } x_i + \overline{x_i} = 1) \\)
 - variables \\( x_i \\) take values \\( 0 = false \\) or \\( 1 = true \\)
 
-Without loss of generality, we will use the [normalized form](https://pure.mpg.de/rest/items/item_1832217_4/component/file_2574300/content#page=7), with all \\( a_i, A\\) non-negative.
 
 In this regard, _Pseudo-Boolean Reasoning_ works on two axioms and four rules, each of which are formally verified in this project using _Lean 4_
 
-[Lean 4](https://lean-lang.org/) is a theorem prover and programming language developed by [Leonardo de Moura](http://leodemoura.github.io/). In the last years it became proeminent in the mathematics community, after [The Xena Project](https://www.ma.imperial.ac.uk/~buzzard/xena/) of Imperial College London formalized Peter Scholze's proof in the area of condensed mathematics in 2021. In 2023 Terence Tao used Lean to formalize a proof of the _Polynomial Freiman-Ruzsa [(PFR)](https://teorth.github.io/pfr/) conjecture_, published in the same year.
+[Lean 4](https://lean-lang.org/) is a theorem prover and programming language developed by [Leonardo de Moura](http://leodemoura.github.io/). In the last years it became proeminent in the mathematics community, after [The Xena Project](https://www.ma.imperial.ac.uk/~buzzard/xena/) of Imperial College London was formed, Peter Scholze's proof in the area of condensed mathematics was formalized in 2021 and in 2023 Terence Tao used Lean to formalize a proof of the _Polynomial Freiman-Ruzsa [(PFR)](https://teorth.github.io/pfr/) conjecture_, published in the same year.
 
 Armed with a Lean 4 proof of the reasoning rules, complex proofs that involve these steps can be guaranteed to be correct.
diff --git a/docs/pseudo_boolean_rules.md b/docs/pseudo_boolean_rules.md
index 6e31796..fca60bf 100644
--- a/docs/pseudo_boolean_rules.md
+++ b/docs/pseudo_boolean_rules.md
@@ -1,18 +1,15 @@
 Pseudo Boolean Rules
 ============
 
-Input Axioms
+Interaction
 -----------------------------
 
-As a starting point for the pseudo boolean reasoning we will get constraints from the user, which are subject to a normalization process
+As a starting point for the pseudo boolean reasoning we will get constraints from the user, which are subject to a normalization process. Then the rules can be applied to produce further inequalities
 
-Literal Axioms
------------------------------
+```lean
+-- code goes here
+```
 
-For any literal \\( l_i\\) we can create a constraint:
-\\[ \frac{}
-    {l_i \ge 0}
-\\]
 
 Addition Rule
 -----------------------------
@@ -22,6 +19,10 @@ Two constraints can be added together, adding the coefficients and the constants
     {\sum_i{(a_i + b_i) l_i} \ge (A+B)}
 \\]
 
+```lean
+-- code goes here
+```
+
 Multiplication Rule
 -----------------------------
 A constraint can be multiplied by any \\( c \in \mathbb{N}^{+} \\):
@@ -30,6 +31,10 @@ A constraint can be multiplied by any \\( c \in \mathbb{N}^{+} \\):
     {\sum_i{c a_i l_i} \ge c A}
 \\]
 
+```lean
+-- code goes here
+```
+
 Division Rule
 -----------------------------
 A constraint can be divided by any \\( c \in \mathbb{N}^{+} \\), and the the ceiling of this division in applied:
@@ -38,6 +43,10 @@ A constraint can be divided by any \\( c \in \mathbb{N}^{+} \\), and the the cei
     {\sum_i{ \lceil \frac{a_i}{c} \rceil l_i} \ge \lceil \frac{A}{c} \rceil}
 \\]
 
+```lean
+-- code goes here
+```
+
 Saturation Rule
 -----------------------------
 A constraint can replace its coefficients by the minimum between them and the constant:
@@ -46,4 +55,8 @@ A constraint can replace its coefficients by the minimum between them and the co
     {\sum_i{ \min(a_i,A)\cdot l_i} \ge A}
 \\]
 
+```lean
+-- code goes here
+```
+
 All these rules can be seen in practice in the Worked Example
\ No newline at end of file

From 1daefd3997c2dc05ddf65b99cf830025c920b3d2 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 19 Jun 2024 17:40:41 +0000
Subject: [PATCH 54/93] Properly credits in the docs

---
 docs/introduction.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/docs/introduction.md b/docs/introduction.md
index 60327b1..1dcab78 100644
--- a/docs/introduction.md
+++ b/docs/introduction.md
@@ -5,7 +5,7 @@ Cutting Planes and Lean 4
 -----------------------------
 
 *Cutting Planes* is a method in mathematical optimizaiton used in the context of discrete combinatorial problems, that can be applied in _Software testing_, _Formal verification_ and _Proof logging_.
-Pseudo-Boolean in the other hand is a logic system introduced by Jakob Nordström in his paper [the paper](thelink.com), where ...
+Pseudo-Boolean in the other hand is a logic system introduced by Jakob Nordström in his paper [A Unified Proof System for Discrete Combinatorial Problems](https://jakobnordstrom.se/docs/presentations/TalkVeriPB_Dagstuhl23.pdf#page=45), which this project aims to implement and verify.
 Our proofs will consist of _Pseudo-Boolean Constraints_, that are 0-1 integer linear inequalities. Without loss of generality, we will use the [normalized form](https://pure.mpg.de/rest/items/item_1832217_4/component/file_2574300/content#page=7).
 
 \\[ \sum_i{a_i l_i} \ge A \\]
@@ -18,6 +18,6 @@ Where we have:
 
 In this regard, _Pseudo-Boolean Reasoning_ works on two axioms and four rules, each of which are formally verified in this project using _Lean 4_
 
-[Lean 4](https://lean-lang.org/) is a theorem prover and programming language developed by [Leonardo de Moura](http://leodemoura.github.io/). In the last years it became proeminent in the mathematics community, after [The Xena Project](https://www.ma.imperial.ac.uk/~buzzard/xena/) of Imperial College London was formed, Peter Scholze's proof in the area of condensed mathematics was formalized in 2021 and in 2023 Terence Tao used Lean to formalize a proof of the _Polynomial Freiman-Ruzsa [(PFR)](https://teorth.github.io/pfr/) conjecture_, published in the same year.
+[Lean 4](https://lean-lang.org/) is a theorem prover and programming language developed by [Leonardo de Moura](http://leodemoura.github.io/). In the last years it became proeminent in the mathematics community, after Kevin Buzzard's [The Xena Project](https://www.ma.imperial.ac.uk/~buzzard/xena/) of Imperial College London was formed, aiming to formalize all undergraduate level math, Peter Scholze's proof in the area of condensed mathematics called [_Liquid Tensor Experiment_](https://github.com/leanprover-community/lean-liquid) was formalized in 2021 and in 2023 Terence Tao used Lean to formalize a proof of the _Polynomial Freiman-Ruzsa [(PFR)](https://teorth.github.io/pfr/) conjecture_, published in the same year.
 
 Armed with a Lean 4 proof of the reasoning rules, complex proofs that involve these steps can be guaranteed to be correct.

From 88e054addd1fa84c5ba6a157a830f1406517e459 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 20 Jun 2024 17:28:52 +0000
Subject: [PATCH 55/93] Fix highlight.js

---
 .gitignore                   |    4 +-
 book/highlight.js            |   53 +
 docs/assets/highlight.js     | 2377 ----------------------------------
 docs/pseudo_boolean_rules.md |    6 +
 4 files changed, 62 insertions(+), 2378 deletions(-)
 create mode 100644 book/highlight.js
 delete mode 100644 docs/assets/highlight.js

diff --git a/.gitignore b/.gitignore
index 438d45e..005e617 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1,4 +1,6 @@
-/book
+# Ignore book files, except for Lean's highlight.js
+/book/*
+!/book/highlight.js
 
 /build
 /lakefile.olean
diff --git a/book/highlight.js b/book/highlight.js
new file mode 100644
index 0000000..3256c00
--- /dev/null
+++ b/book/highlight.js
@@ -0,0 +1,53 @@
+/*
+  Highlight.js 10.1.1 (93fd0d73)
+  License: BSD-3-Clause
+  Copyright (c) 2006-2020, Ivan Sagalaev
+*/
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diff --git a/docs/assets/highlight.js b/docs/assets/highlight.js
deleted file mode 100644
index 903c2d7..0000000
--- a/docs/assets/highlight.js
+++ /dev/null
@@ -1,2377 +0,0 @@
-/*
-  Highlight.js 10.3.2 (31e1fc40)
-  License: BSD-3-Clause
-  Copyright (c) 2006-2020, Ivan Sagalaev
-*/
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-    "use strict"; function e(n) {
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-        ; var t = "function" == typeof n
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-            })), n
-    } class n {
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-        ignoreMatch() { this.ignore = !0 }
-    } function t(e) {
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-    } function r(e, ...n) {
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-            'existsi|10 cases rcases intro intros introv by_cases ' +
-            'refl rfl funext case focus propext exact exacts ' +
-            'refine apply eapply fapply apply_with apply_instance ' +
-            'induction rename assumption revert generalize specialize clear ' +
-            'contradiction by_contradiction by_contra trivial exfalso ' +
-            'symmetry transitivity destruct constructor econstructor ' +
-            'left right split injection injections ' +
-            'repeat skip swap solve1 abstract all_goals any_goals done ' +
-            'fail_if_success success_if_fail guard_target guard_hyp ' +
-            'have replace at suffices show from ' +
-            'congr congr_n congr_arg norm_num ring ',
-        literal:
-            'true false',
-        meta:
-            'noncomputable|10 private protected mutual',
-        strong:
-            'sorry admit',
-    };
-
-    var LEAN_IDENT_RE = /[A-Za-z_][\\w\u207F-\u209C\u1D62-\u1D6A\u2079\'0-9?]*/;
-
-    var DASH_COMMENT = hljs.COMMENT('--', '$');
-    var MULTI_LINE_COMMENT = hljs.COMMENT('/-[^-]', '-/');
-    var DOC_COMMENT = {
-        className: 'doctag',
-        begin: '/-[-!]',
-        end: '-/'
-    };
-
-    var ATTRIBUTE_DECORATOR = {
-        className: 'meta',
-        begin: '@\\[',
-        end: '\\]'
-    };
-
-    var ATTRIBUTE_LINE = {
-        className: 'meta',
-        begin: '^attribute',
-        end: '$'
-    };
-
-    var LEAN_DEFINITION = {
-        className: 'theorem',
-        beginKeywords: 'def theorem lemma class instance structure',
-        end: ':= | where',
-        excludeEnd: true,
-        contains: [
-            {
-                className: 'keyword',
-                begin: /extends/
-            },
-            hljs.inherit(hljs.TITLE_MODE, {
-                begin: LEAN_IDENT_RE
-            }),
-            {
-                className: 'params',
-                begin: /[([{]/, end: /[)\]}]/, endsParent: false,
-                keywords: LEAN_KEYWORDS,
-            },
-            {
-                className: 'symbol',
-                begin: /:/,
-                endsParent: true
-            },
-        ],
-        keywords: LEAN_KEYWORDS
-    };
-    return {
-        name: "lean",
-        keywords: LEAN_KEYWORDS,
-        contains: [
-            hljs.QUOTE_STRING_MODE,
-            hljs.NUMBER_MODE,
-            DASH_COMMENT,
-            MULTI_LINE_COMMENT,
-            DOC_COMMENT,
-            LEAN_DEFINITION,
-            ATTRIBUTE_DECORATOR,
-            ATTRIBUTE_LINE,
-            { begin: /⟨/ } // relevance booster
-        ]
-    };
-});
\ No newline at end of file
diff --git a/docs/pseudo_boolean_rules.md b/docs/pseudo_boolean_rules.md
index fca60bf..10375fc 100644
--- a/docs/pseudo_boolean_rules.md
+++ b/docs/pseudo_boolean_rules.md
@@ -7,7 +7,13 @@ Interaction
 As a starting point for the pseudo boolean reasoning we will get constraints from the user, which are subject to a normalization process. Then the rules can be applied to produce further inequalities
 
 ```lean
+def eval(x) : 3 := 3
 -- code goes here
+  (xs : Fin 4 → Fin 2)
+  (c1 : PBIneq ![(1,0),(2,0),(1,0),(0,0)] xs 2)
+  (c2 : PBIneq ![(1,0),(2,0),(4,0),(2,0)] xs 5)
+  (c3 : PBIneq ![(0,0),(0,0),(0,0),(0,1)] xs 0)
+ 
 ```
 
 

From 58cff0c9aee4f6304d2d68762eb8568f96eb26fa Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 20 Jun 2024 17:36:43 +0000
Subject: [PATCH 56/93] Fix highlight.js in theme folder

---
 docs/pseudo_boolean_rules.md        | 1 +
 {docs/assets => theme}/highlight.js | 0
 2 files changed, 1 insertion(+)
 rename {docs/assets => theme}/highlight.js (100%)

diff --git a/docs/pseudo_boolean_rules.md b/docs/pseudo_boolean_rules.md
index fca60bf..7d248da 100644
--- a/docs/pseudo_boolean_rules.md
+++ b/docs/pseudo_boolean_rules.md
@@ -8,6 +8,7 @@ As a starting point for the pseudo boolean reasoning we will get constraints fro
 
 ```lean
 -- code goes here
+def x := 3
 ```
 
 
diff --git a/docs/assets/highlight.js b/theme/highlight.js
similarity index 100%
rename from docs/assets/highlight.js
rename to theme/highlight.js

From bbbd8f767d5926568c456ec96448413acdb089f6 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 20 Jun 2024 17:38:35 +0000
Subject: [PATCH 57/93] Fix ignore bookk

---
 .gitignore | 1 +
 1 file changed, 1 insertion(+)

diff --git a/.gitignore b/.gitignore
index 438d45e..ec617a2 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1,3 +1,4 @@
+# Ignore book files, except for Lean's highlight.js
 /book
 
 /build

From 81aca2fda45c1116e2076583305bc4e6167f0709 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 20 Jun 2024 17:56:16 +0000
Subject: [PATCH 58/93] Added lean snippets

---
 docs/pseudo_boolean_rules.md | 33 +++++++++++++++++++++------------
 docs/worked_example.md       |  6 ++++--
 2 files changed, 25 insertions(+), 14 deletions(-)

diff --git a/docs/pseudo_boolean_rules.md b/docs/pseudo_boolean_rules.md
index 10375fc..3023861 100644
--- a/docs/pseudo_boolean_rules.md
+++ b/docs/pseudo_boolean_rules.md
@@ -7,13 +7,9 @@ Interaction
 As a starting point for the pseudo boolean reasoning we will get constraints from the user, which are subject to a normalization process. Then the rules can be applied to produce further inequalities
 
 ```lean
-def eval(x) : 3 := 3
--- code goes here
-  (xs : Fin 4 → Fin 2)
-  (c1 : PBIneq ![(1,0),(2,0),(1,0),(0,0)] xs 2)
-  (c2 : PBIneq ![(1,0),(2,0),(4,0),(2,0)] xs 5)
-  (c3 : PBIneq ![(0,0),(0,0),(0,0),(0,1)] xs 0)
- 
+def xs : Fin 2 → Fin 2
+def c1 : PBIneq ![(1,0),(2,0)] xs 2
+def c2 : PBIneq ![(1,0),(2,0)] xs 5
 ```
 
 
@@ -26,7 +22,11 @@ Two constraints can be added together, adding the coefficients and the constants
 \\]
 
 ```lean
--- code goes here
+def xs : Fin 2 → Fin 2
+def c1 : PBIneq ![(1,0),(2,0)] xs 2
+def c2 : PBIneq ![(1,0),(2,0)] xs 5
+-----------------------------------
+def c3 : PBIneq ![(2,0),(4,0)] xs 7 := by apply Addition c1 c2
 ```
 
 Multiplication Rule
@@ -38,7 +38,10 @@ A constraint can be multiplied by any \\( c \in \mathbb{N}^{+} \\):
 \\]
 
 ```lean
--- code goes here
+def xs : Fin 2 → Fin 2
+def c1 : PBIneq ![(1,0),(2,0)] xs 2
+-----------------------------------
+def c2 : PBIneq ![(2,0),(4,0)] xs 4 := by apply Multiplication c1 2
 ```
 
 Division Rule
@@ -50,7 +53,10 @@ A constraint can be divided by any \\( c \in \mathbb{N}^{+} \\), and the the cei
 \\]
 
 ```lean
--- code goes here
+def xs : Fin 2 → Fin 2
+def c1 : PBIneq ![(3,0),(6,0)] xs 7
+-----------------------------------
+def c2 : PBIneq ![(1,0),(2,0)] xs 3 := by apply Division c1 3
 ```
 
 Saturation Rule
@@ -62,7 +68,10 @@ A constraint can replace its coefficients by the minimum between them and the co
 \\]
 
 ```lean
--- code goes here
+def xs : Fin 2 → Fin 2
+def c1 : PBIneq ![(3,0),(6,0)] xs 3
+-----------------------------------
+def c2 : PBIneq ![(3,0),(3,0)] xs 3 := by apply Saturation c1
 ```
 
-All these rules can be seen in practice in the Worked Example
\ No newline at end of file
+All these rules can be seen in practice in the [Worked Example](./worked_example.md)
\ No newline at end of file
diff --git a/docs/worked_example.md b/docs/worked_example.md
index f6fc4da..f571b33 100644
--- a/docs/worked_example.md
+++ b/docs/worked_example.md
@@ -1,10 +1,12 @@
 Worked Example
 ============
 
+This example was taken from the talk "[A Unified Proof System for Discrete Combinatorial Problems](https://jakobnordstrom.se/docs/presentations/TalkVeriPB_Dagstuhl23.pdf#page=58)" at slide 58 by Jakob Nordström. 
+
 Input
 -----------------------------
 
-For this example, in constraints over variables \\(w,x,y,z\\) we will have three input axioms:
+For this example, with constraints over variables \\(w,x,y,z\\), we will have three input axioms:
 
 1. \\( w + 2x + y \ge 2 \\)
 1. \\( w + 2x + 4y + 2z \ge 5 \\)
@@ -47,7 +49,7 @@ Now we add this result to constraint 2:
 ```lean
   let t2 : PBIneq ![(3,0),(6,0),(6,0),(2,0)] xs 9  := by apply Addition t1 c2
 ```
-At this point we want to remove the variable \\(z\\), as it does not appear in our goal. Then we will introduce a \\(\overline{z} \ge 0\\) by the literal axiom and then multiply it by 2:
+At this point we want to remove the variable \\(z\\), as it does not appear in our goal. Then we will use c3 \\(\overline{z} \ge 0\\) then multiply it by 2:
 \\[ \frac
     {\overline{z} \ge 0}
     {2\overline{z} \ge 0}

From 4aa427216309af6b434b98cfe8714b446bd9cd27 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 20 Jun 2024 18:01:07 +0000
Subject: [PATCH 59/93] Minify highligh.js theme

---
 theme/highlight.js | 1890 ++------------------------------------------
 1 file changed, 59 insertions(+), 1831 deletions(-)

diff --git a/theme/highlight.js b/theme/highlight.js
index 903c2d7..3a86992 100644
--- a/theme/highlight.js
+++ b/theme/highlight.js
@@ -6,7 +6,7 @@
 var hljs = function () {
     "use strict"; function e(n) {
         Object.freeze(n)
-        ; var t = "function" == typeof n
+            ; var t = "function" == typeof n
             ; return Object.getOwnPropertyNames(n).forEach((function (r) {
                 !Object.hasOwnProperty.call(n, r) || null === n[r] || "object" != typeof n[r] && "function" != typeof n[r] || t && ("caller" === r || "callee" === r || "arguments" === r) || Object.isFrozen(n[r]) || e(n[r])
             })), n
@@ -42,11 +42,11 @@ var hljs = function () {
             } function g(e) { ("start" === e.event ? c : u)(e.node) }
             for (; e.length || n.length;) {
                 var d = l()
-                ; if (s += t(r.substring(i, d[0].offset)), i = d[0].offset, d === e) {
-                    o.reverse().forEach(u)
-                    ; do { g(d.splice(0, 1)[0]), d = l() } while (d === e && d.length && d[0].offset === i)
-                        ; o.reverse().forEach(c)
-                } else "start" === d[0].event ? o.push(d[0].node) : o.pop(), g(d.splice(0, 1)[0])
+                    ; if (s += t(r.substring(i, d[0].offset)), i = d[0].offset, d === e) {
+                        o.reverse().forEach(u)
+                            ; do { g(d.splice(0, 1)[0]), d = l() } while (d === e && d.length && d[0].offset === i)
+                            ; o.reverse().forEach(c)
+                    } else "start" === d[0].event ? o.push(d[0].node) : o.pop(), g(d.splice(0, 1)[0])
             }
             return s + t(r.substr(i))
         }
@@ -74,7 +74,7 @@ var hljs = function () {
             this.top.children.push(e)
         } openNode(e) {
             const n = { kind: e, children: [] }
-            ; this.add(n), this.stack.push(n)
+                ; this.add(n), this.stack.push(n)
         } closeNode() {
             if (this.stack.length > 1) return this.stack.pop()
         } closeAllNodes() {
@@ -93,7 +93,7 @@ var hljs = function () {
         addKeyword(e, n) { "" !== e && (this.openNode(n), this.addText(e), this.closeNode()) }
         addText(e) { "" !== e && this.add(e) } addSublanguage(e, n) {
             const t = e.root
-            ; t.kind = n, t.sublanguage = !0, this.add(t)
+                ; t.kind = n, t.sublanguage = !0, this.add(t)
         } toHTML() {
             return new o(this, this.options).value()
         } finalize() { return !0 }
@@ -171,17 +171,17 @@ var hljs = function () {
                     }(e) + 1
             } compile() {
                 0 === this.regexes.length && (this.exec = () => null)
-                ; const e = this.regexes.map((e => e[1])); this.matcherRe = n(function (e, n = "|") {
-                    for (var t = /\[(?:[^\\\]]|\\.)*\]|\(\??|\\([1-9][0-9]*)|\\./, r = 0, a = "", i = 0; i < e.length; i++) {
-                        var s = r += 1, o = u(e[i]); for (i > 0 && (a += n), a += "("; o.length > 0;) {
-                            var l = t.exec(o)
-                            ; if (null == l) { a += o; break }
-                            a += o.substring(0, l.index), o = o.substring(l.index + l[0].length),
-                                "\\" === l[0][0] && l[1] ? a += "\\" + (Number(l[1]) + s) : (a += l[0], "(" === l[0] && r++)
-                        } a += ")"
-                    }
-                    return a
-                }(e), !0), this.lastIndex = 0
+                    ; const e = this.regexes.map((e => e[1])); this.matcherRe = n(function (e, n = "|") {
+                        for (var t = /\[(?:[^\\\]]|\\.)*\]|\(\??|\\([1-9][0-9]*)|\\./, r = 0, a = "", i = 0; i < e.length; i++) {
+                            var s = r += 1, o = u(e[i]); for (i > 0 && (a += n), a += "("; o.length > 0;) {
+                                var l = t.exec(o)
+                                    ; if (null == l) { a += o; break }
+                                a += o.substring(0, l.index), o = o.substring(l.index + l[0].length),
+                                    "\\" === l[0][0] && l[1] ? a += "\\" + (Number(l[1]) + s) : (a += l[0], "(" === l[0] && r++)
+                            } a += ")"
+                        }
+                        return a
+                    }(e), !0), this.lastIndex = 0
             } exec(e) {
                 this.matcherRe.lastIndex = this.lastIndex; const n = this.matcherRe.exec(e)
                     ; if (!n) return null
@@ -221,14 +221,14 @@ var hljs = function () {
                     ; if ("object" == typeof s.keywords && (c = s.keywords.$pattern, delete s.keywords.$pattern),
                         s.keywords && (s.keywords = function (e, n) {
                             var t = {}
-                            ; return "string" == typeof e ? r("keyword", e) : Object.keys(e).forEach((function (n) {
-                                r(n, e[n])
-                            })), t; function r(e, r) {
-                                n && (r = r.toLowerCase()), r.split(" ").forEach((function (n) {
-                                    var r = n.split("|")
-                                    ; t[r[0]] = [e, O(r[0], r[1])]
-                                }))
-                            }
+                                ; return "string" == typeof e ? r("keyword", e) : Object.keys(e).forEach((function (n) {
+                                    r(n, e[n])
+                                })), t; function r(e, r) {
+                                    n && (r = r.toLowerCase()), r.split(" ").forEach((function (n) {
+                                        var r = n.split("|")
+                                            ; t[r[0]] = [e, O(r[0], r[1])]
+                                    }))
+                                }
                         }(s.keywords, e.case_insensitive)), s.lexemes && c) throw Error("ERR: Prefer `keywords.$pattern` to `mode.lexemes`, BOTH are not allowed. (see mode reference) ")
                     ; return l.keywordPatternRe = n(s.lexemes || c || /\w+/, !0),
                         o && (s.beginKeywords && (s.begin = "\\b(" + s.beginKeywords.split(" ").join("|") + ")(?=\\b|\\s)",
@@ -251,11 +251,11 @@ var hljs = function () {
                             t(e, l)
                         })), s.starts && t(s.starts, o), l.matcher = function (e) {
                             const n = new a
-                            ; return e.contains.forEach((e => n.addRule(e.begin, {
-                                rule: e, type: "begin"
-                            }))), e.terminator_end && n.addRule(e.terminator_end, {
-                                type: "end"
-                            }), e.illegal && n.addRule(e.illegal, { type: "illegal" }), n
+                                ; return e.contains.forEach((e => n.addRule(e.begin, {
+                                    rule: e, type: "begin"
+                                }))), e.terminator_end && n.addRule(e.terminator_end, {
+                                    type: "end"
+                                }), e.illegal && n.addRule(e.illegal, { type: "illegal" }), n
                         }(l), l
             }(e)
     } function M(e) {
@@ -301,22 +301,22 @@ var hljs = function () {
                     return d.noHighlightRe.test(e)
                 } function f(e, n, t, r) {
                     var a = { code: n, language: e }
-                    ; N("before:highlight", a); var i = a.result ? a.result : p(a.language, a.code, t, r)
+                        ; N("before:highlight", a); var i = a.result ? a.result : p(a.language, a.code, t, r)
                         ; return i.code = a.code, N("after:highlight", i), i
                 } function p(e, t, r, i) {
                     var s = t
-                    ; function l(e, n) {
-                        var t = _.case_insensitive ? n[0].toLowerCase() : n[0]
-                        ; return Object.prototype.hasOwnProperty.call(e.keywords, t) && e.keywords[t]
-                    }
+                        ; function l(e, n) {
+                            var t = _.case_insensitive ? n[0].toLowerCase() : n[0]
+                                ; return Object.prototype.hasOwnProperty.call(e.keywords, t) && e.keywords[t]
+                        }
                     function c() {
                         null != y.subLanguage ? function () {
                             if ("" !== O) {
                                 var e = null
-                                ; if ("string" == typeof y.subLanguage) {
-                                    if (!a[y.subLanguage]) return void M.addText(O)
-                                        ; e = p(y.subLanguage, O, !0, R[y.subLanguage]), R[y.subLanguage] = e.top
-                                } else e = m(O, y.subLanguage.length ? y.subLanguage : null)
+                                    ; if ("string" == typeof y.subLanguage) {
+                                        if (!a[y.subLanguage]) return void M.addText(O)
+                                            ; e = p(y.subLanguage, O, !0, R[y.subLanguage]), R[y.subLanguage] = e.top
+                                    } else e = m(O, y.subLanguage.length ? y.subLanguage : null)
                                     ; y.relevance > 0 && (L += e.relevance), M.addSublanguage(e.emitter, e.language)
                             }
                         }() : function () {
@@ -324,7 +324,7 @@ var hljs = function () {
                                 ; y.keywordPatternRe.lastIndex = 0; let n = y.keywordPatternRe.exec(O), t = ""; for (; n;) {
                                     t += O.substring(e, n.index); const r = l(y, n); if (r) {
                                         const [e, a] = r
-                                        ; M.addText(t), t = "", L += a, M.addKeyword(n[0], e)
+                                            ; M.addText(t), t = "", L += a, M.addKeyword(n[0], e)
                                     } else t += n[0]
                                         ; e = y.keywordPatternRe.lastIndex, n = y.keywordPatternRe.exec(O)
                                 }
@@ -337,11 +337,11 @@ var hljs = function () {
                     } function h(e, t, r) {
                         let a = function (e, n) {
                             var t = e && e.exec(n)
-                            ; return t && 0 === t.index
+                                ; return t && 0 === t.index
                         }(e.endRe, r); if (a) {
                             if (e["on:end"]) {
                                 const r = new n(e)
-                                ; e["on:end"](t, r), r.ignore && (a = !1)
+                                    ; e["on:end"](t, r), r.ignore && (a = !1)
                             } if (a) {
                                 for (; e.endsParent && e.parent;)e = e.parent
                                     ; return e
@@ -358,16 +358,16 @@ var hljs = function () {
                                 g(r.starts)), a.returnEnd ? 0 : n.length
                     } var v = {}; function x(t, a) {
                         var i = a && a[0]
-                        ; if (O += t, null == i) return c(), 0
-                            ; if ("begin" === v.type && "end" === a.type && v.index === a.index && "" === i) {
-                                if (O += s.slice(a.index, a.index + 1), !o) {
-                                    const n = Error("0 width match regex")
-                                    ; throw n.languageName = e, n.badRule = v.rule, n
-                                } return 1
-                            }
+                            ; if (O += t, null == i) return c(), 0
+                                ; if ("begin" === v.type && "end" === a.type && v.index === a.index && "" === i) {
+                                    if (O += s.slice(a.index, a.index + 1), !o) {
+                                        const n = Error("0 width match regex")
+                                            ; throw n.languageName = e, n.badRule = v.rule, n
+                                    } return 1
+                                }
                         if (v = a, "begin" === a.type) return function (e) {
                             var t = e[0], r = e.rule
-                            ; const a = new n(r), i = [r.__beforeBegin, r["on:begin"]]
+                                ; const a = new n(r), i = [r.__beforeBegin, r["on:begin"]]
                                 ; for (const n of i) if (n && (n(e, a), a.ignore)) return f(t)
                                     ; return r && r.endSameAsBegin && (r.endRe = RegExp(t.replace(/[-/\\^$*+?.()|[\]{}]/g, "\\$&"), "m")),
                                         r.skip ? O += t : (r.excludeBegin && (O += t),
@@ -410,7 +410,7 @@ var hljs = function () {
                             relevance: 0,
                             emitter: new d.__emitter(d), value: I(e), illegal: !1, top: g
                         }
-                        ; return n.emitter.addText(e), n
+                            ; return n.emitter.addText(e), n
                     }(e), r = t
                         ; return n.filter(E).filter(w).forEach((function (n) {
                             var a = p(n, e, !1); a.language = n,
@@ -421,11 +421,11 @@ var hljs = function () {
                 } function v(e) {
                     let n = null; const t = function (e) {
                         var n = e.className + " "
-                        ; n += e.parentNode ? e.parentNode.className : ""; const t = d.languageDetectRe.exec(n)
+                            ; n += e.parentNode ? e.parentNode.className : ""; const t = d.languageDetectRe.exec(n)
                             ; if (t) {
                                 var r = E(t[1])
-                                ; return r || (console.warn(u.replace("{}", t[1])), console.warn("Falling back to no-highlight mode for this block.", e)),
-                                    r ? t[1] : "no-highlight"
+                                    ; return r || (console.warn(u.replace("{}", t[1])), console.warn("Falling back to no-highlight mode for this block.", e)),
+                                        r ? t[1] : "no-highlight"
                             } return n.split(/\s+/).find((e => h(e) || E(e)))
                     }(e)
                         ; if (h(t)) return; N("before:highlightBlock", {
@@ -439,8 +439,8 @@ var hljs = function () {
                         block: e, result: a
                     }), e.innerHTML = a.value, e.className = function (e, n, t) {
                         var r = n ? i[n] : t, a = [e.trim()]
-                        ; return e.match(/\bhljs\b/) || a.push("hljs"),
-                            e.includes(r) || a.push(r), a.join(" ").trim()
+                            ; return e.match(/\bhljs\b/) || a.push("hljs"),
+                                e.includes(r) || a.push(r), a.join(" ").trim()
                     }(e.className, t, a.language), e.result = {
                         language: a.language, re: a.relevance,
                         relavance: a.relevance
@@ -451,7 +451,7 @@ var hljs = function () {
                 } const x = () => {
                     if (!x.called) {
                         x.called = !0
-                        ; var e = document.querySelectorAll("pre code"); r.forEach.call(e, v)
+                            ; var e = document.querySelectorAll("pre code"); r.forEach.call(e, v)
                     }
                 }
                 ; function E(e) { return e = (e || "").toLowerCase(), a[e] || a[i[e]] }
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-    }
-}());
 
 // original from https://github.com/leanprover-community/highlightjs-lean/blob/master/src/lean.js, BSD 3, (c) 2020 Patrick Massot
 hljs.registerLanguage("lean", function (hljs) {

From 7dc790840dbcb59267ec40a5e3977bfa424aca79 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 20 Jun 2024 18:05:02 +0000
Subject: [PATCH 60/93] Exclude 'themes' directory for languages stats

---
 .gitattributes | 1 +
 1 file changed, 1 insertion(+)
 create mode 100644 .gitattributes

diff --git a/.gitattributes b/.gitattributes
new file mode 100644
index 0000000..20beb12
--- /dev/null
+++ b/.gitattributes
@@ -0,0 +1 @@
+themes/** linguist-detectable=false
\ No newline at end of file

From 6aeb094e94776e8d9220040dfbb0feff5957d3e8 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 20 Jun 2024 18:06:11 +0000
Subject: [PATCH 61/93] Exclude 'theme' directory for languages stats

---
 .gitattributes | 1 +
 1 file changed, 1 insertion(+)
 create mode 100644 .gitattributes

diff --git a/.gitattributes b/.gitattributes
new file mode 100644
index 0000000..5f387b9
--- /dev/null
+++ b/.gitattributes
@@ -0,0 +1 @@
+theme/** linguist-detectable=false
\ No newline at end of file

From 8c34a05c33ca561e4469bb078e0e75dcd0c87f6c Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Mon, 24 Jun 2024 16:51:37 +0000
Subject: [PATCH 62/93] Increase interaction LaTeX

---
 docs/pseudo_boolean_rules.md | 24 +++++++++++++++++++-----
 1 file changed, 19 insertions(+), 5 deletions(-)

diff --git a/docs/pseudo_boolean_rules.md b/docs/pseudo_boolean_rules.md
index 3023861..89fa5a3 100644
--- a/docs/pseudo_boolean_rules.md
+++ b/docs/pseudo_boolean_rules.md
@@ -5,11 +5,25 @@ Interaction
 -----------------------------
 
 As a starting point for the pseudo boolean reasoning we will get constraints from the user, which are subject to a normalization process. Then the rules can be applied to produce further inequalities
+\\[
+    xs := [0,1]
+\\]
+
+\\[
+    as := [(1,0),(2,0)] \equiv x + 2y 
+\\]
+
+\\[
+    bs := [(0,3),(2,0)] \equiv 3 \overline{x} + 2y
+\\]
 
+\\[
+    {{c1 := \sum_i{as_i xs_i} \ge 2}\qquad {c2 := \sum_i{bs_i xs_i} \ge 5}}
+\\]
 ```lean
-def xs : Fin 2 → Fin 2
+def xs : Fin 2 → Fin 2 := ![0,1]
 def c1 : PBIneq ![(1,0),(2,0)] xs 2
-def c2 : PBIneq ![(1,0),(2,0)] xs 5
+def c2 : PBIneq ![(0,3),(2,0)] xs 5
 ```
 
 
@@ -22,11 +36,11 @@ Two constraints can be added together, adding the coefficients and the constants
 \\]
 
 ```lean
-def xs : Fin 2 → Fin 2
+def xs : Fin 2 → Fin 2 := ![0,1]
 def c1 : PBIneq ![(1,0),(2,0)] xs 2
-def c2 : PBIneq ![(1,0),(2,0)] xs 5
+def c2 : PBIneq ![(0,3),(2,0)] xs 5
 -----------------------------------
-def c3 : PBIneq ![(2,0),(4,0)] xs 7 := by apply Addition c1 c2
+def c3 : PBIneq ![(0,2),(4,0)] xs 6 := by apply Addition c1 c2
 ```
 
 Multiplication Rule

From 567e9093424f576e99fa68b2efb57ddf93ba45f8 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 26 Jun 2024 13:40:23 -0300
Subject: [PATCH 63/93] Created template file

---
 .gitignore                         |    3 +
 article/IEEEtran.cls               | 6347 ++++++++++++++++++++++++++++
 article/IEEEtran_HOWTO.pdf         |  Bin 0 -> 671626 bytes
 article/cutting-planes-article.pdf |  Bin 0 -> 134996 bytes
 article/cutting-planes-article.tex |  288 ++
 article/fig1.png                   |  Bin 0 -> 4535 bytes
 docs/poc.pdf                       |  Bin 60676 -> 0 bytes
 docs/poc.tex                       |   48 -
 8 files changed, 6638 insertions(+), 48 deletions(-)
 create mode 100644 article/IEEEtran.cls
 create mode 100644 article/IEEEtran_HOWTO.pdf
 create mode 100644 article/cutting-planes-article.pdf
 create mode 100644 article/cutting-planes-article.tex
 create mode 100644 article/fig1.png
 delete mode 100644 docs/poc.pdf
 delete mode 100644 docs/poc.tex

diff --git a/.gitignore b/.gitignore
index ec617a2..7d3fd48 100644
--- a/.gitignore
+++ b/.gitignore
@@ -313,3 +313,6 @@ TSWLatexianTemp*
 # option is specified. Footnotes are the stored in a file with suffix Notes.bib.
 # Uncomment the next line to have this generated file ignored.
 #*Notes.bib
+
+# Dowloaded assets
+*.zip
\ No newline at end of file
diff --git a/article/IEEEtran.cls b/article/IEEEtran.cls
new file mode 100644
index 0000000..905a10c
--- /dev/null
+++ b/article/IEEEtran.cls
@@ -0,0 +1,6347 @@
+%%
+%% IEEEtran.cls 2015/08/26 version V1.8b
+%% 
+%% This is the IEEEtran LaTeX class for authors of the Institute of 
+%% Electrical and Electronics Engineers (IEEE) Transactions journals and
+%% conferences.
+%% 
+%% Support sites:
+%% http://www.michaelshell.org/tex/ieeetran/
+%% http://www.ctan.org/pkg/ieeetran
+%% and
+%% http://www.ieee.org/
+%%
+%% Based on the original 1993 IEEEtran.cls, but with many bug fixes
+%% and enhancements (from both JVH and MDS) over the 1996/7 version.
+%%
+%%
+%% Contributors:
+%% Gerry Murray (1993), Silvano Balemi (1993),
+%% Jon Dixon (1996), Peter N"uchter (1996),
+%% Juergen von Hagen (2000), and Michael Shell (2001-2014)
+%% 
+%% 
+%% Copyright (c) 1993-2000 by Gerry Murray, Silvano Balemi, 
+%%                         Jon Dixon, Peter N"uchter,
+%%                         Juergen von Hagen
+%%                         and
+%% Copyright (c) 2001-2015 by Michael Shell
+%%
+%% Current maintainer (V1.3 to V1.8b): Michael Shell
+%%                                     See:
+%%                                     http://www.michaelshell.org/
+%%                                     for current contact information.
+%%
+%% Special thanks to Peter Wilson (CUA) and Donald Arseneau
+%% for allowing the inclusion of the \@ifmtarg command 
+%% from their ifmtarg LaTeX package. 
+%% 
+%%*************************************************************************
+%% Legal Notice:
+%% This code is offered as-is without any warranty either expressed or
+%% implied; without even the implied warranty of MERCHANTABILITY or
+%% FITNESS FOR A PARTICULAR PURPOSE! 
+%% User assumes all risk.
+%% In no event shall the IEEE or any contributor to this code be liable for
+%% any damages or losses, including, but not limited to, incidental,
+%% consequential, or any other damages, resulting from the use or misuse
+%% of any information contained here.
+%%
+%% All comments are the opinions of their respective authors and are not
+%% necessarily endorsed by the IEEE.
+%%
+%% This work is distributed under the LaTeX Project Public License (LPPL)
+%% ( http://www.latex-project.org/ ) version 1.3, and may be freely used,
+%% distributed and modified. A copy of the LPPL, version 1.3, is included
+%% in the base LaTeX documentation of all distributions of LaTeX released
+%% 2003/12/01 or later.
+%% Retain all contribution notices and credits.
+%% ** Modified files should be clearly indicated as such, including  **
+%% ** renaming them and changing author support contact information. **
+%%
+%% File list of work: IEEEtran.cls, IEEEtran_HOWTO.pdf, bare_adv.tex,
+%%                    bare_conf.tex, bare_jrnl.tex, bare_conf_compsoc.tex,
+%%                    bare_jrnl_compsoc.tex
+%% 
+%% Major changes to the user interface should be indicated by an 
+%% increase in the version numbers. If a version is a beta, it will 
+%% be indicated with a BETA suffix, i.e., 1.4 BETA.
+%% Small changes can be indicated by appending letters to the version
+%% such as "IEEEtran_v14a.cls".
+%% In all cases, \Providesclass, any \typeout messages to the user,
+%% \IEEEtransversionmajor and \IEEEtransversionminor must reflect the
+%% correct version information.
+%% The changes should also be documented via source comments.
+%%*************************************************************************
+%%
+%
+% Available class options 
+% e.g., \documentclass[10pt,conference]{IEEEtran} 
+% 
+%             *** choose only one from each category ***
+%
+% 9pt, 10pt, 11pt, 12pt
+%    Sets normal font size. The default is 10pt.
+% 
+% conference, journal, technote, peerreview, peerreviewca
+%    determines format mode - conference papers, journal papers,
+%    correspondence papers (technotes), or peer review papers. The user
+%    should also select 9pt when using technote. peerreview is like
+%    journal mode, but provides for a single-column "cover" title page for
+%    anonymous peer review. The paper title (without the author names) is
+%    repeated at the top of the page after the cover page. For peer review
+%    papers, the \IEEEpeerreviewmaketitle command must be executed (will
+%    automatically be ignored for non-peerreview modes) at the place the
+%    cover page is to end, usually just after the abstract (keywords are
+%    not normally used with peer review papers). peerreviewca is like
+%    peerreview, but allows the author names to be entered and formatted
+%    as with conference mode so that author affiliation and contact
+%    information can be easily seen on the cover page.
+%    The default is journal.
+%
+% draft, draftcls, draftclsnofoot, final
+%    determines if paper is formatted as a widely spaced draft (for
+%    handwritten editor comments) or as a properly typeset final version.
+%    draftcls restricts draft mode to the class file while all other LaTeX
+%    packages (i.e., \usepackage{graphicx}) will behave as final - allows
+%    for a draft paper with visible figures, etc. draftclsnofoot is like
+%    draftcls, but does not display the date and the word "DRAFT" at the foot
+%    of the pages. If using one of the draft modes, the user will probably
+%    also want to select onecolumn.
+%    The default is final.
+%
+% letterpaper, a4paper, cspaper
+%    determines paper size: 8.5in X 11in, 210mm X 297mm or 7.875in X 10.75in.
+%    Changing the paper size in the standard journal and conference modes
+%    will not alter the typesetting of the document - only the margins will
+%    be affected. In particular, documents using the a4paper option will
+%    have reduced side margins (A4 is narrower than US letter) and a longer
+%    bottom margin (A4 is longer than US letter). For both cases, the top
+%    margins will be the same and the text will be horizontally centered.
+%    For the compsoc conference and draft modes, it is the margins that will
+%    remain constant, and thus the text area size will vary, with changes in
+%    the paper size.
+%    The cspaper option is the special ``trim'' paper size (7.875in x 10.75in)
+%    used in the actual publication of Computer Society journals. Under
+%    compsoc journal mode, this option does not alter the typesetting of the
+%    document. Authors should invoke the cspaper option only if requested to
+%    do so by the editors of the specific journal they are submitting to.
+%    For final submission to the IEEE, authors should generally use US letter 
+%    (8.5 X 11in) paper unless otherwise instructed. Note that authors should
+%    ensure that all post-processing (ps, pdf, etc.) uses the same paper
+%    specificiation as the .tex document. Problems here are by far the number
+%    one reason for incorrect margins. IEEEtran will automatically set the
+%    default paper size under pdflatex (without requiring any change to
+%    pdftex.cfg), so this issue is more important to dvips users. Fix
+%    config.ps, config.pdf, or ~/.dvipsrc for dvips, or use the
+%    dvips -t papersize option instead as needed. For the cspaper option,
+%    the corresponding dvips paper name is "ieeecs".
+%    See the testflow documentation
+%    http://www.ctan.org/tex-archive/macros/latex/contrib/IEEEtran/testflow
+%    for more details on dvips paper size configuration.
+%    The default is letterpaper.
+%
+% oneside, twoside
+%    determines if layout follows single sided or two sided (duplex)
+%    printing. The only notable change is with the headings at the top of
+%    the pages.
+%    The default is oneside.
+%
+% onecolumn, twocolumn
+%    determines if text is organized into one or two columns per page. One
+%    column mode is usually used only with draft papers.
+%    The default is twocolumn.
+%
+% comsoc, compsoc, transmag
+%    Use the format of the IEEE Communications Society, IEEE Computer Society
+%    or IEEE Transactions on Magnetics, respectively.
+%
+% romanappendices
+%    Use the "Appendix I" convention when numbering appendices. IEEEtran.cls
+%    now defaults to Alpha "Appendix A" convention - the opposite of what
+%    v1.6b and earlier did.
+%
+% captionsoff
+%    disables the display of the figure/table captions. Some IEEE journals
+%    request that captions be removed and figures/tables be put on pages
+%    of their own at the end of an initial paper submission. The endfloat
+%    package can be used with this class option to achieve this format.
+%
+% nofonttune
+%    turns off tuning of the font interword spacing. Maybe useful to those
+%    not using the standard Times fonts or for those who have already "tuned"
+%    their fonts.
+%    The default is to enable IEEEtran to tune font parameters.
+%
+%
+%----------
+% Available CLASSINPUTs provided (all are macros unless otherwise noted):
+% \CLASSINPUTbaselinestretch
+% \CLASSINPUTinnersidemargin
+% \CLASSINPUToutersidemargin
+% \CLASSINPUTtoptextmargin
+% \CLASSINPUTbottomtextmargin
+%
+% Available CLASSINFOs provided:
+% \ifCLASSINFOpdf                       (TeX if conditional)
+% \CLASSINFOpaperwidth                  (macro)
+% \CLASSINFOpaperheight                 (macro)
+% \CLASSINFOnormalsizebaselineskip      (length)
+% \CLASSINFOnormalsizeunitybaselineskip (length)
+%
+% Available CLASSOPTIONs provided:
+% all class option flags (TeX if conditionals) unless otherwise noted,
+% e.g., \ifCLASSOPTIONcaptionsoff
+% point size options provided as a single macro:
+% \CLASSOPTIONpt
+% which will be defined as 9, 10, 11, or 12 depending on the document's
+% normalsize point size.
+% also, class option peerreviewca implies the use of class option peerreview
+% and classoption draft implies the use of class option draftcls
+
+
+
+
+
+\ProvidesClass{IEEEtran}[2015/08/26 V1.8b by Michael Shell]
+\typeout{-- See the "IEEEtran_HOWTO" manual for usage information.}
+\typeout{-- http://www.michaelshell.org/tex/ieeetran/}
+\NeedsTeXFormat{LaTeX2e}
+
+% IEEEtran.cls version numbers, provided as of V1.3
+% These values serve as a way a .tex file can
+% determine if the new features are provided.
+% The version number of this IEEEtrans.cls can be obtained from 
+% these values. i.e., V1.4
+% KEEP THESE AS INTEGERS! i.e., NO {4a} or anything like that-
+% (no need to enumerate "a" minor changes here)
+\def\IEEEtransversionmajor{1}
+\def\IEEEtransversionminor{8}
+
+
+% hook to allow easy changeover to IEEEtran.cls/tools.sty error reporting
+\def\@IEEEclspkgerror{\ClassError{IEEEtran}}
+
+
+% These do nothing, but provide them like in article.cls
+\newif\if@restonecol
+\newif\if@titlepage
+
+
+% class option conditionals
+\newif\ifCLASSOPTIONonecolumn       \CLASSOPTIONonecolumnfalse
+\newif\ifCLASSOPTIONtwocolumn       \CLASSOPTIONtwocolumntrue
+
+\newif\ifCLASSOPTIONoneside         \CLASSOPTIONonesidetrue
+\newif\ifCLASSOPTIONtwoside         \CLASSOPTIONtwosidefalse
+
+\newif\ifCLASSOPTIONfinal           \CLASSOPTIONfinaltrue
+\newif\ifCLASSOPTIONdraft           \CLASSOPTIONdraftfalse
+\newif\ifCLASSOPTIONdraftcls        \CLASSOPTIONdraftclsfalse
+\newif\ifCLASSOPTIONdraftclsnofoot  \CLASSOPTIONdraftclsnofootfalse
+
+\newif\ifCLASSOPTIONpeerreview      \CLASSOPTIONpeerreviewfalse
+\newif\ifCLASSOPTIONpeerreviewca    \CLASSOPTIONpeerreviewcafalse
+
+\newif\ifCLASSOPTIONjournal         \CLASSOPTIONjournaltrue
+\newif\ifCLASSOPTIONconference      \CLASSOPTIONconferencefalse
+\newif\ifCLASSOPTIONtechnote        \CLASSOPTIONtechnotefalse
+
+\newif\ifCLASSOPTIONnofonttune      \CLASSOPTIONnofonttunefalse
+
+\newif\ifCLASSOPTIONcaptionsoff     \CLASSOPTIONcaptionsofffalse
+
+\newif\ifCLASSOPTIONcomsoc          \CLASSOPTIONcomsocfalse
+\newif\ifCLASSOPTIONcompsoc         \CLASSOPTIONcompsocfalse
+\newif\ifCLASSOPTIONtransmag        \CLASSOPTIONtransmagfalse
+
+\newif\ifCLASSOPTIONromanappendices \CLASSOPTIONromanappendicesfalse
+
+
+% class info conditionals
+
+% indicates if pdf (via pdflatex) output
+\newif\ifCLASSINFOpdf               \CLASSINFOpdffalse
+
+
+% V1.6b internal flag to show if using a4paper
+\newif\if@IEEEusingAfourpaper       \@IEEEusingAfourpaperfalse
+% V1.6b internal flag to show if using cspaper
+\newif\if@IEEEusingcspaper          \@IEEEusingcspaperfalse
+
+
+% IEEEtran class scratch pad registers
+% dimen
+\newdimen\@IEEEtrantmpdimenA
+\newdimen\@IEEEtrantmpdimenB
+\newdimen\@IEEEtrantmpdimenC
+% count
+\newcount\@IEEEtrantmpcountA
+\newcount\@IEEEtrantmpcountB
+\newcount\@IEEEtrantmpcountC
+% token list
+\newtoks\@IEEEtrantmptoksA
+
+% we use \CLASSOPTIONpt so that we can ID the point size (even for 9pt docs)
+% as well as LaTeX's \@ptsize to retain some compatability with some
+% external packages
+\def\@ptsize{0}
+% LaTeX does not support 9pt, so we set \@ptsize to 0 - same as that of 10pt
+\DeclareOption{9pt}{\def\CLASSOPTIONpt{9}\def\@ptsize{0}}
+\DeclareOption{10pt}{\def\CLASSOPTIONpt{10}\def\@ptsize{0}}
+\DeclareOption{11pt}{\def\CLASSOPTIONpt{11}\def\@ptsize{1}}
+\DeclareOption{12pt}{\def\CLASSOPTIONpt{12}\def\@ptsize{2}}
+
+
+
+\DeclareOption{letterpaper}{\setlength{\paperwidth}{8.5in}%
+                            \setlength{\paperheight}{11in}%
+                            \@IEEEusingAfourpaperfalse
+                            \@IEEEusingcspaperfalse
+                            \def\CLASSOPTIONpaper{letter}%
+                            \def\CLASSINFOpaperwidth{8.5in}%
+                            \def\CLASSINFOpaperheight{11in}}
+
+
+\DeclareOption{a4paper}{\setlength{\paperwidth}{210mm}%
+                        \setlength{\paperheight}{297mm}%
+                        \@IEEEusingAfourpapertrue
+                        \@IEEEusingcspaperfalse
+                        \def\CLASSOPTIONpaper{a4}%
+                        \def\CLASSINFOpaperwidth{210mm}%
+                        \def\CLASSINFOpaperheight{297mm}}
+
+% special paper option for compsoc journals
+\DeclareOption{cspaper}{\setlength{\paperwidth}{7.875in}%
+                        \setlength{\paperheight}{10.75in}%
+                        \@IEEEusingcspapertrue
+                        \@IEEEusingAfourpaperfalse
+                        \def\CLASSOPTIONpaper{ieeecs}%
+                        \def\CLASSINFOpaperwidth{7.875in}%
+                        \def\CLASSINFOpaperheight{10.75in}}
+
+\DeclareOption{oneside}{\@twosidefalse\@mparswitchfalse
+                        \CLASSOPTIONonesidetrue\CLASSOPTIONtwosidefalse}
+\DeclareOption{twoside}{\@twosidetrue\@mparswitchtrue
+                        \CLASSOPTIONtwosidetrue\CLASSOPTIONonesidefalse}
+
+\DeclareOption{onecolumn}{\CLASSOPTIONonecolumntrue\CLASSOPTIONtwocolumnfalse}
+\DeclareOption{twocolumn}{\CLASSOPTIONtwocolumntrue\CLASSOPTIONonecolumnfalse}
+
+% If the user selects draft, then this class AND any packages
+% will go into draft mode.
+\DeclareOption{draft}{\CLASSOPTIONdrafttrue\CLASSOPTIONdraftclstrue
+                      \CLASSOPTIONdraftclsnofootfalse} 
+% draftcls is for a draft mode which will not affect any packages
+% used by the document.
+\DeclareOption{draftcls}{\CLASSOPTIONdraftfalse\CLASSOPTIONdraftclstrue
+                         \CLASSOPTIONdraftclsnofootfalse} 
+% draftclsnofoot is like draftcls, but without the footer.
+\DeclareOption{draftclsnofoot}{\CLASSOPTIONdraftfalse\CLASSOPTIONdraftclstrue
+                               \CLASSOPTIONdraftclsnofoottrue} 
+\DeclareOption{final}{\CLASSOPTIONdraftfalse\CLASSOPTIONdraftclsfalse
+                      \CLASSOPTIONdraftclsnofootfalse}
+
+\DeclareOption{journal}{\CLASSOPTIONpeerreviewfalse\CLASSOPTIONpeerreviewcafalse
+                        \CLASSOPTIONjournaltrue\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotefalse}
+
+\DeclareOption{conference}{\CLASSOPTIONpeerreviewfalse\CLASSOPTIONpeerreviewcafalse
+                           \CLASSOPTIONjournalfalse\CLASSOPTIONconferencetrue\CLASSOPTIONtechnotefalse}
+
+\DeclareOption{technote}{\CLASSOPTIONpeerreviewfalse\CLASSOPTIONpeerreviewcafalse
+                         \CLASSOPTIONjournalfalse\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotetrue}
+
+\DeclareOption{peerreview}{\CLASSOPTIONpeerreviewtrue\CLASSOPTIONpeerreviewcafalse
+                           \CLASSOPTIONjournalfalse\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotefalse}
+
+\DeclareOption{peerreviewca}{\CLASSOPTIONpeerreviewtrue\CLASSOPTIONpeerreviewcatrue
+                             \CLASSOPTIONjournalfalse\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotefalse}
+
+\DeclareOption{nofonttune}{\CLASSOPTIONnofonttunetrue}
+
+\DeclareOption{captionsoff}{\CLASSOPTIONcaptionsofftrue}
+
+\DeclareOption{comsoc}{\CLASSOPTIONcomsoctrue\CLASSOPTIONcompsocfalse\CLASSOPTIONtransmagfalse}
+
+\DeclareOption{compsoc}{\CLASSOPTIONcomsocfalse\CLASSOPTIONcompsoctrue\CLASSOPTIONtransmagfalse}
+
+\DeclareOption{transmag}{\CLASSOPTIONtransmagtrue\CLASSOPTIONcomsocfalse\CLASSOPTIONcompsocfalse}
+
+\DeclareOption{romanappendices}{\CLASSOPTIONromanappendicestrue}
+
+
+% default to US letter paper, 10pt, twocolumn, one sided, final, journal
+\ExecuteOptions{letterpaper,10pt,twocolumn,oneside,final,journal}
+% overrride these defaults per user requests
+\ProcessOptions
+
+
+
+%% -- Command Argument Scanning Support Functions --
+
+% Sets the category codes for punctuation to their normal values.
+% For local use with argument scanning.
+\def\IEEEnormalcatcodespunct{\catcode`\!=12 \catcode`\,=12 \catcode`\:=12
+\catcode`\;=12 \catcode`\`=12 \catcode`\'=12 \catcode`\"=12 \catcode`\.=12
+\catcode`\/=12 \catcode`\?=12 \catcode`\*=12 \catcode`\+=12 \catcode`\-=12
+\catcode`\<=12 \catcode`\>=12 \catcode`\(=12 \catcode`\)=12 \catcode`\[=12
+\catcode`\]=12 \catcode`\==12 \catcode`\|=12}
+% Sets the category codes for numbers to their normal values.
+% For local use with argument scanning.
+\def\IEEEnormalcatcodesnum{\catcode`\0=12 \catcode`\1=12 \catcode`\2=12
+\catcode`\3=12 \catcode`\4=12 \catcode`\5=12 \catcode`\6=12 \catcode`\7=12
+\catcode`\8=12 \catcode`\9=12}
+% combined action of \IEEEnormalcatcodespunct and \IEEEnormalcatcodesnum
+\def\IEEEnormalcatcodes{\IEEEnormalcatcodespunct\IEEEnormalcatcodesnum}
+
+
+% usage: \@IEEEextracttoken*{}
+% \@IEEEextracttoken fully expands its argument (which it then stores in
+% \@IEEEextracttokenarg) via \edef and then the meaning of the first
+% nonbrace (but including the empty group) token found is assigned via \let
+% to \@IEEEextractedtoken as well as stored in the macro
+% \@IEEEextractedtokenmacro. Tokens that would otherwise be discarded during
+% the acquisition of the first are stored in \@IEEEextractedtokensdiscarded,
+% however their original relative brace nesting depths are not guaranteed to
+% be preserved.
+% If the argument is empty, or if a first nonbrace token does not exist (or
+% is an empty group), \@IEEEextractedtoken will be \relax and
+% \@IEEEextractedtokenmacro and \@IEEEextractedtokensdiscarded will be empty.
+%
+% For example:
+% \@IEEEextracttoken{{{ab}{cd}}{{ef}g}}
+% results in:
+%
+% \@IEEEextracttokenarg          ==> a macro containing {{ab}{cd}}{{ef}g}
+% \@IEEEextractedtoken           ==> the letter a
+% \@IEEEextractedtokenmacro      ==> a macro containing a
+% \@IEEEextractedtokensdiscarded ==> a macro containing bcd{ef}g
+%
+% the *-star form, \@IEEEextracttoken*, does not expand its argument
+% contents during processing.
+\def\@IEEEextracttoken{\@ifstar{\let\@IEEEextracttokendef=\def\@@IEEEextracttoken}{\let\@IEEEextracttokendef=\edef\@@IEEEextracttoken}}
+
+\def\@@IEEEextracttoken#1{\@IEEEextracttokendef\@IEEEextracttokenarg{#1}\relax
+\def\@IEEEextractedtokensdiscarded{}\relax % initialize to empty
+% if the macro is unchanged after being acquired as a single undelimited argument
+% with anything after it being stripped off as a delimited argument
+% we know we have one token without any enclosing braces. loop until this is true.
+\let\@IEEEextracttokencurgroup\@IEEEextracttokenarg
+\loop
+  % trap case of an empty argument as this would cause a problem with
+  % \@@@IEEEextracttoken's first (nondelimited) argument acquisition
+  \ifx\@IEEEextracttokencurgroup\@empty
+    \def\@IEEEextractedtokenmacro{}\relax
+  \else
+    \expandafter\@@@IEEEextracttoken\@IEEEextracttokencurgroup\@IEEEgeneralsequenceDELIMITER\relax
+  \fi
+  \ifx\@IEEEextractedtokenmacro\@IEEEextracttokencurgroup
+  \else
+    \let\@IEEEextracttokencurgroup=\@IEEEextractedtokenmacro
+\repeat
+% we can safely do a \let= here because there should be at most one token
+% the relax is needed to handle the case of no token found
+\expandafter\let\expandafter\@IEEEextractedtoken\@IEEEextractedtokenmacro\relax}
+
+\def\@@@IEEEextracttoken#1#2\@IEEEgeneralsequenceDELIMITER{\def\@IEEEextractedtokenmacro{#1}\relax
+\def\@@IEEEextractedtokensdiscarded{#2}\expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter
+\@IEEEextractedtokensdiscarded\expandafter\expandafter\expandafter
+{\expandafter\@@IEEEextractedtokensdiscarded\@IEEEextractedtokensdiscarded}}
+%%
+%% -- End of Command Argument Scanning Support Functions --
+
+
+
+% Computer Society conditional execution command
+\long\def\@IEEEcompsoconly#1{\relax\ifCLASSOPTIONcompsoc\relax#1\relax\fi\relax}
+% inverse
+\long\def\@IEEEnotcompsoconly#1{\relax\ifCLASSOPTIONcompsoc\else\relax#1\relax\fi\relax}
+% compsoc conference
+\long\def\@IEEEcompsocconfonly#1{\relax\ifCLASSOPTIONcompsoc\ifCLASSOPTIONconference\relax#1\relax\fi\fi\relax}
+% compsoc not conference
+\long\def\@IEEEcompsocnotconfonly#1{\relax\ifCLASSOPTIONcompsoc\ifCLASSOPTIONconference\else\relax#1\relax\fi\fi\relax}
+
+
+% comsoc verify that newtxmath, mtpro2, mt11p or mathtime has been loaded
+\def\@IEEEcomsocverifymathfont{\typeout{-- Verifying Times compatible math font.}\relax
+  \@ifpackageloaded{newtxmath}{\typeout{-- newtxmath loaded, OK.}}{\@@IEEEcomsocverifymathfont}}
+\def\@@IEEEcomsocverifymathfont{\@ifpackageloaded{mtpro2}{\typeout{-- mtpro2 loaded, OK.}}{\@@@IEEEcomsocverifymathfont}}
+\def\@@@IEEEcomsocverifymathfont{\@ifpackageloaded{mt11p}{\typeout{-- mt11p2 loaded, OK.}}{\@@@@IEEEcomsocverifymathfont}}
+\def\@@@@IEEEcomsocverifymathfont{\@ifpackageloaded{mathtime}{\typeout{-- mathtime loaded, OK.}}{\@IEEEcomsocenforcemathfont}}
+
+% comsoc, if a Times math font was not loaded by user, enforce it
+\def\@IEEEcomsocenforcemathfont{\typeout{** Times compatible math font not found, forcing.}\relax
+\IfFileExists{newtxmath.sty}{\typeout{-- Found newtxmath, loading.}\RequirePackage{newtxmath}}{\@@IEEEcomsocenforcemathfont}}
+\def\@@IEEEcomsocenforcemathfont{\IfFileExists{mtpro2.sty}{\typeout{-- Found mtpro2, loading.}\RequirePackage{mtpro2}}{\@@@IEEEcomsocenforcemathfont}}
+\def\@@@IEEEcomsocenforcemathfont{\IfFileExists{mt11p.sty}{\typeout{-- Found mt11p, loading.}\RequirePackage{mt11p}}{\@@@@IEEEcomsocenforcemathfont}}
+\def\@@@@IEEEcomsocenforcemathfont{\IfFileExists{mathtime.sty}{\typeout{-- Found mathtime, loading.}\RequirePackage{mathtime}}{\@@@@@IEEEcomsocenforcemathfont}}
+% if no acceptable Times math font package found, error with newtxmath requirement
+\def\@@@@@IEEEcomsocenforcemathfont{\typeout{** No Times compatible math font package found. newtxmath is required.}\RequirePackage{newtxmath}}
+
+
+\ifCLASSOPTIONcomsoc
+  % ensure that if newtxmath is used, the cmintegrals option is also invoked
+  \PassOptionsToPackage{cmintegrals}{newtxmath}
+  % comsoc requires a Times like math font
+  % ensure this requirement is satisfied at document start
+  \AtBeginDocument{\@IEEEcomsocverifymathfont}
+\fi
+
+
+
+% The IEEE uses Times Roman font, so we'll default to Times.
+% These three commands make up the entire times.sty package.
+\renewcommand{\sfdefault}{phv}
+\renewcommand{\rmdefault}{ptm}
+\renewcommand{\ttdefault}{pcr}
+
+% V1.7 compsoc nonconference papers, use Palatino/Palladio as the main text font,
+% not Times Roman.
+\@IEEEcompsocnotconfonly{\renewcommand{\rmdefault}{ppl}}
+
+% enable the selected main text font
+\normalfont\selectfont
+
+
+\ifCLASSOPTIONcomsoc
+  \typeout{-- Using IEEE Communications Society mode.}
+\fi
+
+\ifCLASSOPTIONcompsoc
+  \typeout{-- Using IEEE Computer Society mode.}
+\fi
+
+
+% V1.7 conference notice message hook
+\def\@IEEEconsolenoticeconference{\typeout{}%
+\typeout{** Conference Paper **}%
+\typeout{Before submitting the final camera ready copy, remember to:}%
+\typeout{}%
+\typeout{ 1. Manually equalize the lengths of two columns on the last page}%
+\typeout{ of your paper;}%
+\typeout{}%
+\typeout{ 2. Ensure that any PostScript and/or PDF output post-processing}%
+\typeout{ uses only Type 1 fonts and that every step in the generation}%
+\typeout{ process uses the appropriate paper size.}%
+\typeout{}}
+
+
+% we can send console reminder messages to the user here
+\AtEndDocument{\ifCLASSOPTIONconference\@IEEEconsolenoticeconference\fi}
+
+
+% warn about the use of single column other than for draft mode
+\ifCLASSOPTIONtwocolumn\else%
+  \ifCLASSOPTIONdraftcls\else%
+   \typeout{** ATTENTION: Single column mode is not typically used with IEEE publications.}%
+  \fi%
+\fi
+
+
+% V1.7 improved paper size setting code.
+% Set pdfpage and dvips paper sizes. Conditional tests are similar to that
+% of ifpdf.sty. Retain within {} to ensure tested macros are never altered,
+% even if only effect is to set them to \relax.
+% if \pdfoutput is undefined or equal to relax, output a dvips special
+{\@ifundefined{pdfoutput}{\AtBeginDvi{\special{papersize=\CLASSINFOpaperwidth,\CLASSINFOpaperheight}}}{%
+% pdfoutput is defined and not equal to \relax
+% check for pdfpageheight existence just in case someone sets pdfoutput
+% under non-pdflatex. If exists, set them regardless of value of \pdfoutput.
+\@ifundefined{pdfpageheight}{\relax}{\global\pdfpagewidth\paperwidth
+\global\pdfpageheight\paperheight}%
+% if using \pdfoutput=0 under pdflatex, send dvips papersize special
+\ifcase\pdfoutput
+\AtBeginDvi{\special{papersize=\CLASSINFOpaperwidth,\CLASSINFOpaperheight}}%
+\else
+% we are using pdf output, set CLASSINFOpdf flag
+\global\CLASSINFOpdftrue
+\fi}}
+
+% let the user know the selected papersize
+\typeout{-- Using \CLASSINFOpaperwidth\space x \CLASSINFOpaperheight\space
+(\CLASSOPTIONpaper)\space paper.}
+
+\ifCLASSINFOpdf
+\typeout{-- Using PDF output.}
+\else
+\typeout{-- Using DVI output.}
+\fi
+
+
+% The idea hinted here is for LaTeX to generate markleft{} and markright{}
+% automatically for you after you enter \author{}, \journal{},
+% \journaldate{}, journalvol{}, \journalnum{}, etc.
+% However, there may be some backward compatibility issues here as
+% well as some special applications for IEEEtran.cls and special issues
+% that may require the flexible \markleft{}, \markright{} and/or \markboth{}.
+% We'll leave this as an open future suggestion.
+%\newcommand{\journal}[1]{\def\@journal{#1}}
+%\def\@journal{}
+
+
+
+% pointsize values
+% used with ifx to determine the document's normal size
+\def\@IEEEptsizenine{9}
+\def\@IEEEptsizeten{10}
+\def\@IEEEptsizeeleven{11}
+\def\@IEEEptsizetwelve{12}
+
+
+
+% FONT DEFINITIONS (No sizexx.clo file needed) 
+% V1.6 revised font sizes, displayskip values and
+%      revised normalsize baselineskip to reduce underfull vbox problems
+%      on the 58pc = 696pt = 9.5in text height we want
+%      normalsize     #lines/column  baselineskip (aka leading)
+%             9pt     63             11.0476pt (truncated down)
+%            10pt     58             12pt      (exact)
+%            11pt     52             13.3846pt (truncated down)
+%            12pt     50             13.92pt   (exact)
+%
+
+% we need to store the nominal baselineskip for the given font size
+% in case baselinestretch ever changes.
+% this is a dimen, so it will not hold stretch or shrink
+\newdimen\@IEEEnormalsizeunitybaselineskip
+\@IEEEnormalsizeunitybaselineskip\baselineskip
+
+
+
+%% ******* WARNING! *******
+%%
+%% Authors should not alter font sizes, baselineskip ("leading"),
+%% margins or other spacing values in an attempt to squeeze more
+%% material on each page.
+%%
+%% The IEEE's own typesetting software will restore the correct
+%% values when re-typesetting/proofing the submitted document,
+%% possibly resulting in unexpected article over length charges.
+%%
+%% ******* WARNING! *******
+
+
+% 9pt option defaults
+\ifx\CLASSOPTIONpt\@IEEEptsizenine
+\typeout{-- This is a 9 point document.}
+\def\normalsize{\@setfontsize{\normalsize}{9}{11.0476pt}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{11.0476pt}
+\normalsize
+\abovedisplayskip 1.5ex plus 3pt minus 1pt
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0pt plus 3pt
+\belowdisplayshortskip 1.5ex plus 3pt minus 1pt
+\def\small{\@setfontsize{\small}{8.5}{10pt}}
+\def\footnotesize{\@setfontsize{\footnotesize}{8}{9pt}}
+\def\scriptsize{\@setfontsize{\scriptsize}{7}{8pt}}
+\def\tiny{\@setfontsize{\tiny}{5}{6pt}}
+% sublargesize is the same as large - 10pt
+\def\sublargesize{\@setfontsize{\sublargesize}{10}{12pt}}
+\def\large{\@setfontsize{\large}{10}{12pt}}
+\def\Large{\@setfontsize{\Large}{12}{14pt}}
+\def\LARGE{\@setfontsize{\LARGE}{14}{17pt}}
+\def\huge{\@setfontsize{\huge}{17}{20pt}}
+\def\Huge{\@setfontsize{\Huge}{20}{24pt}}
+\fi
+%
+% 10pt option defaults
+\ifx\CLASSOPTIONpt\@IEEEptsizeten
+\typeout{-- This is a 10 point document.}
+\def\normalsize{\@setfontsize{\normalsize}{10}{12.00pt}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{12pt}
+\normalsize
+\abovedisplayskip 1.5ex plus 4pt minus 2pt
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0pt plus 4pt
+\belowdisplayshortskip 1.5ex plus 4pt minus 2pt
+\def\small{\@setfontsize{\small}{9}{10pt}}
+\def\footnotesize{\@setfontsize{\footnotesize}{8}{9pt}}
+\def\scriptsize{\@setfontsize{\scriptsize}{7}{8pt}}
+\def\tiny{\@setfontsize{\tiny}{5}{6pt}}
+% sublargesize is a tad smaller than large - 11pt
+\def\sublargesize{\@setfontsize{\sublargesize}{11}{13.4pt}}
+\def\large{\@setfontsize{\large}{12}{14pt}}
+\def\Large{\@setfontsize{\Large}{14}{17pt}}
+\def\LARGE{\@setfontsize{\LARGE}{17}{20pt}}
+\def\huge{\@setfontsize{\huge}{20}{24pt}}
+\def\Huge{\@setfontsize{\Huge}{24}{28pt}}
+\fi
+%
+% 11pt option defaults
+\ifx\CLASSOPTIONpt\@IEEEptsizeeleven
+\typeout{-- This is an 11 point document.}
+\def\normalsize{\@setfontsize{\normalsize}{11}{13.3846pt}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{13.3846pt}
+\normalsize
+\abovedisplayskip 1.5ex plus 5pt minus 3pt
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0pt plus 5pt
+\belowdisplayshortskip 1.5ex plus 5pt minus 3pt
+\def\small{\@setfontsize{\small}{10}{12pt}}
+\def\footnotesize{\@setfontsize{\footnotesize}{9}{10.5pt}}
+\def\scriptsize{\@setfontsize{\scriptsize}{8}{9pt}}
+\def\tiny{\@setfontsize{\tiny}{6}{7pt}}
+% sublargesize is the same as large - 12pt
+\def\sublargesize{\@setfontsize{\sublargesize}{12}{14pt}}
+\def\large{\@setfontsize{\large}{12}{14pt}}
+\def\Large{\@setfontsize{\Large}{14}{17pt}}
+\def\LARGE{\@setfontsize{\LARGE}{17}{20pt}}
+\def\huge{\@setfontsize{\huge}{20}{24pt}}
+\def\Huge{\@setfontsize{\Huge}{24}{28pt}}
+\fi
+%
+% 12pt option defaults
+\ifx\CLASSOPTIONpt\@IEEEptsizetwelve
+\typeout{-- This is a 12 point document.}
+\def\normalsize{\@setfontsize{\normalsize}{12}{13.92pt}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{13.92pt}
+\normalsize
+\abovedisplayskip 1.5ex plus 6pt minus 4pt
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0pt plus 6pt
+\belowdisplayshortskip 1.5ex plus 6pt minus 4pt
+\def\small{\@setfontsize{\small}{10}{12pt}}
+\def\footnotesize{\@setfontsize{\footnotesize}{9}{10.5pt}}
+\def\scriptsize{\@setfontsize{\scriptsize}{8}{9pt}}
+\def\tiny{\@setfontsize{\tiny}{6}{7pt}}
+% sublargesize is the same as large - 14pt
+\def\sublargesize{\@setfontsize{\sublargesize}{14}{17pt}}
+\def\large{\@setfontsize{\large}{14}{17pt}}
+\def\Large{\@setfontsize{\Large}{17}{20pt}}
+\def\LARGE{\@setfontsize{\LARGE}{20}{24pt}}
+\def\huge{\@setfontsize{\huge}{22}{26pt}}
+\def\Huge{\@setfontsize{\Huge}{24}{28pt}}
+\fi
+
+
+
+% V1.8a compsoc font sizes
+% compsoc font sizes use bp "Postscript" point units (1/72in) 
+% rather than the traditional pt (1/72.27)
+\ifCLASSOPTIONcompsoc
+% -- compsoc defaults --
+% ** will override some of these values later **
+% 9pt
+\ifx\CLASSOPTIONpt\@IEEEptsizenine
+\def\normalsize{\@setfontsize{\normalsize}{9bp}{11bp}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{11bp}
+\normalsize
+\abovedisplayskip 1.5ex plus 3bp minus 1bp
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0bp plus 3bp
+\belowdisplayshortskip 1.5ex plus 3bp minus 1bp
+\def\small{\@setfontsize{\small}{8.5bp}{10bp}}
+\def\footnotesize{\@setfontsize{\footnotesize}{8bp}{9bp}}
+\def\scriptsize{\@setfontsize{\scriptsize}{7bp}{8bp}}
+\def\tiny{\@setfontsize{\tiny}{5bp}{6bp}}
+% sublargesize is the same as large - 10bp
+\def\sublargesize{\@setfontsize{\sublargesize}{10bp}{12bp}}
+\def\large{\@setfontsize{\large}{10bp}{12bp}}
+\def\Large{\@setfontsize{\Large}{12bp}{14bp}}
+\def\LARGE{\@setfontsize{\LARGE}{14bp}{17bp}}
+\def\huge{\@setfontsize{\huge}{17bp}{20bp}}
+\def\Huge{\@setfontsize{\Huge}{20bp}{24bp}}
+\fi
+%
+% 10pt
+\ifx\CLASSOPTIONpt\@IEEEptsizeten
+\def\normalsize{\@setfontsize{\normalsize}{10bp}{12bp}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{12bp}
+\normalsize
+\abovedisplayskip 1.5ex plus 4bp minus 2bp
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0pt plus 4bp
+\belowdisplayshortskip 1.5ex plus 4bp minus 2bp
+\def\small{\@setfontsize{\small}{9bp}{10bp}}
+\def\footnotesize{\@setfontsize{\footnotesize}{8bp}{9bp}}
+\def\scriptsize{\@setfontsize{\scriptsize}{7bp}{8bp}}
+\def\tiny{\@setfontsize{\tiny}{5bp}{6bp}}
+% sublargesize is a tad smaller than large - 11bp
+\def\sublargesize{\@setfontsize{\sublargesize}{11bp}{13.5bp}}
+\def\large{\@setfontsize{\large}{12bp}{14bp}}
+\def\Large{\@setfontsize{\Large}{14bp}{17bp}}
+\def\LARGE{\@setfontsize{\LARGE}{17bp}{20bp}}
+\def\huge{\@setfontsize{\huge}{20bp}{24bp}}
+\def\Huge{\@setfontsize{\Huge}{24bp}{28bp}}
+\fi
+%
+% 11pt
+\ifx\CLASSOPTIONpt\@IEEEptsizeeleven
+\def\normalsize{\@setfontsize{\normalsize}{11bp}{13.5bp}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{13.5bp}
+\normalsize
+\abovedisplayskip 1.5ex plus 5bp minus 3bp
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0pt plus 5bp
+\belowdisplayshortskip 1.5ex plus 5bp minus 3bp
+\def\small{\@setfontsize{\small}{10bp}{12bp}}
+\def\footnotesize{\@setfontsize{\footnotesize}{9bp}{10.5bp}}
+\def\scriptsize{\@setfontsize{\scriptsize}{8bp}{9bp}}
+\def\tiny{\@setfontsize{\tiny}{6bp}{7bp}}
+% sublargesize is the same as large - 12bp
+\def\sublargesize{\@setfontsize{\sublargesize}{12bp}{14bp}}
+\def\large{\@setfontsize{\large}{12bp}{14bp}}
+\def\Large{\@setfontsize{\Large}{14bp}{17bp}}
+\def\LARGE{\@setfontsize{\LARGE}{17bp}{20bp}}
+\def\huge{\@setfontsize{\huge}{20bp}{24bp}}
+\def\Huge{\@setfontsize{\Huge}{24bp}{28bp}}
+\fi
+%
+% 12pt
+\ifx\CLASSOPTIONpt\@IEEEptsizetwelve
+\def\normalsize{\@setfontsize{\normalsize}{12bp}{14bp}}%
+\setlength{\@IEEEnormalsizeunitybaselineskip}{14bp}%
+\normalsize
+\abovedisplayskip 1.5ex plus 6bp minus 4bp
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0pt plus 6bp
+\belowdisplayshortskip 1.5ex plus 6bp minus 4bp
+\def\small{\@setfontsize{\small}{10bp}{12bp}}
+\def\footnotesize{\@setfontsize{\footnotesize}{9bp}{10.5bp}}
+\def\scriptsize{\@setfontsize{\scriptsize}{8bp}{9bp}}
+\def\tiny{\@setfontsize{\tiny}{6bp}{7bp}}
+% sublargesize is the same as large - 14bp
+\def\sublargesize{\@setfontsize{\sublargesize}{14bp}{17bp}}
+\def\large{\@setfontsize{\large}{14bp}{17bp}}
+\def\Large{\@setfontsize{\Large}{17bp}{20bp}}
+\def\LARGE{\@setfontsize{\LARGE}{20bp}{24bp}}
+\def\huge{\@setfontsize{\huge}{22bp}{26bp}}
+\def\Huge{\@setfontsize{\Huge}{24bp}{28bp}}
+\fi
+%
+% -- override defaults: compsoc journals use special normalsizes --
+\ifCLASSOPTIONconference
+% 
+% compsoc conferences
+% 9pt
+\ifx\CLASSOPTIONpt\@IEEEptsizenine
+\def\normalsize{\@setfontsize{\normalsize}{9bp}{10.8bp}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{10.8bp}
+\normalsize
+\abovedisplayskip 1.5ex plus 3bp minus 1bp
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0bp plus 3bp
+\belowdisplayshortskip 1.5ex plus 3bp minus 1bp
+\fi
+% 10pt
+\ifx\CLASSOPTIONpt\@IEEEptsizeten
+\def\normalsize{\@setfontsize{\normalsize}{10bp}{11.2bp}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{11.2bp}
+\normalsize
+\abovedisplayskip 1.5ex plus 4bp minus 2bp
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0pt plus 4bp
+\belowdisplayshortskip 1.5ex plus 4bp minus 2bp
+\fi
+% 11pt
+\ifx\CLASSOPTIONpt\@IEEEptsizeeleven
+\def\normalsize{\@setfontsize{\normalsize}{11bp}{13.2bp}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{13.2bp}
+\normalsize
+\abovedisplayskip 1.5ex plus 5bp minus 3bp
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0pt plus 5bp
+\belowdisplayshortskip 1.5ex plus 5bp minus 3bp
+\fi
+% 12pt
+\ifx\CLASSOPTIONpt\@IEEEptsizetwelve
+\def\normalsize{\@setfontsize{\normalsize}{12bp}{14.4bp}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{14.4bp}
+\normalsize
+\abovedisplayskip 1.5ex plus 6bp minus 4bp
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0pt plus 6bp
+\belowdisplayshortskip 1.5ex plus 6bp minus 4bp
+\fi
+% 
+% compsoc nonconferences
+\else
+% 9pt
+\ifx\CLASSOPTIONpt\@IEEEptsizenine
+\def\normalsize{\@setfontsize{\normalsize}{9bp}{10.8bp}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{10.8bp}
+\normalsize
+\abovedisplayskip 1.5ex plus 3bp minus 1bp
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0bp plus 3bp
+\belowdisplayshortskip 1.5ex plus 3bp minus 1bp
+\fi
+% 10pt
+\ifx\CLASSOPTIONpt\@IEEEptsizeten
+% the official spec is 9.5bp with 11.4bp leading for 10pt,
+% but measurements of proofs suggest upto 11.723bp leading
+% here we'll use 11.54bp which gives 61 lines per column
+% with the standard compsoc margins
+\def\normalsize{\@setfontsize{\normalsize}{9.5bp}{11.54bp}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{11.54bp}
+\normalsize
+\abovedisplayskip 1.5ex plus 4bp minus 2bp
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0pt plus 4bp
+\belowdisplayshortskip 1.5ex plus 4bp minus 2bp
+\fi
+% 11pt
+\ifx\CLASSOPTIONpt\@IEEEptsizeeleven
+\def\normalsize{\@setfontsize{\normalsize}{11bp}{13.2bp}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{13.2bp}
+\normalsize
+\abovedisplayskip 1.5ex plus 5bp minus 3bp
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0pt plus 5bp
+\belowdisplayshortskip 1.5ex plus 5bp minus 3bp
+\fi
+% 12pt
+\ifx\CLASSOPTIONpt\@IEEEptsizetwelve
+\def\normalsize{\@setfontsize{\normalsize}{12bp}{14.4bp}}
+\setlength{\@IEEEnormalsizeunitybaselineskip}{14.4bp}
+\normalsize
+\abovedisplayskip 1.5ex plus 6bp minus 4bp
+\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip 0pt plus 6bp
+\belowdisplayshortskip 1.5ex plus 6bp minus 4bp
+\fi
+\fi\fi
+
+
+
+
+% V1.6 The Computer Modern Fonts will issue a substitution warning for
+% 24pt titles (24.88pt is used instead, but the default and correct
+% Times font will scale exactly as needed) increase the substitution
+% tolerance to turn off this warning.
+% 
+% V1.8a, the compsoc bp font sizes can also cause bogus font substitution
+% warnings with footnote or scriptsize math and the $\bullet$ itemized
+% list of \IEEEcompsocitemizethanks. So, increase this to 1.5pt or more.
+\def\fontsubfuzz{1.7bp}
+
+
+% warn the user in case they forget to use the 9pt option with
+% technote
+\ifCLASSOPTIONtechnote%
+ \ifx\CLASSOPTIONpt\@IEEEptsizenine\else%
+  \typeout{** ATTENTION: Technotes are normally 9pt documents.}%
+ \fi%
+\fi
+
+
+% V1.7
+% Improved \textunderscore to provide a much better fake _ when used with
+% OT1 encoding. Under OT1, detect use of pcr or cmtt \ttfamily and use
+% available true _ glyph for those two typewriter fonts.
+\def\@IEEEstringptm{ptm} % Times Roman family
+\def\@IEEEstringppl{ppl} % Palatino Roman family
+\def\@IEEEstringphv{phv} % Helvetica Sans Serif family
+\def\@IEEEstringpcr{pcr} % Courier typewriter family
+\def\@IEEEstringcmtt{cmtt} % Computer Modern typewriter family
+\DeclareTextCommandDefault{\textunderscore}{\leavevmode
+\ifx\f@family\@IEEEstringpcr\string_\else
+\ifx\f@family\@IEEEstringcmtt\string_\else
+\ifx\f@family\@IEEEstringptm\kern 0em\vbox{\hrule\@width 0.5em\@height 0.5pt\kern -0.3ex}\else
+\ifx\f@family\@IEEEstringppl\kern 0em\vbox{\hrule\@width 0.5em\@height 0.5pt\kern -0.3ex}\else
+\ifx\f@family\@IEEEstringphv\kern -0.03em\vbox{\hrule\@width 0.62em\@height 0.52pt\kern -0.33ex}\kern -0.03em\else
+\kern 0.09em\vbox{\hrule\@width 0.6em\@height 0.44pt\kern -0.63pt\kern -0.42ex}\kern 0.09em\fi\fi\fi\fi\fi\relax}
+
+
+
+
+% set the default \baselinestretch
+\def\baselinestretch{1}
+\ifCLASSOPTIONdraftcls
+  \def\baselinestretch{1.5}% default baselinestretch for draft modes
+\fi 
+
+
+% process CLASSINPUT baselinestretch
+\ifx\CLASSINPUTbaselinestretch\@IEEEundefined
+\else
+  \edef\baselinestretch{\CLASSINPUTbaselinestretch} % user CLASSINPUT override
+  \typeout{** ATTENTION: Overriding \string\baselinestretch\space to
+           \baselinestretch\space via \string\CLASSINPUT.}
+\fi
+
+\small\normalsize % make \baselinestretch take affect
+
+
+
+
+% store the normalsize baselineskip
+\newdimen\CLASSINFOnormalsizebaselineskip
+\CLASSINFOnormalsizebaselineskip=\baselineskip\relax
+% and the normalsize unity (baselinestretch=1) baselineskip
+% we could save a register by giving the user access to
+% \@IEEEnormalsizeunitybaselineskip. However, let's protect
+% its read only internal status
+\newdimen\CLASSINFOnormalsizeunitybaselineskip
+\CLASSINFOnormalsizeunitybaselineskip=\@IEEEnormalsizeunitybaselineskip\relax
+% store the nominal value of jot
+\newdimen\IEEEnormaljot
+\IEEEnormaljot=0.25\baselineskip\relax
+
+% set \jot
+\jot=\IEEEnormaljot\relax
+
+
+
+
+% V1.6, we are now going to fine tune the interword spacing
+% The default interword glue for Times under TeX appears to use a
+% nominal interword spacing of 25% (relative to the font size, i.e., 1em)
+% a maximum of 40% and a minimum of 19%.
+% For example, 10pt text uses an interword glue of:
+% 
+% 2.5pt plus 1.49998pt minus 0.59998pt
+% 
+% However, the IEEE allows for a more generous range which reduces the need
+% for hyphenation, especially for two column text. Furthermore, the IEEE
+% tends to use a little bit more nominal space between the words.
+% The IEEE's interword spacing percentages appear to be:
+% 35% nominal
+% 23% minimum
+% 50% maximum
+% (They may even be using a tad more for the largest fonts such as 24pt.)
+% 
+% for bold text, the IEEE increases the spacing a little more:
+% 37.5% nominal
+% 23% minimum
+% 55% maximum
+
+% here are the interword spacing ratios we'll use
+% for medium (normal weight)
+\def\@IEEEinterspaceratioM{0.35}
+\def\@IEEEinterspaceMINratioM{0.23}
+\def\@IEEEinterspaceMAXratioM{0.50}
+
+% for bold
+\def\@IEEEinterspaceratioB{0.375}
+\def\@IEEEinterspaceMINratioB{0.23}
+\def\@IEEEinterspaceMAXratioB{0.55}
+
+
+% compsoc nonconference papers use Palatino,
+% tweak settings to better match the proofs
+\ifCLASSOPTIONcompsoc
+\ifCLASSOPTIONconference\else
+% for medium (normal weight)
+\def\@IEEEinterspaceratioM{0.28}
+\def\@IEEEinterspaceMINratioM{0.21}
+\def\@IEEEinterspaceMAXratioM{0.47}
+% for bold
+\def\@IEEEinterspaceratioB{0.305}
+\def\@IEEEinterspaceMINratioB{0.21}
+\def\@IEEEinterspaceMAXratioB{0.52}
+\fi\fi
+
+
+% command to revise the interword spacing for the current font under TeX:
+% \fontdimen2 = nominal interword space
+% \fontdimen3 = interword stretch
+% \fontdimen4 = interword shrink
+% since all changes to the \fontdimen are global, we can enclose these commands
+% in braces to confine any font attribute or length changes
+\def\@@@IEEEsetfontdimens#1#2#3{{%
+\setlength{\@IEEEtrantmpdimenB}{\f@size pt}% grab the font size in pt, could use 1em instead.
+\setlength{\@IEEEtrantmpdimenA}{#1\@IEEEtrantmpdimenB}%
+\fontdimen2\font=\@IEEEtrantmpdimenA\relax
+\addtolength{\@IEEEtrantmpdimenA}{-#2\@IEEEtrantmpdimenB}%
+\fontdimen3\font=-\@IEEEtrantmpdimenA\relax
+\setlength{\@IEEEtrantmpdimenA}{#1\@IEEEtrantmpdimenB}%
+\addtolength{\@IEEEtrantmpdimenA}{-#3\@IEEEtrantmpdimenB}%
+\fontdimen4\font=\@IEEEtrantmpdimenA\relax}}
+
+% revise the interword spacing for each font weight
+\def\@@IEEEsetfontdimens{{%
+\mdseries
+\@@@IEEEsetfontdimens{\@IEEEinterspaceratioM}{\@IEEEinterspaceMAXratioM}{\@IEEEinterspaceMINratioM}%
+\bfseries
+\@@@IEEEsetfontdimens{\@IEEEinterspaceratioB}{\@IEEEinterspaceMAXratioB}{\@IEEEinterspaceMINratioB}%
+}}
+
+% revise the interword spacing for each font shape
+% \slshape is not often used for IEEE work and is not altered here. The \scshape caps are
+% already a tad too large in the free LaTeX fonts (as compared to what the IEEE uses) so we
+% won't alter these either.
+\def\@IEEEsetfontdimens{{%
+\normalfont
+\@@IEEEsetfontdimens
+\normalfont\itshape
+\@@IEEEsetfontdimens
+}}
+
+% command to revise the interword spacing for each font size (and shape
+% and weight). Only the \rmfamily is done here as \ttfamily uses a 
+% fixed spacing and \sffamily is not used as the main text of IEEE papers.
+\def\@IEEEtunefonts{{\selectfont\rmfamily
+\tiny\@IEEEsetfontdimens
+\scriptsize\@IEEEsetfontdimens
+\footnotesize\@IEEEsetfontdimens
+\small\@IEEEsetfontdimens
+\normalsize\@IEEEsetfontdimens
+\sublargesize\@IEEEsetfontdimens
+\large\@IEEEsetfontdimens
+\LARGE\@IEEEsetfontdimens
+\huge\@IEEEsetfontdimens
+\Huge\@IEEEsetfontdimens}}
+
+% if the nofonttune class option is not given, revise the interword spacing
+% now - in case IEEEtran makes any default length measurements, and make
+% sure all the default fonts are loaded
+\ifCLASSOPTIONnofonttune\else
+\@IEEEtunefonts
+\fi
+
+% and again at the start of the document in case the user loaded different fonts
+\AtBeginDocument{\ifCLASSOPTIONnofonttune\else\@IEEEtunefonts\fi}
+
+
+
+
+
+% -- V1.8a page setup commands --
+
+% The default sample text for calculating margins
+% Note that IEEE publications use \scriptsize for headers and footers.
+\def\IEEEdefaultsampletext{\normalfont\normalsize gT}
+\def\IEEEdefaultheadersampletext{\normalfont\scriptsize T}% IEEE headers default to uppercase
+\def\IEEEdefaultfootersampletext{\normalfont\scriptsize gT}
+
+
+
+% usage: \IEEEsettextwidth{inner margin}{outer margin}
+% Sets \textwidth to allow the specified inner and outer margins
+% for the current \paperwidth.
+\def\IEEEsettextwidth#1#2{\@IEEEtrantmpdimenA\paperwidth
+\@IEEEtrantmpdimenB#1\relax
+\advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB
+\@IEEEtrantmpdimenB#2\relax
+\advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB
+\textwidth\@IEEEtrantmpdimenA}
+
+
+
+% usage: \IEEEsetsidemargin{mode: i, o, c, a}{margin/offset}
+% Sets \oddsidemargin and \evensidemargin to yield the specified margin
+% of the given mode.
+% The available modes are:
+% i = inner margin
+% o = outer margin
+% c = centered, with the given offset
+% a = adjust the margins using the given offset
+% For the offsets, positive values increase the inner margin.
+% \textwidth should be set properly for the given margins before calling this
+% function.
+\def\IEEEsetsidemargin#1#2{\@IEEEtrantmpdimenA #2\relax
+\@IEEEextracttoken{#1}\relax
+% check for mode errors
+\ifx\@IEEEextractedtokenmacro\@empty
+  \@IEEEclspkgerror{Empty mode type in \string\IEEEsetsidemargin\space (line \the\inputlineno).\MessageBreak
+  Defaulting to `i'}{Valid modes for \string\IEEEsetsidemargin\space are: i, o, c and a.}\relax
+  \let\@IEEEextractedtoken=i\relax
+  \def\@IEEEextractedtokenmacro{i}\relax
+\else
+  \ifx\@IEEEextractedtokensdiscarded\@empty\else
+    \typeout{** WARNING: \string\IEEEsetsidemargin\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+  \fi
+\fi
+% handle each mode
+\if\@IEEEextractedtoken a\relax
+  \advance\oddsidemargin by \@IEEEtrantmpdimenA\relax
+\else
+\if\@IEEEextractedtoken c\relax
+  \oddsidemargin\paperwidth
+  \advance\oddsidemargin by -\textwidth
+  \divide\oddsidemargin by 2\relax
+  \advance\oddsidemargin by -1in\relax
+  \advance\oddsidemargin by \@IEEEtrantmpdimenA\relax
+\else
+\if\@IEEEextractedtoken o\relax
+  \oddsidemargin\paperwidth
+  \advance\oddsidemargin by -\textwidth
+  \advance\oddsidemargin by -\@IEEEtrantmpdimenA
+  \advance\oddsidemargin by -1in\relax
+\else
+  \if\@IEEEextractedtoken i\relax
+  \else
+    \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEsetsidemargin\space (line \the\inputlineno).\MessageBreak
+    Defaulting to `i'}%
+    {Valid modes for \string\IEEEsetsidemargin\space are: i, o, c  and a.}%
+  \fi
+  \oddsidemargin\@IEEEtrantmpdimenA
+  \advance\oddsidemargin by -1in\relax
+\fi\fi\fi
+% odd and even side margins both mean "inner" for single sided pages
+\evensidemargin\oddsidemargin
+% but are mirrors of each other when twosided is in effect
+\if@twoside
+  \evensidemargin\paperwidth
+  \advance\evensidemargin by -\textwidth
+  \advance\evensidemargin by -\oddsidemargin
+  % have to compensate for both the builtin 1in LaTex offset
+  % and the fact we already subtracted this offset from \oddsidemargin
+  \advance\evensidemargin -2in\relax
+\fi}
+
+
+
+% usage: \IEEEsettextheight[sample text]{top text margin}{bottom text margin}
+% Sets \textheight based on the specified top margin and bottom margin.
+% Takes into consideration \paperheight, \topskip, and (by default) the
+% the actual height and depth of the \IEEEdefaultsampletext text. 
+\def\IEEEsettextheight{\@ifnextchar [{\@IEEEsettextheight}{\@IEEEsettextheight[\IEEEdefaultsampletext]}}
+\def\@IEEEsettextheight[#1]#2#3{\textheight\paperheight\relax
+  \@IEEEtrantmpdimenA #2\relax
+  \advance \textheight by -\@IEEEtrantmpdimenA% subtract top margin
+  \@IEEEtrantmpdimenA #3\relax
+  \advance \textheight by -\@IEEEtrantmpdimenA% subtract bottom margin
+  \advance \textheight by \topskip% add \topskip
+  % subtract off everything above the top, and below the bottom, baselines
+  \settoheight{\@IEEEtrantmpdimenA}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance \textheight by -\@IEEEtrantmpdimenA
+  \settodepth{\@IEEEtrantmpdimenA}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance \textheight by -\@IEEEtrantmpdimenA}
+
+
+
+\newdimen\IEEEquantizedlength
+\IEEEquantizedlength 0sp\relax
+\newdimen\IEEEquantizedlengthdiff
+\IEEEquantizedlengthdiff 0sp\relax
+\def\IEEEquantizedlengthint{0}
+
+% usage: \IEEEquantizelength{mode: d, c, i}{base unit}{length}
+% Sets the length \IEEEquantizedlength to be an integer multiple of the given
+% (nonzero) base unit such that \IEEEquantizedlength approximates the given
+% length.
+% \IEEEquantizedlengthdiff is a length equal to the difference between the
+% \IEEEquantizedlength and the given length.
+% \IEEEquantizedlengthint is a macro containing the integer number of base units
+% in \IEEEquantizedlength.
+% i.e., \IEEEquantizedlength =  \IEEEquantizedlengthint * base unit
+% The mode determines how \IEEEquantizedlength is quantized:
+% d = always decrease (always round down \IEEEquantizeint)
+% c = use the closest match
+% i = always increase (always round up \IEEEquantizeint)
+% In anycase, if the given length is already quantized,
+% \IEEEquantizedlengthdiff will be set to zero.
+\def\IEEEquantizelength#1#2#3{\begingroup
+% work in isolation so as not to externally disturb the \@IEEEtrantmp
+% variables
+% load the argument values indirectly via \IEEEquantizedlengthdiff
+% in case the user refers to our \@IEEEtrantmpdimenX, \IEEEquantizedlength,
+% etc. in the arguments. we also will work with these as counters,
+% i.e., in sp units
+% A has the base unit
+\IEEEquantizedlengthdiff #2\relax\relax\relax\relax
+\@IEEEtrantmpcountA\IEEEquantizedlengthdiff
+% B has the input length
+\IEEEquantizedlengthdiff #3\relax\relax\relax\relax
+\@IEEEtrantmpcountB\IEEEquantizedlengthdiff
+\@IEEEtrantmpdimenA\the\@IEEEtrantmpcountA sp\relax
+\@IEEEtrantmpdimenB\the\@IEEEtrantmpcountB sp\relax
+% \@IEEEtrantmpcountC will have the quantized int
+% \IEEEquantizedlength will have the quantized length
+% \@IEEEtrantmpdimenC will have the quantized diff
+% initialize them to zero as this is what will be
+% exported if an error occurs
+\@IEEEtrantmpcountC 0\relax
+\IEEEquantizedlength 0sp\relax
+\@IEEEtrantmpdimenC 0sp\relax
+% extract mode
+\@IEEEextracttoken{#1}\relax
+% check for mode errors
+\ifx\@IEEEextractedtokenmacro\@empty
+  \@IEEEclspkgerror{Empty mode type in \string\IEEEquantizelength\space (line \the\inputlineno).\MessageBreak
+  Defaulting to `d'}{Valid modes for \string\IEEEquantizelength\space are: d, c and i.}\relax
+  \let\@IEEEextractedtoken=d\relax
+  \def\@IEEEextractedtokenmacro{d}\relax
+\else
+  \ifx\@IEEEextractedtokensdiscarded\@empty\else
+    \typeout{** WARNING: \string\IEEEquantizelength\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+  \fi
+\fi
+% check for base unit is zero error
+\ifnum\@IEEEtrantmpcountA=0\relax
+\@IEEEclspkgerror{Base unit is zero in \string\IEEEquantizelength\space (line \the\inputlineno).\MessageBreak
+  \string\IEEEquantizedlength\space and \string\IEEEquantizedlengthdiff\space are set to zero}{Division by zero is not allowed.}\relax
+\else% base unit is nonzero
+  % \@IEEEtrantmpcountC carries the number of integer units
+  % in the quantized length (integer length \ base)
+  \@IEEEtrantmpcountC\@IEEEtrantmpcountB\relax
+  \divide\@IEEEtrantmpcountC by \@IEEEtrantmpcountA\relax
+  % \IEEEquantizedlength has the (rounded down) quantized length
+  % = base * int
+  \IEEEquantizedlength\@IEEEtrantmpdimenA\relax
+  \multiply\IEEEquantizedlength by \@IEEEtrantmpcountC\relax
+  % \@IEEEtrantmpdimenC has the difference
+  % = quantized length - length
+  \@IEEEtrantmpdimenC\IEEEquantizedlength\relax
+  \advance\@IEEEtrantmpdimenC by -\@IEEEtrantmpdimenB\relax
+  % trap special case of length being already quantized
+  % to avoid a roundup under i option
+  \ifdim\@IEEEtrantmpdimenC=0sp\relax
+  \else % length not is already quantized
+    % set dimenA to carry the upper quantized (absolute value) difference:
+    % quantizedlength + base - length
+    \advance\@IEEEtrantmpdimenA by \IEEEquantizedlength\relax
+    \advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB\relax
+    % set dimenB to carry the lower quantized (absolute value) difference:
+    % length - quantizedlength
+    \advance\@IEEEtrantmpdimenB by -\IEEEquantizedlength\relax
+    % handle each mode
+    \if\@IEEEextractedtoken c\relax
+      % compare upper and lower amounts, select upper if lower > upper
+      \ifdim\@IEEEtrantmpdimenB>\@IEEEtrantmpdimenA\relax
+        % use upper
+        \advance\IEEEquantizedlength by \the\@IEEEtrantmpcountA sp\relax
+        \advance\@IEEEtrantmpcountC by 1\relax
+        \@IEEEtrantmpdimenC\@IEEEtrantmpdimenA
+      \else% <=. uselower
+        % no need to do anything for lower, use output values already setup
+      \fi
+    \else% not mode c
+    \if\@IEEEextractedtoken i\relax
+      % always round up under i mode
+      \advance\IEEEquantizedlength by \the\@IEEEtrantmpcountA sp\relax
+      \advance\@IEEEtrantmpcountC by 1\relax
+      \@IEEEtrantmpdimenC\@IEEEtrantmpdimenA
+    \else
+      \if\@IEEEextractedtoken d\relax
+      \else
+        \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEquantizelength\space (line \the\inputlineno).\MessageBreak
+        Defaulting to `d'}%
+        {Valid modes for \string\IEEEquantizelength\space are: d, c, and i.}\relax
+      \fi % if d
+      % no need to do anything for d, use output values already setup
+    \fi\fi % if i, c
+  \fi % if length is already quantized
+\fi% if base unit is zero
+% globally assign the results to macros we use here to escape the enclosing
+% group without needing to call \global on any of the \@IEEEtrantmp variables.
+% \@IEEEtrantmpcountC has the quantized int
+% \IEEEquantizedlength has the quantized length
+% \@IEEEtrantmpdimenC has the quantized diff
+\xdef\@IEEEquantizedlengthintmacro{\the\@IEEEtrantmpcountC}\relax
+\@IEEEtrantmpcountC\IEEEquantizedlength\relax
+\xdef\@IEEEquantizedlengthmacro{\the\@IEEEtrantmpcountC}\relax
+\@IEEEtrantmpcountC\@IEEEtrantmpdimenC\relax
+\xdef\@IEEEquantizedlengthdiffmacro{\the\@IEEEtrantmpcountC}\relax
+\endgroup
+% locally assign the outputs here from the macros
+\expandafter\IEEEquantizedlength\@IEEEquantizedlengthmacro sp\relax
+\expandafter\IEEEquantizedlengthdiff\@IEEEquantizedlengthdiffmacro sp\relax
+\edef\IEEEquantizedlengthint{\@IEEEquantizedlengthintmacro}\relax}
+
+
+
+\newdimen\IEEEquantizedtextheightdiff
+\IEEEquantizedtextheightdiff 0sp\relax
+
+% usage: \IEEEquantizetextheight[base unit]{mode: d, c, i}
+% Sets \textheight to be an integer multiple of the current \baselineskip
+% (or the optionally specified base unit) plus the first (\topskip) line.
+% \IEEEquantizedtextheightdiff is a length equal to the difference between
+% the new quantized and original \textheight.
+% \IEEEquantizedtextheightlpc is a macro containing the integer number of
+% lines per column under the quantized \textheight. i.e.,
+% \textheight = \IEEEquantizedtextheightlpc * \baselineskip + \topskip
+% The mode determines how \textheight is quantized:
+% d = always decrease (always round down the number of lines per column)
+% c = use the closest match
+% i = always increase (always round up the number of lines per column)
+% In anycase, if \textheight is already quantized, it will remain unchanged,
+% and \IEEEquantizedtextheightdiff will be set to zero.
+% Depends on: \IEEEquantizelength
+\def\IEEEquantizetextheight{\@ifnextchar [{\@IEEEquantizetextheight}{\@IEEEquantizetextheight[\baselineskip]}}
+\def\@IEEEquantizetextheight[#1]#2{\begingroup
+% use our \IEEEquantizedtextheightdiff as a scratch pad
+% we need to subtract off \topskip before quantization
+\IEEEquantizedtextheightdiff\textheight
+\advance\IEEEquantizedtextheightdiff by -\topskip\relax
+\IEEEquantizelength{#2}{#1}{\IEEEquantizedtextheightdiff}
+% add back \topskip line
+\advance\IEEEquantizedlength by \topskip
+\@IEEEtrantmpcountC\IEEEquantizedlengthint\relax
+\advance\@IEEEtrantmpcountC by 1\relax
+% globally assign the results to macros we use here to escape the enclosing
+% group without needing to call \global on any of the \@IEEEtrantmp variables.
+\xdef\@IEEEquantizedtextheightlpcmacro{\the\@IEEEtrantmpcountC}\relax
+\@IEEEtrantmpcountC\IEEEquantizedlength\relax
+\xdef\@IEEEquantizedtextheightmacro{\the\@IEEEtrantmpcountC}\relax
+\@IEEEtrantmpcountC\IEEEquantizedlengthdiff\relax
+\xdef\@IEEEquantizedtextheightdiffmacro{\the\@IEEEtrantmpcountC}\relax
+\endgroup
+% locally assign the outputs here from the macros
+\textheight\@IEEEquantizedtextheightmacro sp\relax
+\IEEEquantizedtextheightdiff\@IEEEquantizedtextheightdiffmacro sp\relax
+\edef\IEEEquantizedtextheightlpc{\@IEEEquantizedtextheightlpcmacro}}
+
+
+
+% usage: \IEEEsettopmargin[sample text]{mode: t, b, c, a, q}{margin/offset}
+% Sets \topmargin based on the specified vertical margin.
+% Takes into consideration the base 1in offset, \headheight, \headsep,
+% \topskip, and (by default) the the actual height (or, for the bottom, depth)
+% of the \IEEEdefaultsampletext text.
+% The available modes are:
+% t = top margin
+% b = bottom margin
+% c = vertically centered, with the given offset
+% a = adjust the vertical margins using the given offset
+% q = adjust the margins using \IEEEquantizedtextheightdiff and the given offset
+% For the offsets, positive values increase the top margin.
+% \headheight, \headsep, \topskip and \textheight should be set properly for the
+% given margins before calling this function.
+\def\IEEEsettopmargin{\@ifnextchar [{\@IEEEsettopmargin}{\@IEEEsettopmargin[\IEEEdefaultsampletext]}}
+\def\@IEEEsettopmargin[#1]#2#3{\@IEEEtrantmpdimenA #3\relax
+\@IEEEextracttoken{#2}\relax
+% check for mode errors
+\ifx\@IEEEextractedtokenmacro\@empty
+  \@IEEEclspkgerror{Empty mode type in \string\IEEEsettopmargin\space (line \the\inputlineno).\MessageBreak
+  Defaulting to `t'}{Valid modes for \string\IEEEsettopmargin\space are: t, b, c, a and q.}\relax
+  \let\@IEEEextractedtoken=t\relax
+  \def\@IEEEextractedtokenmacro{t}\relax
+\else
+  \ifx\@IEEEextractedtokensdiscarded\@empty\else
+    \typeout{** WARNING: \string\IEEEsettopmargin\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+  \fi
+\fi
+% handle each mode
+\if\@IEEEextractedtoken a\relax
+  \advance\topmargin by \@IEEEtrantmpdimenA\relax
+\else
+\if\@IEEEextractedtoken q\relax
+  % we need to adjust by half the \IEEEquantizedtextheightdiff value
+  \@IEEEtrantmpdimenB\IEEEquantizedtextheightdiff\relax
+  \divide\@IEEEtrantmpdimenB by 2\relax
+  % a positive \IEEEquantizedtextheightdiff means we need to reduce \topmargin
+  % because \textheight has been lenghtened
+  \advance\topmargin by -\@IEEEtrantmpdimenB\relax
+  \advance\topmargin by \@IEEEtrantmpdimenA\relax
+\else
+\if\@IEEEextractedtoken c\relax
+  \topmargin\paperheight
+  \advance\topmargin by -\textheight
+  % \textheight includes \topskip, but we should not count topskip whitespace here, backout
+  \advance \topmargin by \topskip
+  \settoheight{\@IEEEtrantmpdimenB}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance\topmargin by -\@IEEEtrantmpdimenB\relax
+  \settodepth{\@IEEEtrantmpdimenB}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance\topmargin by -\@IEEEtrantmpdimenB\relax
+  \divide\topmargin by 2\relax
+  \advance\topmargin by \@IEEEtrantmpdimenA\relax
+\else
+\if\@IEEEextractedtoken b\relax
+  \topmargin\paperheight
+  \advance\topmargin by -\textheight
+  % \textheight includes \topskip, but we should not count topskip whitespace here, backout
+  \advance \topmargin by \topskip
+  \settodepth{\@IEEEtrantmpdimenB}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance\topmargin by -\@IEEEtrantmpdimenB\relax
+  \advance\topmargin by -\@IEEEtrantmpdimenA\relax
+\else
+  \if\@IEEEextractedtoken t\relax
+  \else
+    \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEsettopmargin\space (line \the\inputlineno).\MessageBreak
+    Defaulting to `t'}%
+    {Valid modes for \string\IEEEsettopmargin\space are: t, b, c, a and q.}\relax
+  \fi
+  \topmargin\@IEEEtrantmpdimenA\relax
+  \settoheight{\@IEEEtrantmpdimenB}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance\topmargin by \@IEEEtrantmpdimenB\relax
+\fi\fi % if t, b, c
+% convert desired top margin into actual \topmargin
+% this is not done for the q or a modes because they are only adjustments
+\advance \topmargin by -\topskip
+\advance \topmargin by -1in
+\advance \topmargin by -\headheight
+\advance \topmargin by -\headsep
+\fi\fi % if q, a
+}
+
+
+
+% usage: \IEEEsetheadermargin[header sample][text sample]{mode: t, b, c, a}{margin/offset}
+% Differentially adjusts \topmargin and \headsep (such that their sum is unchanged)
+% based on the specified header margin.
+% Takes into consideration the base 1in offset, \headheight, \topskip, and (by default)
+% the actual height (or depth) of the \IEEEdefaultheadersampletext and
+% \IEEEdefaultsampletext text.
+% The available modes are:
+% t = top margin (top of the header text to the top of the page)
+% b = bottom margin (bottom of the header text to the top of the main text)
+% c = vertically centered between the main text and the top of the page,
+%     with the given offset
+% a = adjust the vertical position using the given offset
+% For the offsets, positive values move the header downward.
+% \headheight, \headsep, \topskip and \topmargin should be set properly before
+% calling this function.
+\def\IEEEsetheadermargin{\@ifnextchar [{\@IEEEsetheadermargin}{\@IEEEsetheadermargin[\IEEEdefaultheadersampletext]}}
+\def\@IEEEsetheadermargin[#1]{\@ifnextchar [{\@@IEEEsetheadermargin[#1]}{\@@IEEEsetheadermargin[#1][\IEEEdefaultsampletext]}}
+\def\@@IEEEsetheadermargin[#1][#2]#3#4{\@IEEEtrantmpdimenA #4\relax
+\@IEEEextracttoken{#3}\relax
+% check for mode errors
+\ifx\@IEEEextractedtokenmacro\@empty
+  \@IEEEclspkgerror{Empty mode type in \string\IEEEsetheadermargin\space (line \the\inputlineno).\MessageBreak
+  Defaulting to `t'}{Valid modes for \string\IEEEsetheadermargin\space are: t, b, c, and a.}\relax
+  \let\@IEEEextractedtoken=t\relax
+  \def\@IEEEextractedtokenmacro{t}\relax
+\else
+  \ifx\@IEEEextractedtokensdiscarded\@empty\else
+    \typeout{** WARNING: \string\IEEEsetheadermargin\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+  \fi
+\fi
+% handle each mode
+\if\@IEEEextractedtoken a\relax
+  % No need to do anything here and can pass through the adjustment
+  % value as is. The end adjustment of \topmargin and \headsep will
+  % do all that is needed
+\else
+\if\@IEEEextractedtoken c\relax
+  % get the bottom margin
+  \@IEEEtrantmpdimenB\headsep\relax
+  \settodepth{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+  \advance\@IEEEtrantmpdimenB by \topskip
+  \settoheight{\@IEEEtrantmpdimenC}{\begingroup #2\relax\relax\relax\endgroup}\relax
+  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+  % at this point \@IEEEtrantmpdimenB has the actual header bottom margin
+  % subtract from it the top header margin
+  \advance\@IEEEtrantmpdimenB -1in\relax % take into consideration the system 1in offset of the top margin
+  \advance\@IEEEtrantmpdimenB by -\topmargin
+  \advance\@IEEEtrantmpdimenB by -\headheight
+  \settoheight{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance\@IEEEtrantmpdimenB by \@IEEEtrantmpdimenC
+  % at this point \@IEEEtrantmpdimenB has the difference between the bottom and top margins
+  % we need to adjust by half this amount to center the header
+  \divide\@IEEEtrantmpdimenB by 2\relax
+  % and add to offset
+  \advance\@IEEEtrantmpdimenA by \@IEEEtrantmpdimenB
+\else
+\if\@IEEEextractedtoken b\relax
+  \@IEEEtrantmpdimenB\headsep\relax
+  \settodepth{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+  \advance\@IEEEtrantmpdimenB by \topskip
+  \settoheight{\@IEEEtrantmpdimenC}{\begingroup #2\relax\relax\relax\endgroup}\relax
+  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+  % at this point \@IEEEtrantmpdimenB has the actual header bottom margin
+  % get the difference between the actual and the desired
+  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenA
+  \@IEEEtrantmpdimenA\@IEEEtrantmpdimenB
+\else
+  \if\@IEEEextractedtoken t\relax
+  \else
+    \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEsetheadermargin\space (line \the\inputlineno).\MessageBreak
+    Defaulting to `t'}%
+    {Valid modes for \string\IEEEsetheadermargin\space are: t, b, c and a.}\relax
+  \fi
+  \@IEEEtrantmpdimenB 1in\relax % take into consideration the system 1in offset of the top margin
+  \advance\@IEEEtrantmpdimenB by \topmargin
+  \advance\@IEEEtrantmpdimenB by \headheight
+  \settoheight{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+  % at this point \@IEEEtrantmpdimenB has the actual header top margin
+  % get the difference between the desired and the actual
+  \advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB
+\fi\fi % if t, b, c
+\fi % if a
+% advance \topmargin by the needed amount and reduce \headsep by the same
+% so as not to disturb the location of the main text
+\advance\topmargin by \@IEEEtrantmpdimenA\relax
+\advance\headsep by -\@IEEEtrantmpdimenA\relax
+}
+
+
+
+% usage: \IEEEsetfootermargin[footer sample][text sample]{mode: t, b, c, a}{margin/offset}
+% Adjusts \footskip based on the specified footer margin.
+% Takes into consideration the base 1in offset, \paperheight, \headheight,
+% \headsep, \textheight and (by default) the actual height (or depth) of the
+% \IEEEdefaultfootersampletext and \IEEEdefaultsampletext text.
+% The available modes are:
+% t = top margin (top of the footer text to the bottom of the main text)
+% b = bottom margin (bottom of the footer text to the bottom of page)
+% c = vertically centered between the main text and the bottom of the page,
+%     with the given offset
+% a = adjust the vertical position using the given offset
+% For the offsets, positive values move the footer downward.
+% \headheight, \headsep, \topskip, \topmargin, and \textheight should be set
+% properly before calling this function.
+\def\IEEEsetfootermargin{\@ifnextchar [{\@IEEEsetfootermargin}{\@IEEEsetfootermargin[\IEEEdefaultfootersampletext]}}
+\def\@IEEEsetfootermargin[#1]{\@ifnextchar [{\@@IEEEsetfootermargin[#1]}{\@@IEEEsetfootermargin[#1][\IEEEdefaultsampletext]}}
+\def\@@IEEEsetfootermargin[#1][#2]#3#4{\@IEEEtrantmpdimenA #4\relax
+\@IEEEextracttoken{#3}\relax
+% check for mode errors
+\ifx\@IEEEextractedtokenmacro\@empty
+  \@IEEEclspkgerror{Empty mode type in \string\IEEEsetfootermargin\space (line \the\inputlineno).\MessageBreak
+  Defaulting to `t'}{Valid modes for \string\IEEEsetfootermargin\space are: t, b, c, and a.}\relax
+  \let\@IEEEextractedtoken=t\relax
+  \def\@IEEEextractedtokenmacro{t}\relax
+\else
+  \ifx\@IEEEextractedtokensdiscarded\@empty\else
+    \typeout{** WARNING: \string\IEEEsetfootermargin\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+  \fi
+\fi
+% handle each mode
+\if\@IEEEextractedtoken a\relax
+  % No need to do anything here and can pass through the adjustment
+  % value as is. The end adjustment of \footskip will do all that
+  % is needed
+\else
+\if\@IEEEextractedtoken c\relax
+  % calculate the bottom margin
+  \@IEEEtrantmpdimenB 1in\relax % system 1in offset
+  \advance\@IEEEtrantmpdimenB\topmargin\relax
+  \advance\@IEEEtrantmpdimenB\headheight\relax
+  \advance\@IEEEtrantmpdimenB\headsep\relax
+  \advance\@IEEEtrantmpdimenB\textheight\relax
+  \advance\@IEEEtrantmpdimenB\footskip\relax
+  \settodepth{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance\@IEEEtrantmpdimenC by \@IEEEtrantmpdimenB
+  \@IEEEtrantmpdimenB\paperheight
+  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+  % at this point \@IEEEtrantmpdimenB has the actual footer bottom margin
+  % now subtract off the footer top margin
+  \advance\@IEEEtrantmpdimenB -\footskip\relax
+  \settodepth{\@IEEEtrantmpdimenC}{\begingroup #2\relax\relax\relax\endgroup}\relax
+  \advance\@IEEEtrantmpdimenB by \@IEEEtrantmpdimenC
+  \settoheight{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance\@IEEEtrantmpdimenB by \@IEEEtrantmpdimenC
+  % at this point \@IEEEtrantmpdimenB has the difference between the bottom
+  % and top footer margins
+  % our adjustment must be half this value to center the footer
+  \divide\@IEEEtrantmpdimenB by 2\relax
+  % add to the offset
+  \advance\@IEEEtrantmpdimenA by \@IEEEtrantmpdimenB
+\else
+\if\@IEEEextractedtoken b\relax
+  % calculate the bottom margin
+  \@IEEEtrantmpdimenB 1in\relax % system 1in offset
+  \advance\@IEEEtrantmpdimenB\topmargin\relax
+  \advance\@IEEEtrantmpdimenB\headheight\relax
+  \advance\@IEEEtrantmpdimenB\headsep\relax
+  \advance\@IEEEtrantmpdimenB\textheight\relax
+  \advance\@IEEEtrantmpdimenB\footskip\relax
+  \settodepth{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance\@IEEEtrantmpdimenC by \@IEEEtrantmpdimenB
+  \@IEEEtrantmpdimenB\paperheight
+  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+  % at this point \@IEEEtrantmpdimenB has the actual footer bottom margin
+  % get the difference between the actual and the desired
+  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenA
+  \@IEEEtrantmpdimenA\@IEEEtrantmpdimenB
+\else
+  \if\@IEEEextractedtoken t\relax
+  \else
+    \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEsetfootermargin\space (line \the\inputlineno).\MessageBreak
+    Defaulting to `t'}%
+    {Valid modes for \string\IEEEsetfootermargin\space are: t, b, c and a.}\relax
+  \fi
+  \@IEEEtrantmpdimenB\footskip\relax
+  \settodepth{\@IEEEtrantmpdimenC}{\begingroup #2\relax\relax\relax\endgroup}\relax
+  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+  \settoheight{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+  % at this point \@IEEEtrantmpdimenB has the actual footer top margin
+  % get the difference between the desired and the actual
+  \advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB
+\fi\fi % if t, b, c
+\fi % if a
+% advance \footskip by the needed amount
+\advance\footskip by \@IEEEtrantmpdimenA\relax
+}
+
+% -- End V1.8a page setup commands --
+
+
+
+
+
+% V1.6
+% LaTeX is a little to quick to use hyphenations
+% So, we increase the penalty for their use and raise
+% the badness level that triggers an underfull hbox
+% warning. The author may still have to tweak things,
+% but the appearance will be much better "right out
+% of the box" than that under V1.5 and prior.
+% TeX default is 50
+\hyphenpenalty=750
+\ifCLASSOPTIONcompsoc
+\hyphenpenalty 500
+\fi
+% If we didn't adjust the interword spacing, 2200 might be better.
+% The TeX default is 1000
+\hbadness=1350
+% The IEEE does not use extra spacing after punctuation
+\frenchspacing
+
+% V1.7 increase this a tad to discourage equation breaks
+\binoppenalty=1000 % default 700
+\relpenalty=800     % default 500
+
+% v1.8a increase these to discourage widows and orphans
+\clubpenalty=1000          % default 150
+\widowpenalty=1000         % default 150
+\displaywidowpenalty=1000 % default  50
+
+
+% margin note stuff
+\marginparsep      10pt
+\marginparwidth    20pt
+\marginparpush     25pt
+
+
+% if things get too close, go ahead and let them touch
+\lineskip            0pt
+\normallineskip      0pt
+\lineskiplimit       0pt
+\normallineskiplimit 0pt
+
+% The distance from the lower edge of the text body to the
+% footline
+\footskip 0.4in
+
+% normally zero, should be relative to font height.
+% put in a little rubber to help stop some bad breaks (underfull vboxes)
+\parskip 0ex plus 0.2ex minus 0.1ex
+
+\parindent    1.0em
+\ifCLASSOPTIONcompsoc
+  \parindent    1.5em
+\fi
+
+\headheight   12pt
+\headsep      18pt
+% use the normal font baselineskip
+% so that \topskip is unaffected by changes in \baselinestretch
+\topskip=\@IEEEnormalsizeunitybaselineskip
+
+
+% V1.8 \maxdepth defaults to 4pt, but should be font size dependent
+\maxdepth=0.5\@IEEEnormalsizeunitybaselineskip
+\textheight       58pc  % 9.63in, 696pt
+
+% set the default top margin to 58pt
+% which results in a \topmargin of -49.59pt for 10pt documents
+\IEEEsettopmargin{t}{58pt}
+% tweak textheight to a perfect integer number of lines/column.
+% standard is: 9pt/63 lpc; 10pt/58 lpc; 11pt/52 lpc; 12pt/50 lpc
+\IEEEquantizetextheight{c}
+% tweak top margin so that the error is shared equally at the top and bottom
+\IEEEsettopmargin{q}{0sp}
+
+
+\columnsep         1pc
+\textwidth        43pc   % 2 x 21pc + 1pc = 43pc
+
+% set the default side margins to center the text
+\IEEEsetsidemargin{c}{0pt}
+
+
+% adjust margins for default conference mode
+\ifCLASSOPTIONconference
+ \textheight        9.25in % The standard for conferences (668.4975pt)
+ \IEEEsettopmargin{t}{0.75in}
+ % tweak textheight to a perfect integer number of lines/page.
+ % standard is: 9pt/61 lpc; 10pt/56 lpc; 11pt/50 lpc; 12pt/48 lpc
+ \IEEEquantizetextheight{c}
+ % tweak top margin so that the error is shared equally at the top and bottom
+ \IEEEsettopmargin{q}{0sp}
+\fi
+
+
+% compsoc text sizes, margins and spacings
+\ifCLASSOPTIONcompsoc
+ \columnsep         12bp
+ % CS specs for \textwdith are 6.875in
+ % \textwidth         6.875in
+ % however, measurements from proofs show they are using 3.5in columns
+ \textwidth         7in
+ \advance\textwidth by \columnsep
+ % set the side margins to center the text
+ \IEEEsetsidemargin{c}{0pt}
+ % top/bottom margins to center
+ % could just set \textheight to 9.75in for all the different paper sizes
+ % and then quantize, but we'll do it the long way here to allow for easy
+ % future per-paper size adjustments
+ \IEEEsettextheight{0.625in}{0.625in}% 11in - 2 * 0.625in = 9.75in is the standard text height for compsoc journals
+ \IEEEsettopmargin{t}{0.625in}
+ \if@IEEEusingcspaper
+   \IEEEsettextheight{0.5in}{0.5in}% 10.75in - 2 * 0.5in = 9.75in
+   \IEEEsettopmargin{t}{0.5in}
+ \fi
+ \if@IEEEusingAfourpaper
+   \IEEEsettextheight{24.675mm}{24.675mm}% 297mm - 2 * 24.675mm = 247.650mm (9.75in)
+   \IEEEsettopmargin{t}{24.675mm}
+ \fi
+ % tweak textheight to a perfect integer number of lines/page.
+ % standard is: 9pt/65 lpc; 10pt/61 lpc; 11pt/53 lpc; 12pt/49 lpc
+ \IEEEquantizetextheight{c}
+ % tweak top margin so that the error is shared equally at the top and bottom
+ \IEEEsettopmargin{q}{0sp}
+
+% compsoc conference
+ \ifCLASSOPTIONconference
+  % compsoc conference use a larger value for columnsep
+  \columnsep 0.25in
+  \IEEEsettextwidth{0.75in}{0.75in}
+  % set the side margins to center the text (0.75in for letterpaper)
+  \IEEEsetsidemargin{c}{0pt}
+  % compsoc conferences want 1in top and bottom margin
+  \IEEEsettextheight{1in}{1in}
+  \IEEEsettopmargin{t}{1in}
+  % tweak textheight to a perfect integer number of lines/page.
+  % standard is: 9pt/58 lpc; 10pt/53 lpc; 11pt/48 lpc; 12pt/46 lpc
+  \IEEEquantizetextheight{c}
+  % tweak top margin so that the error is shared equally at the top and bottom
+  \IEEEsettopmargin{q}{0sp}
+ \fi
+\fi
+
+
+
+% draft mode settings override that of all other modes
+% provides a nice 1in margin all around the paper and extra
+% space between the lines for editor's comments
+\ifCLASSOPTIONdraftcls
+  % we want 1in side margins regardless of paper type
+  \IEEEsettextwidth{1in}{1in}
+  \IEEEsetsidemargin{c}{0pt}
+  % want 1in top and bottom margins
+  \IEEEsettextheight{1in}{1in}
+  \IEEEsettopmargin{t}{1in}
+  % digitize textheight to be an integer number of lines.
+  % this may cause the top and bottom margins to be off a tad
+  \IEEEquantizetextheight{c}
+  % tweak top margin so that the error is shared equally at the top and bottom
+  \IEEEsettopmargin{q}{0sp}
+\fi
+
+
+
+% process CLASSINPUT inner/outer margin
+% if inner margin defined, but outer margin not, set outer to inner.
+\ifx\CLASSINPUTinnersidemargin\@IEEEundefined
+\else
+  \ifx\CLASSINPUToutersidemargin\@IEEEundefined
+    \edef\CLASSINPUToutersidemargin{\CLASSINPUTinnersidemargin}
+  \fi
+\fi
+
+\ifx\CLASSINPUToutersidemargin\@IEEEundefined
+\else
+  % if outer margin defined, but inner margin not, set inner to outer.
+  \ifx\CLASSINPUTinnersidemargin\@IEEEundefined
+    \edef\CLASSINPUTinnersidemargin{\CLASSINPUToutersidemargin}
+  \fi
+  \IEEEsettextwidth{\CLASSINPUTinnersidemargin}{\CLASSINPUToutersidemargin}
+  \IEEEsetsidemargin{i}{\CLASSINPUTinnersidemargin}
+  \typeout{** ATTENTION: Overriding inner side margin to \CLASSINPUTinnersidemargin\space and 
+           outer side margin to \CLASSINPUToutersidemargin\space via \string\CLASSINPUT.}
+\fi
+
+
+
+% process CLASSINPUT top/bottom text margin
+% if toptext margin defined, but bottomtext margin not, set bottomtext to toptext margin
+\ifx\CLASSINPUTtoptextmargin\@IEEEundefined
+\else
+  \ifx\CLASSINPUTbottomtextmargin\@IEEEundefined
+    \edef\CLASSINPUTbottomtextmargin{\CLASSINPUTtoptextmargin}
+  \fi
+\fi
+
+\ifx\CLASSINPUTbottomtextmargin\@IEEEundefined
+\else
+  % if bottomtext margin defined, but toptext margin not, set toptext to bottomtext margin
+  \ifx\CLASSINPUTtoptextmargin\@IEEEundefined
+    \edef\CLASSINPUTtoptextmargin{\CLASSINPUTbottomtextmargin}
+  \fi
+  \IEEEsettextheight{\CLASSINPUTtoptextmargin}{\CLASSINPUTbottomtextmargin}
+  \IEEEsettopmargin{t}{\CLASSINPUTtoptextmargin}
+  \typeout{** ATTENTION: Overriding top text margin to \CLASSINPUTtoptextmargin\space and 
+           bottom text margin to \CLASSINPUTbottomtextmargin\space via \string\CLASSINPUT.}
+\fi
+
+
+
+% default to center header and footer text in the margins
+\IEEEsetheadermargin{c}{0pt}
+\IEEEsetfootermargin{c}{0pt}
+
+% adjust header and footer positions for compsoc journals
+\ifCLASSOPTIONcompsoc
+  \ifCLASSOPTIONjournal
+    \IEEEsetheadermargin{b}{\@IEEEnormalsizeunitybaselineskip}
+    \IEEEsetfootermargin{t}{\@IEEEnormalsizeunitybaselineskip}
+  \fi
+\fi
+
+
+% V1.8a display lines per column info message on user's console
+\def\IEEEdisplayinfolinespercolumn{\@IEEEtrantmpdimenA=\textheight
+% topskip represents only one line even if > baselineskip
+\advance\@IEEEtrantmpdimenA by -1\topskip
+\@IEEEtrantmpcountA=\@IEEEtrantmpdimenA
+\@IEEEtrantmpcountB=\@IEEEtrantmpdimenA
+\divide\@IEEEtrantmpcountB by \baselineskip
+% need to add one line to include topskip (first) line
+\advance\@IEEEtrantmpcountB by 1
+% save lines per column value as text
+\edef\@IEEEnumlinespercolumninfotxt{\the\@IEEEtrantmpcountB}
+% backout topskip advance to allow direct \@IEEEtrantmpcountA comparison
+\advance\@IEEEtrantmpcountB by -1
+% restore value as text height (without topskip) rather than just as number of lines
+\multiply\@IEEEtrantmpcountB by \baselineskip
+% is the column height an integer number of lines per column?
+\ifnum\@IEEEtrantmpcountA=\@IEEEtrantmpcountB
+\edef\@IEEEnumlinespercolumnexactinfotxt{exact}
+\else
+\@IEEEtrantmpdimenA\@IEEEtrantmpcountA sp\relax
+\advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpcountB sp\relax
+\edef\@IEEEnumlinespercolumnexactinfotxt{approximate, difference = \the\@IEEEtrantmpdimenA}
+\fi
+\typeout{-- Lines per column: \@IEEEnumlinespercolumninfotxt\space (\@IEEEnumlinespercolumnexactinfotxt).}}
+% delay execution till start of document to allow for user changes
+\AtBeginDocument{\IEEEdisplayinfolinespercolumn}
+
+
+
+% LIST SPACING CONTROLS
+
+% Controls the amount of EXTRA spacing
+% above and below \trivlist 
+% Both \list and IED lists override this.
+% However, \trivlist will use this as will most
+% things built from \trivlist like the \center
+% environment.
+\topsep           0.5\baselineskip
+
+% Controls the additional spacing around lists preceded
+% or followed by blank lines. the IEEE does not increase
+% spacing before or after paragraphs so it is set to zero.
+% \z@ is the same as zero, but faster.
+\partopsep          \z@
+
+% Controls the spacing between paragraphs in lists. 
+% The IEEE does not increase spacing before or after paragraphs
+% so this is also zero. 
+% With IEEEtran.cls, global changes to
+% this value DO affect lists (but not IED lists).
+\parsep             \z@
+
+% Controls the extra spacing between list items. 
+% The IEEE does not put extra spacing between items.
+% With IEEEtran.cls, global changes to this value DO affect
+% lists (but not IED lists).
+\itemsep            \z@
+
+% \itemindent is the amount to indent the FIRST line of a list
+% item. It is auto set to zero within the \list environment. To alter
+% it, you have to do so when you call the \list.
+% However, the IEEE uses this for the theorem environment
+% There is an alternative value for this near \leftmargini below
+\itemindent         -1em
+
+% \leftmargin, the spacing from the left margin of the main text to
+% the left of the main body of a list item is set by \list.
+% Hence this statement does nothing for lists.
+% But, quote and verse do use it for indention.
+\leftmargin         2em
+
+% we retain this stuff from the older IEEEtran.cls so that \list
+% will work the same way as before. However, itemize, enumerate and
+% description (IED) could care less about what these are as they
+% all are overridden.
+\leftmargini        2em
+%\itemindent         2em  % Alternative values: sometimes used.
+%\leftmargini        0em
+\leftmarginii       1em
+\leftmarginiii    1.5em
+\leftmarginiv     1.5em
+\leftmarginv      1.0em
+\leftmarginvi     1.0em
+\labelsep         0.5em 
+\labelwidth         \z@
+
+
+% The old IEEEtran.cls behavior of \list is retained.
+% However, the new V1.3 IED list environments override all the
+% @list stuff (\@listX is called within \list for the
+% appropriate level just before the user's list_decl is called). 
+% \topsep is now 2pt as the IEEE puts a little extra space around
+% lists - used by those non-IED macros that depend on \list.
+% Note that \parsep and \itemsep are not redefined as in 
+% the sizexx.clo \@listX (which article.cls uses) so global changes
+% of these values DO affect \list
+% 
+\def\@listi{\leftmargin\leftmargini \topsep 2pt plus 1pt minus 1pt}
+\let\@listI\@listi
+\def\@listii{\leftmargin\leftmarginii\labelwidth\leftmarginii%
+    \advance\labelwidth-\labelsep \topsep 2pt}
+\def\@listiii{\leftmargin\leftmarginiii\labelwidth\leftmarginiii%
+    \advance\labelwidth-\labelsep \topsep 2pt}
+\def\@listiv{\leftmargin\leftmarginiv\labelwidth\leftmarginiv%
+    \advance\labelwidth-\labelsep \topsep 2pt}
+\def\@listv{\leftmargin\leftmarginv\labelwidth\leftmarginv%
+    \advance\labelwidth-\labelsep \topsep 2pt}
+\def\@listvi{\leftmargin\leftmarginvi\labelwidth\leftmarginvi%
+    \advance\labelwidth-\labelsep \topsep 2pt}
+
+
+% The IEEE uses 5) not 5.
+\def\labelenumi{\theenumi)}     \def\theenumi{\arabic{enumi}}
+
+% The IEEE uses a) not (a)
+\def\labelenumii{\theenumii)}  \def\theenumii{\alph{enumii}}
+
+% The IEEE uses iii) not iii.
+\def\labelenumiii{\theenumiii)} \def\theenumiii{\roman{enumiii}}
+
+% The IEEE uses A) not A.
+\def\labelenumiv{\theenumiv)}   \def\theenumiv{\Alph{enumiv}}
+
+% exactly the same as in article.cls
+\def\p@enumii{\theenumi}
+\def\p@enumiii{\theenumi(\theenumii)}
+\def\p@enumiv{\p@enumiii\theenumiii}
+
+% itemized list label styles
+\def\labelitemi{$\scriptstyle\bullet$}
+\def\labelitemii{\textbf{--}}
+\def\labelitemiii{$\ast$}
+\def\labelitemiv{$\cdot$}
+
+
+
+% **** V1.3 ENHANCEMENTS ****
+% Itemize, Enumerate and Description (IED) List Controls
+% ***************************
+% 
+% 
+% The IEEE seems to use at least two different values by
+% which ITEMIZED list labels are indented to the right
+% For The Journal of Lightwave Technology (JLT) and The Journal
+% on Selected Areas in Communications (JSAC), they tend to use
+% an indention equal to \parindent. For Transactions on Communications
+% they tend to indent ITEMIZED lists a little more--- 1.3\parindent.
+% We'll provide both values here for you so that you can choose 
+% which one you like in your document using a command such as:
+% setlength{\IEEEilabelindent}{\IEEEilabelindentB}
+\newdimen\IEEEilabelindentA
+\IEEEilabelindentA \parindent
+
+\newdimen\IEEEilabelindentB
+\IEEEilabelindentB 1.3\parindent
+% However, we'll default to using \parindent
+% which makes more sense to me
+\newdimen\IEEEilabelindent
+\IEEEilabelindent \IEEEilabelindentA
+
+
+% This controls the default amount the enumerated list labels
+% are indented to the right.
+% Normally, this is the same as the paragraph indention
+\newdimen\IEEEelabelindent
+\IEEEelabelindent \parindent
+
+% This controls the default amount the description list labels
+% are indented to the right.
+% Normally, this is the same as the paragraph indention
+\newdimen\IEEEdlabelindent
+\IEEEdlabelindent \parindent
+
+% This is the value actually used within the IED lists.
+% The IED environments automatically set its value to
+% one of the three values above, so global changes do 
+% not have any effect
+\newdimen\IEEElabelindent
+\IEEElabelindent \parindent
+
+% The actual amount labels will be indented is
+% \IEEElabelindent multiplied by the factor below
+% corresponding to the level of nesting depth
+% This provides a means by which the user can
+% alter the effective \IEEElabelindent for deeper
+% levels
+% There may not be such a thing as correct "standard IEEE"
+% values. What the IEEE actually does may depend on the specific
+% circumstances.
+% The first list level almost always has full indention.
+% The second levels I've seen have only 75% of the normal indentation
+% Three level or greater nestings are very rare. I am guessing
+% that they don't use any indentation.
+\def\IEEElabelindentfactori{1.0}   % almost always one
+\def\IEEElabelindentfactorii{0.75} % 0.0 or 1.0 may be used in some cases
+\def\IEEElabelindentfactoriii{0.0} % 0.75? 0.5? 0.0?
+\def\IEEElabelindentfactoriv{0.0}
+\def\IEEElabelindentfactorv{0.0}
+\def\IEEElabelindentfactorvi{0.0}
+
+% value actually used within IED lists, it is auto
+% set to one of the 6 values above
+% global changes here have no effect
+\def\IEEElabelindentfactor{1.0}
+
+% This controls the default spacing between the end of the IED
+% list labels and the list text, when normal text is used for
+% the labels.
+% compsoc uses a larger value here, but we'll set that later
+% in the class so that this code block area can be extracted
+% as-is for IEEEtrantools.sty
+\newdimen\IEEEiednormlabelsep
+\IEEEiednormlabelsep 0.6em
+
+% This controls the default spacing between the end of the IED
+% list labels and the list text, when math symbols are used for
+% the labels (nomenclature lists). The IEEE usually increases the 
+% spacing in these cases
+\newdimen\IEEEiedmathlabelsep
+\IEEEiedmathlabelsep 1.2em
+
+% This controls the extra vertical separation put above and
+% below each IED list. the IEEE usually puts a little extra spacing
+% around each list. However, this spacing is barely noticeable.
+% compsoc uses a larger value here, but we'll set that later
+% in the class so that this code block area can be extracted
+% as-is for IEEEtrantools.sty
+\newskip\IEEEiedtopsep
+\IEEEiedtopsep 2pt plus 1pt minus 1pt
+
+
+% This command is executed within each IED list environment
+% at the beginning of the list. You can use this to set the 
+% parameters for some/all your IED list(s) without disturbing 
+% global parameters that affect things other than lists.
+% i.e., renewcommand{\IEEEiedlistdecl}{\setlength{\labelsep}{5em}}
+% will alter the \labelsep for the next list(s) until 
+% \IEEEiedlistdecl is redefined. 
+\def\IEEEiedlistdecl{\relax}
+
+% This command provides an easy way to set \leftmargin based
+% on the \labelwidth, \labelsep and the argument \IEEElabelindent
+% Usage: \IEEEcalcleftmargin{width-to-indent-the-label}
+% output is in the \leftmargin variable, i.e., effectively:
+% \leftmargin = argument + \labelwidth + \labelsep
+% Note controlled spacing here, shield end of lines with %
+\def\IEEEcalcleftmargin#1{\setlength{\leftmargin}{#1}%
+\addtolength{\leftmargin}{\labelwidth}%
+\addtolength{\leftmargin}{\labelsep}}
+
+% This command provides an easy way to set \labelwidth to the
+% width of the given text. It is the same as
+% \settowidth{\labelwidth}{label-text}
+% and useful as a shorter alternative.
+% Typically used to set \labelwidth to be the width
+% of the longest label in the list
+\def\IEEEsetlabelwidth#1{\settowidth{\labelwidth}{#1}}
+
+% When this command is executed, IED lists will use the 
+% IEEEiedmathlabelsep label separation rather than the normal
+% spacing. To have an effect, this command must be executed via
+% the \IEEEiedlistdecl or within the option of the IED list
+% environments.
+\def\IEEEusemathlabelsep{\setlength{\labelsep}{\IEEEiedmathlabelsep}}
+
+% A flag which controls whether the IED lists automatically
+% calculate \leftmargin from \IEEElabelindent, \labelwidth and \labelsep
+% Useful if you want to specify your own \leftmargin
+% This flag must be set (\IEEEnocalcleftmargintrue or \IEEEnocalcleftmarginfalse) 
+% via the \IEEEiedlistdecl or within the option of the IED list
+% environments to have an effect.
+\newif\ifIEEEnocalcleftmargin
+\IEEEnocalcleftmarginfalse
+
+% A flag which controls whether \IEEElabelindent is multiplied by
+% the \IEEElabelindentfactor for each list level.
+% This flag must be set via the \IEEEiedlistdecl or within the option 
+% of the IED list environments to have an effect.
+\newif\ifIEEEnolabelindentfactor
+\IEEEnolabelindentfactorfalse
+
+
+% internal variable to indicate type of IED label
+% justification
+% 0 - left; 1 - center; 2 - right
+\def\@IEEEiedjustify{0}
+
+
+% commands to allow the user to control IED
+% label justifications. Use these commands within
+% the IED environment option or in the \IEEEiedlistdecl
+% Note that changing the normal list justifications
+% is nonstandard and the IEEE may not like it if you do so!
+% I include these commands as they may be helpful to
+% those who are using these enhanced list controls for
+% other non-IEEE related LaTeX work.
+% itemize and enumerate automatically default to right
+% justification, description defaults to left.
+\def\IEEEiedlabeljustifyl{\def\@IEEEiedjustify{0}}%left
+\def\IEEEiedlabeljustifyc{\def\@IEEEiedjustify{1}}%center
+\def\IEEEiedlabeljustifyr{\def\@IEEEiedjustify{2}}%right
+
+
+
+
+% commands to save to and restore from the list parameter copies
+% this allows us to set all the list parameters within
+% the list_decl and prevent \list (and its \@list) 
+% from overriding any of our parameters
+% V1.6 use \edefs instead of dimen's to conserve dimen registers
+% Note controlled spacing here, shield end of lines with %
+\def\@IEEEsavelistparams{\edef\@IEEEiedtopsep{\the\topsep}%
+\edef\@IEEEiedlabelwidth{\the\labelwidth}%
+\edef\@IEEEiedlabelsep{\the\labelsep}%
+\edef\@IEEEiedleftmargin{\the\leftmargin}%
+\edef\@IEEEiedpartopsep{\the\partopsep}%
+\edef\@IEEEiedparsep{\the\parsep}%
+\edef\@IEEEieditemsep{\the\itemsep}%
+\edef\@IEEEiedrightmargin{\the\rightmargin}%
+\edef\@IEEEiedlistparindent{\the\listparindent}%
+\edef\@IEEEieditemindent{\the\itemindent}}
+
+% Note controlled spacing here
+\def\@IEEErestorelistparams{\topsep\@IEEEiedtopsep\relax%
+\labelwidth\@IEEEiedlabelwidth\relax%
+\labelsep\@IEEEiedlabelsep\relax%
+\leftmargin\@IEEEiedleftmargin\relax%
+\partopsep\@IEEEiedpartopsep\relax%
+\parsep\@IEEEiedparsep\relax%
+\itemsep\@IEEEieditemsep\relax%
+\rightmargin\@IEEEiedrightmargin\relax%
+\listparindent\@IEEEiedlistparindent\relax%
+\itemindent\@IEEEieditemindent\relax}
+
+
+% v1.6b provide original LaTeX IED list environments
+% note that latex.ltx defines \itemize and \enumerate, but not \description
+% which must be created by the base classes
+% save original LaTeX itemize and enumerate
+\let\LaTeXitemize\itemize
+\let\endLaTeXitemize\enditemize
+\let\LaTeXenumerate\enumerate
+\let\endLaTeXenumerate\endenumerate
+
+% provide original LaTeX description environment from article.cls
+\newenvironment{LaTeXdescription}
+               {\list{}{\labelwidth\z@ \itemindent-\leftmargin
+                        \let\makelabel\descriptionlabel}}
+               {\endlist}
+\newcommand*\descriptionlabel[1]{\hspace\labelsep
+                                 \normalfont\bfseries #1}
+
+
+% override LaTeX's default IED lists
+\def\itemize{\@IEEEitemize}
+\def\enditemize{\@endIEEEitemize}
+\def\enumerate{\@IEEEenumerate}
+\def\endenumerate{\@endIEEEenumerate}
+\def\description{\@IEEEdescription}
+\def\enddescription{\@endIEEEdescription}
+
+% provide the user with aliases - may help those using packages that
+% override itemize, enumerate, or description
+\def\IEEEitemize{\@IEEEitemize}
+\def\endIEEEitemize{\@endIEEEitemize}
+\def\IEEEenumerate{\@IEEEenumerate}
+\def\endIEEEenumerate{\@endIEEEenumerate}
+\def\IEEEdescription{\@IEEEdescription}
+\def\endIEEEdescription{\@endIEEEdescription}
+
+
+% V1.6 we want to keep the IEEEtran IED list definitions as our own internal
+% commands so they are protected against redefinition
+\def\@IEEEitemize{\@ifnextchar[{\@@IEEEitemize}{\@@IEEEitemize[\relax]}}
+\def\@IEEEenumerate{\@ifnextchar[{\@@IEEEenumerate}{\@@IEEEenumerate[\relax]}}
+\def\@IEEEdescription{\@ifnextchar[{\@@IEEEdescription}{\@@IEEEdescription[\relax]}}
+\def\@endIEEEitemize{\endlist}
+\def\@endIEEEenumerate{\endlist}
+\def\@endIEEEdescription{\endlist}
+
+
+% DO NOT ALLOW BLANK LINES TO BE IN THESE IED ENVIRONMENTS
+% AS THIS WILL FORCE NEW PARAGRAPHS AFTER THE IED LISTS
+% IEEEtran itemized list MDS 1/2001
+% Note controlled spacing here, shield end of lines with %
+\def\@@IEEEitemize[#1]{%
+                \ifnum\@itemdepth>3\relax\@toodeep\else%
+                \ifnum\@listdepth>5\relax\@toodeep\else%
+                \advance\@itemdepth\@ne%
+                \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
+                % get the IEEElabelindentfactor for this level
+                \advance\@listdepth\@ne% we need to know what the level WILL be
+                \edef\IEEElabelindentfactor{\csname IEEElabelindentfactor\romannumeral\the\@listdepth\endcsname}%
+                \advance\@listdepth-\@ne% undo our increment
+                \def\@IEEEiedjustify{2}% right justified labels are default
+                % set other defaults
+                \IEEEnocalcleftmarginfalse%
+                \IEEEnolabelindentfactorfalse%
+                \topsep\IEEEiedtopsep%
+                \IEEElabelindent\IEEEilabelindent%
+                \labelsep\IEEEiednormlabelsep%
+                \partopsep 0ex%
+                \parsep 0ex%
+                \itemsep 0ex%
+                \rightmargin 0em%
+                \listparindent 0em%
+                \itemindent 0em%
+                % calculate the label width
+                % the user can override this later if
+                % they specified a \labelwidth
+                \settowidth{\labelwidth}{\csname labelitem\romannumeral\the\@itemdepth\endcsname}%
+                \@IEEEsavelistparams% save our list parameters
+                \list{\csname\@itemitem\endcsname}{%
+                \@IEEErestorelistparams% override any list{} changes
+                                       % to our globals
+                \let\makelabel\@IEEEiedmakelabel% v1.6b setup \makelabel
+                \IEEEiedlistdecl% let user alter parameters
+                #1\relax%
+                % If the user has requested not to use the
+                % IEEElabelindent factor, don't revise \IEEElabelindent
+                \ifIEEEnolabelindentfactor\relax%
+                \else\IEEElabelindent=\IEEElabelindentfactor\IEEElabelindent%
+                \fi%
+                % Unless the user has requested otherwise,
+                % calculate our left margin based
+                % on \IEEElabelindent, \labelwidth and
+                % \labelsep
+                \ifIEEEnocalcleftmargin\relax%
+                \else\IEEEcalcleftmargin{\IEEElabelindent}%
+                \fi}\fi\fi}%
+
+
+% DO NOT ALLOW BLANK LINES TO BE IN THESE IED ENVIRONMENTS
+% AS THIS WILL FORCE NEW PARAGRAPHS AFTER THE IED LISTS
+% IEEEtran enumerate list MDS 1/2001
+% Note controlled spacing here, shield end of lines with %
+\def\@@IEEEenumerate[#1]{%
+                \ifnum\@enumdepth>3\relax\@toodeep\else%
+                \ifnum\@listdepth>5\relax\@toodeep\else%
+                \advance\@enumdepth\@ne%
+                \edef\@enumctr{enum\romannumeral\the\@enumdepth}%
+                % get the IEEElabelindentfactor for this level
+                \advance\@listdepth\@ne% we need to know what the level WILL be
+                \edef\IEEElabelindentfactor{\csname IEEElabelindentfactor\romannumeral\the\@listdepth\endcsname}%
+                \advance\@listdepth-\@ne% undo our increment
+                \def\@IEEEiedjustify{2}% right justified labels are default
+                % set other defaults
+                \IEEEnocalcleftmarginfalse%
+                \IEEEnolabelindentfactorfalse%
+                \topsep\IEEEiedtopsep%
+                \IEEElabelindent\IEEEelabelindent%
+                \labelsep\IEEEiednormlabelsep%
+                \partopsep 0ex%
+                \parsep 0ex%
+                \itemsep 0ex%
+                \rightmargin 0em%
+                \listparindent 0em%
+                \itemindent 0em%
+                % calculate the label width
+                % We'll set it to the width suitable for all labels using
+                % normalfont 1) to 9)
+                % The user can override this later
+                \settowidth{\labelwidth}{9)}%
+                \@IEEEsavelistparams% save our list parameters
+                \list{\csname label\@enumctr\endcsname}{\usecounter{\@enumctr}%
+                \@IEEErestorelistparams% override any list{} changes
+                                       % to our globals
+                \let\makelabel\@IEEEiedmakelabel% v1.6b setup \makelabel
+                \IEEEiedlistdecl% let user alter parameters 
+                #1\relax%
+                % If the user has requested not to use the
+                % IEEElabelindent factor, don't revise \IEEElabelindent
+                \ifIEEEnolabelindentfactor\relax%
+                \else\IEEElabelindent=\IEEElabelindentfactor\IEEElabelindent%
+                \fi%
+                % Unless the user has requested otherwise,
+                % calculate our left margin based
+                % on \IEEElabelindent, \labelwidth and
+                % \labelsep
+                \ifIEEEnocalcleftmargin\relax%
+                \else\IEEEcalcleftmargin{\IEEElabelindent}%
+                \fi}\fi\fi}%
+
+
+% DO NOT ALLOW BLANK LINES TO BE IN THESE IED ENVIRONMENTS
+% AS THIS WILL FORCE NEW PARAGRAPHS AFTER THE IED LISTS
+% IEEEtran description list MDS 1/2001
+% Note controlled spacing here, shield end of lines with %
+\def\@@IEEEdescription[#1]{%
+                \ifnum\@listdepth>5\relax\@toodeep\else%
+                % get the IEEElabelindentfactor for this level
+                \advance\@listdepth\@ne% we need to know what the level WILL be
+                \edef\IEEElabelindentfactor{\csname IEEElabelindentfactor\romannumeral\the\@listdepth\endcsname}%
+                \advance\@listdepth-\@ne% undo our increment
+                \def\@IEEEiedjustify{0}% left justified labels are default
+                % set other defaults
+                \IEEEnocalcleftmarginfalse%
+                \IEEEnolabelindentfactorfalse%
+                \topsep\IEEEiedtopsep% 
+                \IEEElabelindent\IEEEdlabelindent%
+                % assume normal labelsep
+                \labelsep\IEEEiednormlabelsep%
+                \partopsep 0ex%
+                \parsep 0ex%
+                \itemsep 0ex%
+                \rightmargin 0em%
+                \listparindent 0em%
+                \itemindent 0em%
+                % Bogus label width in case the user forgets
+                % to set it.
+                % TIP: If you want to see what a variable's width is you
+                % can use the TeX command \showthe\width-variable to 
+                % display it on the screen during compilation 
+                % (This might be helpful to know when you need to find out
+                % which label is the widest)
+                \settowidth{\labelwidth}{Hello}%
+                \@IEEEsavelistparams% save our list parameters
+                \list{}{\@IEEErestorelistparams% override any list{} changes
+                                               % to our globals
+                \let\makelabel\@IEEEiedmakelabel% v1.6b setup \makelabel
+                \IEEEiedlistdecl% let user alter parameters 
+                #1\relax%
+                % If the user has requested not to use the
+                % labelindent factor, don't revise \IEEElabelindent
+                \ifIEEEnolabelindentfactor\relax%
+                \else\IEEElabelindent=\IEEElabelindentfactor\IEEElabelindent%
+                \fi%
+                % Unless the user has requested otherwise,
+                % calculate our left margin based
+                % on \IEEElabelindent, \labelwidth and
+                % \labelsep
+                \ifIEEEnocalcleftmargin\relax%
+                \else\IEEEcalcleftmargin{\IEEElabelindent}\relax%
+                \fi}\fi}
+
+% v1.6b we use one makelabel that does justification as needed.
+\def\@IEEEiedmakelabel#1{\relax\if\@IEEEiedjustify 0\relax
+\makebox[\labelwidth][l]{\normalfont #1}\else
+\if\@IEEEiedjustify 1\relax
+\makebox[\labelwidth][c]{\normalfont #1}\else
+\makebox[\labelwidth][r]{\normalfont #1}\fi\fi}
+
+
+% compsoc uses a larger value for the normal labelsep
+% and also extra spacing above and below each list
+\ifCLASSOPTIONcompsoc
+  \IEEEiednormlabelsep 1.2em
+  \IEEEiedtopsep 6pt plus 3pt minus 3pt
+\fi
+
+
+% VERSE and QUOTE
+% V1.7 define environments with newenvironment
+\newenvironment{verse}{\let\\=\@centercr
+    \list{}{\itemsep\z@ \itemindent -1.5em \listparindent \itemindent
+    \rightmargin\leftmargin\advance\leftmargin 1.5em}\item\relax}
+    {\endlist}
+\newenvironment{quotation}{\list{}{\listparindent 1.5em \itemindent\listparindent
+    \rightmargin\leftmargin \parsep 0pt plus 1pt}\item\relax}
+    {\endlist}
+\newenvironment{quote}{\list{}{\rightmargin\leftmargin}\item\relax}
+    {\endlist}
+
+
+% \titlepage
+% provided only for backward compatibility. \maketitle is the correct
+% way to create the title page. 
+\def\titlepage{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+    \else \newpage \fi \thispagestyle{empty}\c@page\z@}
+\def\endtitlepage{\if@restonecol\twocolumn \else \newpage \fi}
+
+% standard values from article.cls
+\arraycolsep     5pt
+\arrayrulewidth .4pt
+\doublerulesep   2pt
+
+\tabcolsep       6pt
+\tabbingsep      0.5em
+
+
+%% FOOTNOTES
+%
+%\skip\footins 10pt plus 4pt minus 2pt
+% V1.6 respond to changes in font size
+% space added above the footnotes (if present)
+\skip\footins 0.9\baselineskip  plus 0.4\baselineskip  minus 0.2\baselineskip
+
+% V1.6, we need to make \footnotesep responsive to changes
+% in \baselineskip or strange spacings will result when in
+% draft mode. Here is a little LaTeX secret - \footnotesep
+% determines the height of an invisible strut that is placed
+% *above* the baseline of footnotes after the first. Since
+% LaTeX considers the space for characters to be 0.7\baselineskip
+% above the baseline and 0.3\baselineskip below it, we need to
+% use 0.7\baselineskip as a \footnotesep to maintain equal spacing
+% between all the lines of the footnotes. The IEEE often uses a tad
+% more, so use 0.8\baselineskip. This slightly larger value also helps
+% the text to clear the footnote marks. Note that \thanks in IEEEtran
+% uses its own value of \footnotesep which is set in \maketitle.
+{\footnotesize
+\global\footnotesep 0.8\baselineskip}
+
+
+\skip\@mpfootins = \skip\footins
+\fboxsep = 3pt
+\fboxrule = .4pt
+% V1.6 use 1em, then use LaTeX2e's \@makefnmark
+% Note that the IEEE normally *left* aligns the footnote marks, so we don't need
+% box resizing tricks here.
+\long\def\@makefntext#1{\parindent 1em\indent\hbox{\@makefnmark}#1}% V1.6 use 1em
+% V1.7 compsoc does not use superscipts for footnote marks
+\ifCLASSOPTIONcompsoc
+\def\@IEEEcompsocmakefnmark{\hbox{\normalfont\@thefnmark.\ }}
+\long\def\@makefntext#1{\parindent 1em\indent\hbox{\@IEEEcompsocmakefnmark}#1}
+\fi
+
+% The IEEE does not use footnote rules
+\def\footnoterule{}
+
+% V1.7 for compsoc, the IEEE uses a footnote rule only for \thanks. We devise a "one-shot"
+% system to implement this.
+\newif\if@IEEEenableoneshotfootnoterule
+\@IEEEenableoneshotfootnoterulefalse
+\ifCLASSOPTIONcompsoc
+\def\footnoterule{\relax\if@IEEEenableoneshotfootnoterule
+\kern-5pt
+\hbox to \columnwidth{\hfill\vrule width 0.5\columnwidth height 0.4pt\hfill}
+\kern4.6pt
+\global\@IEEEenableoneshotfootnoterulefalse
+\else
+\relax
+\fi}
+\fi
+
+% V1.6 do not allow LaTeX to break a footnote across multiple pages
+\interfootnotelinepenalty=10000
+
+% V1.6 discourage breaks within equations
+% Note that amsmath normally sets this to 10000,
+% but LaTeX2e normally uses 100.
+\interdisplaylinepenalty=2500
+
+% default allows section depth up to /paragraph
+\setcounter{secnumdepth}{4}
+
+% technotes do not allow /paragraph
+\ifCLASSOPTIONtechnote
+   \setcounter{secnumdepth}{3}
+\fi
+% neither do compsoc conferences
+\@IEEEcompsocconfonly{\setcounter{secnumdepth}{3}}
+
+
+\newcounter{section}
+\newcounter{subsection}[section]
+\newcounter{subsubsection}[subsection]
+\newcounter{paragraph}[subsubsection]
+
+% used only by IEEEtran's IEEEeqnarray as other packages may
+% have their own, different, implementations
+\newcounter{IEEEsubequation}[equation]
+
+% as shown when called by user from \ref, \label and in table of contents
+\def\theequation{\arabic{equation}}                          % 1
+\def\theIEEEsubequation{\theequation\alph{IEEEsubequation}}  % 1a (used only by IEEEtran's IEEEeqnarray)
+\ifCLASSOPTIONcompsoc
+% compsoc is all arabic
+\def\thesection{\arabic{section}}                
+\def\thesubsection{\thesection.\arabic{subsection}}
+\def\thesubsubsection{\thesubsection.\arabic{subsubsection}}
+\def\theparagraph{\thesubsubsection.\arabic{paragraph}}
+\else
+\def\thesection{\Roman{section}}                             % I
+% V1.7, \mbox prevents breaks around - 
+\def\thesubsection{\mbox{\thesection-\Alph{subsection}}}     % I-A
+% V1.7 use I-A1 format used by the IEEE rather than I-A.1
+\def\thesubsubsection{\thesubsection\arabic{subsubsection}}  % I-A1
+\def\theparagraph{\thesubsubsection\alph{paragraph}}         % I-A1a
+\fi
+
+% From Heiko Oberdiek. Because of the \mbox in \thesubsection, we need to
+% tell hyperref to disable the \mbox command when making PDF bookmarks.
+% This done already with hyperref.sty version 6.74o and later, but
+% it will not hurt to do it here again for users of older versions.
+\@ifundefined{pdfstringdefPreHook}{\let\pdfstringdefPreHook\@empty}{}%
+\g@addto@macro\pdfstringdefPreHook{\let\mbox\relax}
+
+
+% Main text forms (how shown in main text headings)
+% V1.6, using \thesection in \thesectiondis allows changes
+% in the former to automatically appear in the latter
+\ifCLASSOPTIONcompsoc
+  \ifCLASSOPTIONconference% compsoc conference
+    \def\thesectiondis{\thesection.}
+    \def\thesubsectiondis{\thesectiondis\arabic{subsection}.}
+    \def\thesubsubsectiondis{\thesubsectiondis\arabic{subsubsection}.}
+    \def\theparagraphdis{\thesubsubsectiondis\arabic{paragraph}.}
+  \else% compsoc not conferencs
+    \def\thesectiondis{\thesection}
+    \def\thesubsectiondis{\thesectiondis.\arabic{subsection}}
+    \def\thesubsubsectiondis{\thesubsectiondis.\arabic{subsubsection}}
+    \def\theparagraphdis{\thesubsubsectiondis.\arabic{paragraph}}
+  \fi
+\else% not compsoc
+  \def\thesectiondis{\thesection.}                   % I.
+  \def\thesubsectiondis{\Alph{subsection}.}          % B.
+  \def\thesubsubsectiondis{\arabic{subsubsection})}  % 3)
+  \def\theparagraphdis{\alph{paragraph})}            % d)
+\fi
+
+% just like LaTeX2e's \@eqnnum
+\def\theequationdis{{\normalfont \normalcolor (\theequation)}}% (1)
+% IEEEsubequation used only by IEEEtran's IEEEeqnarray
+\def\theIEEEsubequationdis{{\normalfont \normalcolor (\theIEEEsubequation)}}% (1a)
+% redirect LaTeX2e's equation number display and all that depend on
+% it, through IEEEtran's \theequationdis
+\def\@eqnnum{\theequationdis}
+
+
+
+% V1.7 provide string macros as article.cls does
+\def\contentsname{Contents}
+\def\listfigurename{List of Figures}
+\def\listtablename{List of Tables}
+\def\refname{References}
+\def\indexname{Index}
+\def\figurename{Fig.}
+\def\tablename{TABLE}
+\@IEEEcompsocconfonly{\def\figurename{Figure}}
+\def\partname{Part}
+\def\appendixname{Appendix}
+\def\abstractname{Abstract}
+% IEEE specific names
+\def\IEEEkeywordsname{Index Terms}
+\def\IEEEproofname{Proof}
+
+
+% LIST OF FIGURES AND TABLES AND TABLE OF CONTENTS
+%
+\def\@pnumwidth{1.55em}
+\def\@tocrmarg{2.55em}
+\def\@dotsep{4.5}
+\setcounter{tocdepth}{3}
+
+% adjusted some spacings here so that section numbers will not easily 
+% collide with the section titles. 
+% VIII; VIII-A; and VIII-A.1 are usually the worst offenders.
+% MDS 1/2001
+\def\tableofcontents{\section*{\contentsname}\@starttoc{toc}}
+\def\l@section#1#2{\addpenalty{\@secpenalty}\addvspace{1.0em plus 1pt}%
+    \@tempdima 2.75em \begingroup \parindent \z@ \rightskip \@pnumwidth%
+    \parfillskip-\@pnumwidth {\bfseries\leavevmode #1}\hfil\hbox to\@pnumwidth{\hss #2}\par%
+    \endgroup}
+% argument format #1:level, #2:labelindent,#3:labelsep
+\def\l@subsection{\@dottedtocline{2}{2.75em}{3.75em}}
+\def\l@subsubsection{\@dottedtocline{3}{6.5em}{4.5em}}
+% must provide \l@ defs for ALL sublevels EVEN if tocdepth
+% is such as they will not appear in the table of contents
+% these defs are how TOC knows what level these things are!
+\def\l@paragraph{\@dottedtocline{4}{6.5em}{5.5em}}
+\def\l@subparagraph{\@dottedtocline{5}{6.5em}{6.5em}}
+\def\listoffigures{\section*{\listfigurename}\@starttoc{lof}}
+\def\l@figure{\@dottedtocline{1}{0em}{2.75em}}
+\def\listoftables{\section*{\listtablename}\@starttoc{lot}}
+\let\l@table\l@figure
+
+
+% Definitions for floats
+%
+% Normal Floats
+% V1.8 floatsep et al. revised down by 0.15\baselineskip
+% to account for the sideeffects of \topskip compensation
+\floatsep 0.85\baselineskip plus  0.2\baselineskip minus  0.2\baselineskip
+\textfloatsep 1.55\baselineskip plus  0.2\baselineskip minus  0.4\baselineskip
+\@fptop 0pt plus 1fil
+\@fpsep 0.75\baselineskip plus 2fil
+\@fpbot 0pt plus 1fil
+\def\topfraction{0.9}
+\def\bottomfraction{0.4}
+\def\floatpagefraction{0.8}
+% V1.7, let top floats approach 90% of page
+\def\textfraction{0.1}
+
+% Double Column Floats
+\dblfloatsep 0.85\baselineskip plus  0.2\baselineskip minus  0.2\baselineskip
+
+\dbltextfloatsep 1.55\baselineskip plus  0.2\baselineskip minus  0.4\baselineskip
+% Note that it would be nice if the rubber here actually worked in LaTeX2e.
+% There is a long standing limitation in LaTeX, first discovered (to the best
+% of my knowledge) by Alan Jeffrey in 1992. LaTeX ignores the stretchable
+% portion of \dbltextfloatsep, and as a result, double column figures can and
+% do result in an non-integer number of lines in the main text columns with
+% underfull vbox errors as a consequence. A post to comp.text.tex
+% by Donald Arseneau confirms that this had not yet been fixed in 1998.
+% IEEEtran V1.6 will fix this problem for you in the titles, but it doesn't
+% protect you from other double floats. Happy vspace'ing.
+
+\@dblfptop 0pt plus 1fil
+\@dblfpsep 0.75\baselineskip plus 2fil
+\@dblfpbot 0pt plus 1fil
+\def\dbltopfraction{0.8}
+\def\dblfloatpagefraction{0.8}
+\setcounter{dbltopnumber}{4}
+
+\intextsep 0.85\baselineskip plus 0.2\baselineskip minus  0.2\baselineskip
+\setcounter{topnumber}{2}
+\setcounter{bottomnumber}{2}
+\setcounter{totalnumber}{4}
+
+
+
+% article class provides these, we should too.
+\newlength\abovecaptionskip
+\newlength\belowcaptionskip
+% but only \abovecaptionskip is used above figure captions and *below* table
+% captions
+\setlength\abovecaptionskip{0.5\baselineskip}
+% compsoc journals are a little more generous
+\ifCLASSOPTIONcompsoc\ifCLASSOPTIONjournal
+  \setlength\abovecaptionskip{0.75\baselineskip}
+\fi\fi
+\setlength\belowcaptionskip{0pt}
+% V1.6 create hooks in case the caption spacing ever needs to be
+% overridden by a user
+\def\@IEEEfigurecaptionsepspace{\vskip\abovecaptionskip\relax}%
+\def\@IEEEtablecaptionsepspace{\vskip\abovecaptionskip\relax}%
+
+
+% 1.6b revise caption system so that \@makecaption uses two arguments
+% as with LaTeX2e. Otherwise, there will be problems when using hyperref.
+\def\@IEEEtablestring{table}
+
+
+% V1.8 compensate for \topskip so top of top figures align with tops of the first lines of main text
+% here we calculate a space equal to the amount \topskip exceeds the main text height
+% we hook in at \@floatboxreset
+\def\@IEEEfiguretopskipspace{\ifdim\prevdepth=-1000pt\relax
+\setlength{\@IEEEtrantmpdimenA}{1\topskip}\relax
+\addtolength{\@IEEEtrantmpdimenA}{-0.7\@IEEEnormalsizeunitybaselineskip}\relax
+\vspace*{\@IEEEtrantmpdimenA}\fi}
+% V1.8 compensate for \topskip at the top of top tables so caption text is on main text baseline
+% use a strut set on the caption baseline within \@makecaption
+\def\@IEEEtabletopskipstrut{\ifdim\prevdepth=-1000pt\rule{0pt}{\topskip}\fi}
+% the \ifdim\prevdepth checks are always expected to be true for IEEE style float caption ordering
+% because top of figure content and top of captions in tables is the first thing on the vertical
+% list of these floats
+% thanks to Donald Arseneau for his 2000/11/11 post "Re: caption hacking" with info on this topic.
+
+
+\ifCLASSOPTIONcompsoc
+% V1.7 compsoc \@makecaption
+\ifCLASSOPTIONconference% compsoc conference
+\long\def\@makecaption#1#2{%
+% test if is a for a figure or table
+\ifx\@captype\@IEEEtablestring%
+% if a table, do table caption
+\footnotesize\bgroup\par\centering\@IEEEtabletopskipstrut{\normalfont\footnotesize {#1.}\nobreakspace\scshape #2}\par\addvspace{0.5\baselineskip}\egroup%
+\@IEEEtablecaptionsepspace
+% if not a table, format it as a figure
+\else
+\@IEEEfigurecaptionsepspace
+\setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}\nobreakspace #2}%
+\ifdim \wd\@tempboxa >\hsize%
+% if caption is longer than a line, let it wrap around
+\setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}\nobreakspace}%
+\parbox[t]{\hsize}{\normalfont\footnotesize \noindent\unhbox\@tempboxa#2}%
+% if caption is shorter than a line, center
+\else%
+\hbox to\hsize{\normalfont\footnotesize\hfil\box\@tempboxa\hfil}%
+\fi\fi}
+%
+\else% nonconference compsoc
+\long\def\@makecaption#1#2{%
+% test if is a for a figure or table
+\ifx\@captype\@IEEEtablestring%
+% if a table, do table caption
+\footnotesize\bgroup\par\centering\@IEEEtabletopskipstrut{\normalfont\sffamily\footnotesize #1}\\{\normalfont\sffamily\footnotesize #2}\par\addvspace{0.5\baselineskip}\egroup%
+\@IEEEtablecaptionsepspace
+% if not a table, format it as a figure
+\else
+\@IEEEfigurecaptionsepspace
+\setbox\@tempboxa\hbox{\normalfont\sffamily\footnotesize {#1.}\nobreakspace #2}%
+\ifdim \wd\@tempboxa >\hsize%
+% if caption is longer than a line, let it wrap around
+\setbox\@tempboxa\hbox{\normalfont\sffamily\footnotesize {#1.}\nobreakspace}%
+\parbox[t]{\hsize}{\normalfont\sffamily\footnotesize \noindent\unhbox\@tempboxa#2}%
+% if caption is shorter than a line, left justify
+\else%
+\hbox to\hsize{\normalfont\sffamily\footnotesize\box\@tempboxa\hfil}%
+\fi\fi}
+\fi
+%
+\else% traditional noncompsoc \@makecaption
+\long\def\@makecaption#1#2{%
+% test if is a for a figure or table
+\ifx\@captype\@IEEEtablestring%
+% if a table, do table caption
+\footnotesize\bgroup\par\centering\@IEEEtabletopskipstrut{\normalfont\footnotesize #1}\\{\normalfont\footnotesize\scshape #2}\par\addvspace{0.5\baselineskip}\egroup%
+\@IEEEtablecaptionsepspace
+% if not a table, format it as a figure
+\else
+\@IEEEfigurecaptionsepspace
+% 3/2001 use footnotesize, not small; use two nonbreaking spaces, not one
+\setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}\nobreakspace\nobreakspace #2}%
+\ifdim \wd\@tempboxa >\hsize%
+% if caption is longer than a line, let it wrap around
+\setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}\nobreakspace\nobreakspace}%
+\parbox[t]{\hsize}{\normalfont\footnotesize\noindent\unhbox\@tempboxa#2}%
+% if caption is shorter than a line, center if conference, left justify otherwise
+\else%
+\ifCLASSOPTIONconference \hbox to\hsize{\normalfont\footnotesize\hfil\box\@tempboxa\hfil}%
+\else \hbox to\hsize{\normalfont\footnotesize\box\@tempboxa\hfil}%
+\fi\fi\fi}
+\fi
+
+
+
+% V1.7 disable captions class option, do so in a way that retains operation of \label
+% within \caption
+\ifCLASSOPTIONcaptionsoff
+\long\def\@makecaption#1#2{\vspace*{2em}\footnotesize\bgroup\par\addvspace{0.5\baselineskip}\centering{\footnotesize #1}\par\addvspace{0.5\baselineskip}\egroup%
+\let\@IEEEtemporiglabeldefsave\label
+\let\@IEEEtemplabelargsave\relax
+\def\label##1{\gdef\@IEEEtemplabelargsave{##1}}%
+\setbox\@tempboxa\hbox{#2}%
+\let\label\@IEEEtemporiglabeldefsave
+\ifx\@IEEEtemplabelargsave\relax\else\label{\@IEEEtemplabelargsave}\fi}
+\fi
+
+
+% V1.7 define end environments with \def not \let so as to work OK with
+% preview-latex
+\newcounter{figure}
+\def\thefigure{\@arabic\c@figure}
+\def\fps@figure{tbp}
+\def\ftype@figure{1}
+\def\ext@figure{lof}
+\def\fnum@figure{\figurename\nobreakspace\thefigure}
+% V1.8 within figures add \@IEEEfiguretopskipspace compensation to LaTeX2e's \@floatboxreset
+\def\figure{\def\@floatboxreset{\reset@font\normalsize\@setminipage\@IEEEfiguretopskipspace}\@float{figure}}
+\def\endfigure{\end@float}
+% V1.8 also add \@IEEEfiguretopskipspace compensation to \figure*
+\@namedef{figure*}{\def\@floatboxreset{\reset@font\normalsize\@setminipage\@IEEEfiguretopskipspace}\@dblfloat{figure}}
+\@namedef{endfigure*}{\end@dblfloat}
+
+\newcounter{table}
+\ifCLASSOPTIONcompsoc
+\def\thetable{\arabic{table}}
+\else
+\def\thetable{\@Roman\c@table}
+\fi
+\def\fps@table{tbp}
+\def\ftype@table{2}
+\def\ext@table{lot}
+\def\fnum@table{\tablename\nobreakspace\thetable}
+% V1.6 The IEEE uses 8pt text for tables
+% within tables alter LaTeX2e's \@floatboxreset to use \footnotesize
+\def\table{\def\@floatboxreset{\reset@font\footnotesize\@setminipage}\@float{table}}
+\def\endtable{\end@float}
+% v1.6b double column tables need to default to footnotesize as well.
+\@namedef{table*}{\def\@floatboxreset{\reset@font\footnotesize\@setminipage}\@dblfloat{table}}
+\@namedef{endtable*}{\end@dblfloat}
+
+
+
+
+%% -- Command Argument Scanning Support Functions --
+%% V1.8a 
+
+% usage: \@IEEEstripouterbraces*{}
+% \@IEEEstripouterbraces fully expands its argument (which it then stores
+% in \@IEEEstripouterbracesarg) via \edef, then removes any outer enclosing
+% braces, and finally stores the result in the macro
+% \@IEEEstrippedouterbraces.
+%
+% For example:
+% \@IEEEstripouterbraces{{{{ab}c}}}
+% results in:
+%
+% \@IEEEstripouterbracesarg ==> a macro containing {{{ab}c}}
+% \@IEEEstrippedouterbraces ==> a macro containing {ab}c
+%
+% the *-star form,\@IEEEstripouterbraces*, does not expand the argument
+% contents during processing
+\def\@IEEEstripouterbraces{\@ifstar{\let\@IEEEstripouterbracesdef=\def\@@IEEEstripouterbraces}{\let\@IEEEstripouterbracesdef=\edef\@@IEEEstripouterbraces}}
+
+\def\@@IEEEstripouterbraces#1{\@IEEEstripouterbracesdef\@IEEEstripouterbracesarg{#1}\relax
+% If the macro is unchanged after being acquired as a single delimited
+% argument, we know we have one sequence of tokens without any enclosing
+% braces. Loop until this is true.
+\loop
+  \expandafter\@@@IEEEstripouterbraces\@IEEEstripouterbracesarg\@IEEEgeneralsequenceDELIMITER
+\ifx\@IEEEstrippedouterbraces\@IEEEstripouterbracesarg
+\else
+  \let\@IEEEstripouterbracesarg\@IEEEstrippedouterbraces
+\repeat}
+
+\def\@@@IEEEstripouterbraces#1\@IEEEgeneralsequenceDELIMITER{\def\@IEEEstrippedouterbraces{#1}}
+
+
+
+% usage: \@IEEEextractgroup*{}
+% \@IEEEextractgroup fully expands its argument (which it then stores in
+% \@IEEEextractgrouparg) via \edef and then assigns the first "brace group"
+% of tokens to the macro \@IEEEextractedgroup.
+% The remaining groups, if any, are stored in the macro
+% \@IEEEextractedgroupremain. If the argument does not contain the requisite
+% groups, the respective macros will be defined to be empty.
+% There is an asymmetry in that \@IEEEextractedgroup is stripped of its first
+% outer grouping while \@IEEEextractedgroupremain retains even the outer
+% grouping (if present) that originally identified it as a group.
+%
+% For example:
+% \@IEEEextractgroup{{{ab}}{c{de}}}
+% results in:
+%
+% \@IEEEextractgrouparg       ==> a macro containing {{ab}}{c{de}}
+% \@IEEEextractedgroup        ==> a macro containing {ab}
+% \@IEEEextractedgroupremain  ==> a macro containing {c{de}}
+%
+% The *-star form, \@IEEEextractgroup*, does not expand its argument
+% contents during processing.
+\def\@IEEEextractgroup{\@ifstar{\let\@IEEEextractgroupdef=\def\@@IEEEextractgroup}{\let\@IEEEextractgroupdef=\edef\@@IEEEextractgroup}}
+
+\def\@@IEEEextractgroup#1{\@IEEEextractgroupdef\@IEEEextractgrouparg{#1}\relax
+% trap the case of an empty extracted group as this would cause problems with
+% \@IEEEextractgroupremain's argument acquisition
+\ifx\@IEEEextractgrouparg\@empty
+  \def\@IEEEextractedgroup{}\relax
+  \def\@IEEEextractedgroupremain{}\relax
+\else
+  % We have to use some dirty tricks here. We want to insert {} around
+  % whatever remains after the first group so that TeX's argument scanner
+  % will preserve any originally enclosing braces as well as provide an
+  % empty argument to acquire even if there isn't a second group.
+  % In this first of two dirty tricks, we put a } at the end of the structure
+  % we are going to extract from. The \ifnum0=`{\fi keeps TeX happy to allow
+  % what would otherwise be an unbalanced macro definition for
+  % \@@IEEEextractgroup to be acceptable to it.
+  \ifnum0=`{\fi\expandafter\@IEEEextractgroupremain\@IEEEextractgrouparg}\relax
+\fi}
+
+% In the second part of the dirty tricks, we insert a leading { right after
+% the first group is acquired, but before the remainder is. Again, the
+% \ifnum0=`}\fi keeps TeX happy during definition time, but will disappear
+% during run time.
+\def\@IEEEextractgroupremain#1{\def\@IEEEextractedgroup{#1}\expandafter\@@IEEEextractgroupremain\expandafter{\ifnum0=`}\fi}
+
+\def\@@IEEEextractgroupremain#1{\def\@IEEEextractedgroupremain{#1}}
+
+
+
+% \@IEEEextracttoken relocated at top because margin setting commands rely on it
+
+
+
+% usage: \@IEEEextracttokengroups*{}
+% \@IEEEextracttokengroups fully expands its argument (which it then stores
+% in \@IEEEextracttokengroupsarg) and then assigns the first "brace group" of
+% tokens (with the outermost braces removed) to the macro
+% \@IEEEextractedfirstgroup.
+% The meaning of the first nonbrace (but including the empty group) token
+% within this first group is assigned via \let to \@IEEEextractedfirsttoken
+% as well as stored in the macro \@IEEEextractedfirsttokenmacro. If a first
+% nonbrace token does not exist (or is an empty group), these will be \relax
+% and empty, respectively.  Tokens that would otherwise be discarded during
+% the acquisition of the first token in the first group are stored in
+% \@IEEEextractedfirsttokensdiscarded, however their original relative brace
+% nesting depths are not guaranteed to be preserved.
+% The first group within this first group is stored in the macro
+% \@IEEEextractedfirstfirstgroup.
+% Likewise for the next group after the first: \@IEEEextractednextgroup,
+% \@IEEEextractednextfirstgroup, \@IEEEextractednextgroupfirsttoken,
+% \@IEEEextractednextgroupfirsttokenmacro, and 
+% \@IEEEextractednextfirsttokensdiscarded.
+% All tokens/groups after the first group, including any enclosing braces,
+% are stored in the macro \@IEEEextractedafterfirstgroupremain which will
+% be empty if none exist.
+%
+% For example:
+% \@IEEEextracttokengroups{{{ab}{cd}}{{ef}g}}
+% will result in:
+%
+% \@IEEEextracttokengroupsarg             ==> a macro containing {{ab}{cd}}{{ef}g}
+% \@IEEEextractedfirstgroup               ==> a macro containing {ab}{cd}
+% \@IEEEextractedafterfirstgroupremain    ==> a macro containing {{ef}g}
+% \@IEEEextractedfirsttoken               ==> the letter a
+% \@IEEEextractedfirsttokenmacro          ==> a macro containing a
+% \@IEEEextractedfirsttokensdiscarded     ==> a macro containing bcd
+% \@IEEEextractedfirstfirstgroup          ==> a macro containing ab
+% \@IEEEextractednextgroup                ==> a macro containing {ef}g
+% \@IEEEextractednextfirsttoken           ==> the letter e
+% \@IEEEextractednextfirsttokenmacro      ==> a macro containing e
+% \@IEEEextractednextfirsttokensdiscarded ==> a macro containing fg
+% \@IEEEextractednextfirstgroup           ==> a macro containing ef
+%
+% If given an empty argument, \@IEEEextractedfirsttoken and
+% \@IEEEextractednextfirsttoken will be set to \relax
+% and all the macros will be empty.
+% the *-star form, \@IEEEextracttokengroups*, does not expand its argument
+% contents during processing.
+%
+% Depends on: \@IEEEextractgroup, \@IEEEextracttoken
+\def\@IEEEextracttokengroups{\@ifstar{\let\@IEEEextracttokengroupsdef=\def\@@IEEEextracttokengroups}{\let\@IEEEextracttokengroupsdef=\edef\@@IEEEextracttokengroups}}
+\def\@@IEEEextracttokengroups#1{\@IEEEextracttokengroupsdef\@IEEEextracttokengroupsarg{#1}\relax
+% begin extraction, these functions are safe with empty arguments
+% first group
+\expandafter\@IEEEextractgroup\expandafter*\expandafter{\@IEEEextracttokengroupsarg}\relax
+\let\@IEEEextractedfirstgroup\@IEEEextractedgroup
+\let\@IEEEextractedafterfirstgroupremain\@IEEEextractedgroupremain
+\expandafter\@IEEEextracttoken\expandafter*\expandafter{\@IEEEextractedfirstgroup}\relax
+\let\@IEEEextractedfirsttoken\@IEEEextractedtoken
+\let\@IEEEextractedfirsttokenmacro\@IEEEextractedtokenmacro
+\let\@IEEEextractedfirsttokensdiscarded\@IEEEextractedtokensdiscarded
+% first first group
+\expandafter\@IEEEextractgroup\expandafter*\expandafter{\@IEEEextractedfirstgroup}\relax
+\let\@IEEEextractedfirstfirstgroup\@IEEEextractedgroup
+% next group
+\expandafter\@IEEEextractgroup\expandafter*\expandafter{\@IEEEextractedafterfirstgroupremain}\relax
+\let\@IEEEextractednextgroup\@IEEEextractedgroup
+\expandafter\@IEEEextracttoken\expandafter*\expandafter{\@IEEEextractednextgroup}\relax
+\let\@IEEEextractednextfirsttoken\@IEEEextractedtoken
+\let\@IEEEextractednextfirsttokenmacro\@IEEEextractedtokenmacro
+\let\@IEEEextractednextfirsttokensdiscarded\@IEEEextractedtokensdiscarded
+% next first group
+\expandafter\@IEEEextractgroup\expandafter*\expandafter{\@IEEEextractednextgroup}\relax
+\let\@IEEEextractednextfirstgroup\@IEEEextractedgroup}
+
+
+%% -- End of Command Argument Scanning Support Functions --
+
+
+
+
+%%
+%% START OF IEEEeqnarray DEFINITIONS
+%%
+%% Inspired by the concepts, examples, and previous works of LaTeX 
+%% coders and developers such as Donald Arseneau, Fred Bartlett, 
+%% David Carlisle, Tony Liu, Frank Mittelbach, Piet van Oostrum, 
+%% Roland Winkler and Mark Wooding.
+%% I don't make the claim that my work here is even near their calibre. ;)
+
+
+\newif\if@IEEEeqnarrayboxnojot% flag to indicate if the environment was called as the star form
+\@IEEEeqnarrayboxnojotfalse
+
+\newif\if@advanceIEEEeqncolcnt% tracks if the environment should advance the col counter
+% allows a way to make an \IEEEeqnarraybox that can be used within an \IEEEeqnarray
+% used by IEEEeqnarraymulticol so that it can work properly in both
+\@advanceIEEEeqncolcnttrue
+
+\newcount\@IEEEeqnnumcols % tracks how many IEEEeqnarray cols are defined
+\newcount\@IEEEeqncolcnt  % tracks how many IEEEeqnarray cols the user actually used
+
+
+% The default math style used by the columns
+\def\IEEEeqnarraymathstyle{\displaystyle}
+% The default text style used by the columns
+% default to using the current font
+\def\IEEEeqnarraytextstyle{\relax}
+
+% like the iedlistdecl but for \IEEEeqnarray
+\def\IEEEeqnarraydecl{\relax}
+\def\IEEEeqnarrayboxdecl{\relax}
+
+
+
+% V1.8 flags to indicate that equation numbering is to persist
+\newif\if@IEEEeqnumpersist%
+\@IEEEeqnumpersistfalse
+\newif\if@IEEEsubeqnumpersist%
+\@IEEEsubeqnumpersistfalse
+%
+% V1.8 flags to indicate if (sub)equation number of last line was preadvanced
+\newif\if@IEEEeqnumpreadv%
+\@IEEEeqnumpreadvfalse
+\newif\if@IEEEsubeqnumpreadv%
+\@IEEEsubeqnumpreadvfalse
+
+\newcount\@IEEEsubeqnnumrollback% saves previous value of IEEEsubequation number in case we need to restore it
+
+% \yesnumber is the opposite of \nonumber
+% a novel concept with the same def as the equationarray package
+% However, we give IEEE versions too since some LaTeX packages such as 
+% the MDWtools mathenv.sty redefine \nonumber to something else.
+% This command is intended for use in non-IEEEeqnarray math environments
+\providecommand{\yesnumber}{\global\@eqnswtrue}
+
+
+% IEEEyes/nonumber 
+% V1.8 add persistant * forms
+% These commands can alter the type of equation an IEEEeqnarray line is.
+\def\IEEEyesnumber{\@ifstar{\global\@IEEEeqnumpersisttrue\global\@IEEEsubeqnumpersistfalse\@IEEEyesnumber}{\@IEEEyesnumber}}
+
+\def\@IEEEyesnumber{\global\@eqnswtrue
+\if@IEEEeqnarrayISinner% alter counters and label only inside an IEEEeqnarray
+\ifnum\c@IEEEsubequation>0\relax
+   \stepcounter{equation}\setcounter{IEEEsubequation}{0}\gdef\@currentlabel{\p@equation\theequation}\relax
+   \gdef\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
+\fi
+% even if we reached this eqn num via a preadv, it is legit now
+\global\@IEEEeqnumpreadvfalse\global\@IEEEsubeqnumpreadvfalse
+\fi}
+
+\def\IEEEnonumber{\@ifstar{\global\@IEEEeqnumpersistfalse\global\@IEEEsubeqnumpersistfalse\global\@eqnswfalse}{\global\@eqnswfalse}}
+
+
+\def\IEEEyessubnumber{\@ifstar{\global\@IEEEsubeqnumpersisttrue\@IEEEyessubnumber}{\@IEEEyessubnumber}}
+%
+\def\@IEEEyessubnumber{\if@IEEEeqnarrayISinner% alter counters and label only inside an IEEEeqnarray
+  \ifnum\c@IEEEsubequation>0\relax% if it already is a subequation, we are good to go as-is
+  \else% if we are a regular equation we have to watch out for two cases
+    \if@IEEEeqnumpreadv% if this equation is the result of a preadvance, backout and bump the sub eqnnum
+       \global\advance\c@equation\m@ne\global\c@IEEEsubequation=\@IEEEsubeqnnumrollback\addtocounter{IEEEsubequation}{1}\relax
+    \else% non-preadvanced equations just need initialization of their sub eqnnum
+       \setcounter{IEEEsubequation}{1}\relax
+    \fi
+  \fi% fi already is subequation
+  \gdef\@currentlabel{\p@IEEEsubequation\theIEEEsubequation}\relax
+  \gdef\@currentHref{\@IEEEtheHrefsubequation}% setup hyperref label
+  \global\@IEEEeqnumpreadvfalse\global\@IEEEsubeqnumpreadvfalse% no longer a preadv anymore
+  \global\@eqnswtrue
+\fi}
+
+
+\def\IEEEnosubnumber{\@ifstar{\global\@IEEEsubeqnumpersistfalse\@IEEEnosubnumber}{\@IEEEnosubnumber}}
+%
+\def\@IEEEnosubnumber{\if@IEEEeqnarrayISinner% alter counters and label only inside an IEEEeqnarray
+  \if@eqnsw % we do nothing unless we know we will display because we play with the counters here
+    % if it currently is a subequation, bump up to the next equation number and turn off the subequation
+    \ifnum\c@IEEEsubequation>0\relax\addtocounter{equation}{1}\setcounter{IEEEsubequation}{0}\relax
+    \fi
+    \global\@IEEEeqnumpreadvfalse\global\@IEEEsubeqnumpreadvfalse% no longer a preadv anymore
+    \gdef\@currentlabel{\p@equation\theequation}\relax
+    \gdef\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
+  \fi
+\fi}
+
+
+
+% allows users to "push away" equations that get too close to the equation numbers
+\def\IEEEeqnarraynumspace{\hphantom{\ifnum\c@IEEEsubequation>0\relax\theIEEEsubequationdis\else\theequationdis\fi}}
+
+% provides a way to span multiple columns within IEEEeqnarray environments
+% will consider \if@advanceIEEEeqncolcnt before globally advancing the
+% column counter - so as to work within \IEEEeqnarraybox
+% usage: \IEEEeqnarraymulticol{number cols. to span}{col type}{cell text}
+\long\def\IEEEeqnarraymulticol#1#2#3{\multispan{#1}\relax
+% check if column is defined for the precolumn definition
+% We have to be careful here because TeX scans for & even within an \iffalse
+% where it does not expand macros. So, if we used only one \ifx and a #3
+% appeared in the false branch and the user inserted another alignment
+% structure that uses & in the \IEEEeqnarraymulticol{}, TeX will not see that
+% there is an inner alignment in the false branch yet still will see any &
+% there and will think that they apply to the outer alignment resulting in an
+% incomplete \ifx error.
+% So, here we use separate checks for the pre and post parts in order to keep
+% the #3 outside of all conditionals.
+\relax\expandafter\ifx\csname @IEEEeqnarraycolDEF#2\endcsname\@IEEEeqnarraycolisdefined\relax
+\csname @IEEEeqnarraycolPRE#2\endcsname
+\else% if not, error and use default type
+\@IEEEclspkgerror{Invalid column type "#2" in \string\IEEEeqnarraymulticol.\MessageBreak
+Using a default centering column instead}%
+{You must define IEEEeqnarray column types before use.}%
+\csname @IEEEeqnarraycolPRE@IEEEdefault\endcsname
+\fi
+% The ten \relax are to help prevent misleading error messages in case a user
+% accidently inserted a macro that tries to acquire additional arguments.
+#3\relax\relax\relax\relax\relax\relax\relax\relax\relax\relax
+% check if column is defined for the postcolumn definition
+\expandafter\ifx\csname @IEEEeqnarraycolDEF#2\endcsname\@IEEEeqnarraycolisdefined\relax
+\csname @IEEEeqnarraycolPOST#2\endcsname
+\else% if not, use the default type
+\csname @IEEEeqnarraycolPOST@IEEEdefault\endcsname
+\fi
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by #1\relax\fi}
+
+% like \omit, but maintains track of the column counter for \IEEEeqnarray
+\def\IEEEeqnarrayomit{\omit\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by 1\relax\fi}
+
+
+% provides a way to define a letter referenced column type
+% usage: \IEEEeqnarraydefcol{col. type letter/name}{pre insertion text}{post insertion text}
+\def\IEEEeqnarraydefcol#1#2#3{\expandafter\def\csname @IEEEeqnarraycolPRE#1\endcsname{#2}%
+\expandafter\def\csname @IEEEeqnarraycolPOST#1\endcsname{#3}%
+\expandafter\def\csname @IEEEeqnarraycolDEF#1\endcsname{1}}
+
+
+% provides a way to define a numerically referenced inter-column glue types
+% usage: \IEEEeqnarraydefcolsep{col. glue number}{glue definition}
+\def\IEEEeqnarraydefcolsep#1#2{\expandafter\def\csname @IEEEeqnarraycolSEP\romannumeral #1\endcsname{#2}%
+\expandafter\def\csname @IEEEeqnarraycolSEPDEF\romannumeral #1\endcsname{1}}
+
+
+\def\@IEEEeqnarraycolisdefined{1}% just a macro for 1, used for checking undefined column types
+
+
+% expands and appends the given argument to the \@IEEEtrantmptoksA token list
+% used to build up the \halign preamble
+\def\@IEEEappendtoksA#1{\edef\@@IEEEappendtoksA{\@IEEEtrantmptoksA={\the\@IEEEtrantmptoksA #1}}%
+\@@IEEEappendtoksA}
+
+% also appends to \@IEEEtrantmptoksA, but does not expand the argument
+% uses \toks8 as a scratchpad register
+\def\@IEEEappendNOEXPANDtoksA#1{\toks8={#1}%
+\edef\@@IEEEappendNOEXPANDtoksA{\@IEEEtrantmptoksA={\the\@IEEEtrantmptoksA\the\toks8}}%
+\@@IEEEappendNOEXPANDtoksA}
+
+% define some common column types for the user
+% math
+\IEEEeqnarraydefcol{l}{$\IEEEeqnarraymathstyle}{$\hfil}
+\IEEEeqnarraydefcol{c}{\hfil$\IEEEeqnarraymathstyle}{$\hfil}
+\IEEEeqnarraydefcol{r}{\hfil$\IEEEeqnarraymathstyle}{$}
+\IEEEeqnarraydefcol{L}{$\IEEEeqnarraymathstyle{}}{{}$\hfil}
+\IEEEeqnarraydefcol{C}{\hfil$\IEEEeqnarraymathstyle{}}{{}$\hfil}
+\IEEEeqnarraydefcol{R}{\hfil$\IEEEeqnarraymathstyle{}}{{}$}
+% text
+\IEEEeqnarraydefcol{s}{\IEEEeqnarraytextstyle}{\hfil}
+\IEEEeqnarraydefcol{t}{\hfil\IEEEeqnarraytextstyle}{\hfil}
+\IEEEeqnarraydefcol{u}{\hfil\IEEEeqnarraytextstyle}{}
+
+% vertical rules
+\IEEEeqnarraydefcol{v}{}{\vrule width\arrayrulewidth}
+\IEEEeqnarraydefcol{vv}{\vrule width\arrayrulewidth\hfil}{\hfil\vrule width\arrayrulewidth}
+\IEEEeqnarraydefcol{V}{}{\vrule width\arrayrulewidth\hskip\doublerulesep\vrule width\arrayrulewidth}
+\IEEEeqnarraydefcol{VV}{\vrule width\arrayrulewidth\hskip\doublerulesep\vrule width\arrayrulewidth\hfil}%
+{\hfil\vrule width\arrayrulewidth\hskip\doublerulesep\vrule width\arrayrulewidth}
+
+% horizontal rules
+\IEEEeqnarraydefcol{h}{}{\leaders\hrule height\arrayrulewidth\hfil}
+\IEEEeqnarraydefcol{H}{}{\leaders\vbox{\hrule width\arrayrulewidth\vskip\doublerulesep\hrule width\arrayrulewidth}\hfil}
+
+% plain
+\IEEEeqnarraydefcol{x}{}{}
+\IEEEeqnarraydefcol{X}{$}{$}
+
+% the default column type to use in the event a column type is not defined
+\IEEEeqnarraydefcol{@IEEEdefault}{\hfil$\IEEEeqnarraymathstyle}{$\hfil}
+
+
+% a zero tabskip (used for "-" col types)
+\def\@IEEEeqnarraycolSEPzero{0pt plus 0pt minus 0pt}
+% a centering tabskip (used for "+" col types)
+\def\@IEEEeqnarraycolSEPcenter{1000pt plus 0pt minus 1000pt}
+
+% top level default tabskip glues for the start, end, and inter-column
+% may be reset within environments not always at the top level, e.g., \IEEEeqnarraybox
+\edef\@IEEEeqnarraycolSEPdefaultstart{\@IEEEeqnarraycolSEPcenter}% default start glue
+\edef\@IEEEeqnarraycolSEPdefaultend{\@IEEEeqnarraycolSEPcenter}% default end glue
+\edef\@IEEEeqnarraycolSEPdefaultmid{\@IEEEeqnarraycolSEPzero}% default inter-column glue
+
+
+
+% creates a vertical rule that extends from the bottom to the top a a cell
+% Provided in case other packages redefine \vline some other way.
+% usage: \IEEEeqnarrayvrule[rule thickness]
+% If no argument is provided, \arrayrulewidth will be used for the rule thickness. 
+\newcommand\IEEEeqnarrayvrule[1][\arrayrulewidth]{\vrule\@width#1\relax}
+
+% creates a blank separator row
+% usage: \IEEEeqnarrayseprow[separation length][font size commands]
+% default is \IEEEeqnarrayseprow[0.25\normalbaselineskip][\relax]
+% blank arguments inherit the default values
+% uses \skip5 as a scratch register - calls \@IEEEeqnarraystrutsize which uses more scratch registers
+\def\IEEEeqnarrayseprow{\relax\@ifnextchar[{\@IEEEeqnarrayseprow}{\@IEEEeqnarrayseprow[0.25\normalbaselineskip]}}
+\def\@IEEEeqnarrayseprow[#1]{\relax\@ifnextchar[{\@@IEEEeqnarrayseprow[#1]}{\@@IEEEeqnarrayseprow[#1][\relax]}}
+\def\@@IEEEeqnarrayseprow[#1][#2]{\def\@IEEEeqnarrayseprowARGONE{#1}%
+\ifx\@IEEEeqnarrayseprowARGONE\@empty%
+% get the skip value, based on the font commands
+% use skip5 because \IEEEeqnarraystrutsize uses \skip0, \skip2, \skip3
+% assign within a bogus box to confine the font changes
+{\setbox0=\hbox{#2\relax\global\skip5=0.25\normalbaselineskip}}%
+\else%
+{\setbox0=\hbox{#2\relax\global\skip5=#1}}%
+\fi%
+\@IEEEeqnarrayhoptolastcolumn\IEEEeqnarraystrutsize{\skip5}{0pt}[\relax]\relax}
+
+% creates a blank separator row, but omits all the column templates
+% usage: \IEEEeqnarrayseprowcut[separation length][font size commands]
+% default is \IEEEeqnarrayseprowcut[0.25\normalbaselineskip][\relax]
+% blank arguments inherit the default values
+% uses \skip5 as a scratch register - calls \@IEEEeqnarraystrutsize which uses more scratch registers
+\def\IEEEeqnarrayseprowcut{\multispan{\@IEEEeqnnumcols}\relax% span all the cols
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+\@ifnextchar[{\@IEEEeqnarrayseprowcut}{\@IEEEeqnarrayseprowcut[0.25\normalbaselineskip]}}
+\def\@IEEEeqnarrayseprowcut[#1]{\relax\@ifnextchar[{\@@IEEEeqnarrayseprowcut[#1]}{\@@IEEEeqnarrayseprowcut[#1][\relax]}}
+\def\@@IEEEeqnarrayseprowcut[#1][#2]{\def\@IEEEeqnarrayseprowARGONE{#1}%
+\ifx\@IEEEeqnarrayseprowARGONE\@empty%
+% get the skip value, based on the font commands
+% use skip5 because \IEEEeqnarraystrutsize uses \skip0, \skip2, \skip3
+% assign within a bogus box to confine the font changes
+{\setbox0=\hbox{#2\relax\global\skip5=0.25\normalbaselineskip}}%
+\else%
+{\setbox0=\hbox{#2\relax\global\skip5=#1}}%
+\fi%
+\IEEEeqnarraystrutsize{\skip5}{0pt}[\relax]\relax}
+
+
+
+% draws a single rule across all the columns optional
+% argument determines the rule width, \arrayrulewidth is the default
+% updates column counter as needed and turns off struts
+% usage: \IEEEeqnarrayrulerow[rule line thickness]
+\def\IEEEeqnarrayrulerow{\multispan{\@IEEEeqnnumcols}\relax% span all the cols
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+\@ifnextchar[{\@IEEEeqnarrayrulerow}{\@IEEEeqnarrayrulerow[\arrayrulewidth]}}
+\def\@IEEEeqnarrayrulerow[#1]{\leaders\hrule height#1\hfil\relax% put in our rule 
+% turn off any struts
+\IEEEeqnarraystrutsize{0pt}{0pt}[\relax]\relax}
+
+
+% draws a double rule by using a single rule row, a separator row, and then
+% another single rule row 
+% first optional argument determines the rule thicknesses, \arrayrulewidth is the default
+% second optional argument determines the rule spacing, \doublerulesep is the default
+% usage: \IEEEeqnarraydblrulerow[rule line thickness][rule spacing]
+\def\IEEEeqnarraydblrulerow{\multispan{\@IEEEeqnnumcols}\relax% span all the cols
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+\@ifnextchar[{\@IEEEeqnarraydblrulerow}{\@IEEEeqnarraydblrulerow[\arrayrulewidth]}}
+\def\@IEEEeqnarraydblrulerow[#1]{\relax\@ifnextchar[{\@@IEEEeqnarraydblrulerow[#1]}%
+{\@@IEEEeqnarraydblrulerow[#1][\doublerulesep]}}
+\def\@@IEEEeqnarraydblrulerow[#1][#2]{\def\@IEEEeqnarraydblrulerowARG{#1}%
+% we allow the user to say \IEEEeqnarraydblrulerow[][]
+\ifx\@IEEEeqnarraydblrulerowARG\@empty%
+\@IEEEeqnarrayrulerow[\arrayrulewidth]%
+\else%
+\@IEEEeqnarrayrulerow[#1]\relax%
+\fi%
+\def\@IEEEeqnarraydblrulerowARG{#2}%
+\ifx\@IEEEeqnarraydblrulerowARG\@empty%
+\\\IEEEeqnarrayseprow[\doublerulesep][\relax]%
+\else%
+\\\IEEEeqnarrayseprow[#2][\relax]%
+\fi%
+\\\multispan{\@IEEEeqnnumcols}%
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+\def\@IEEEeqnarraydblrulerowARG{#1}%
+\ifx\@IEEEeqnarraydblrulerowARG\@empty%
+\@IEEEeqnarrayrulerow[\arrayrulewidth]%
+\else%
+\@IEEEeqnarrayrulerow[#1]%
+\fi%
+}
+
+% draws a double rule by using a single rule row, a separator (cutting) row, and then
+% another single rule row 
+% first optional argument determines the rule thicknesses, \arrayrulewidth is the default
+% second optional argument determines the rule spacing, \doublerulesep is the default
+% usage: \IEEEeqnarraydblrulerow[rule line thickness][rule spacing]
+\def\IEEEeqnarraydblrulerowcut{\multispan{\@IEEEeqnnumcols}\relax% span all the cols
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+\@ifnextchar[{\@IEEEeqnarraydblrulerowcut}{\@IEEEeqnarraydblrulerowcut[\arrayrulewidth]}}
+\def\@IEEEeqnarraydblrulerowcut[#1]{\relax\@ifnextchar[{\@@IEEEeqnarraydblrulerowcut[#1]}%
+{\@@IEEEeqnarraydblrulerowcut[#1][\doublerulesep]}}
+\def\@@IEEEeqnarraydblrulerowcut[#1][#2]{\def\@IEEEeqnarraydblrulerowARG{#1}%
+% we allow the user to say \IEEEeqnarraydblrulerow[][]
+\ifx\@IEEEeqnarraydblrulerowARG\@empty%
+\@IEEEeqnarrayrulerow[\arrayrulewidth]%
+\else%
+\@IEEEeqnarrayrulerow[#1]%
+\fi%
+\def\@IEEEeqnarraydblrulerowARG{#2}%
+\ifx\@IEEEeqnarraydblrulerowARG\@empty%
+\\\IEEEeqnarrayseprowcut[\doublerulesep][\relax]%
+\else%
+\\\IEEEeqnarrayseprowcut[#2][\relax]%
+\fi%
+\\\multispan{\@IEEEeqnnumcols}%
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+\def\@IEEEeqnarraydblrulerowARG{#1}%
+\ifx\@IEEEeqnarraydblrulerowARG\@empty%
+\@IEEEeqnarrayrulerow[\arrayrulewidth]%
+\else%
+\@IEEEeqnarrayrulerow[#1]%
+\fi%
+}
+
+
+
+% inserts a full row's worth of &'s
+% relies on \@IEEEeqnnumcols to provide the correct number of columns
+% uses \@IEEEtrantmptoksA, \count0 as scratch registers
+\def\@IEEEeqnarrayhoptolastcolumn{\@IEEEtrantmptoksA={}\count0=1\relax%
+\loop% add cols if the user did not use them all
+\ifnum\count0<\@IEEEeqnnumcols\relax%
+\@IEEEappendtoksA{&}%
+\advance\count0 by 1\relax% update the col count
+\repeat%
+\the\@IEEEtrantmptoksA%execute the &'s
+}
+
+
+
+\newif\if@IEEEeqnarrayISinner % flag to indicate if we are within the lines
+\@IEEEeqnarrayISinnerfalse    % of an IEEEeqnarray - after the IEEEeqnarraydecl
+
+\edef\@IEEEeqnarrayTHEstrutheight{0pt} % height and depth of IEEEeqnarray struts
+\edef\@IEEEeqnarrayTHEstrutdepth{0pt}
+
+\edef\@IEEEeqnarrayTHEmasterstrutheight{0pt} % default height and depth of
+\edef\@IEEEeqnarrayTHEmasterstrutdepth{0pt}  % struts within an IEEEeqnarray
+
+\edef\@IEEEeqnarrayTHEmasterstrutHSAVE{0pt} % saved master strut height
+\edef\@IEEEeqnarrayTHEmasterstrutDSAVE{0pt} % and depth
+
+\newif\if@IEEEeqnarrayusemasterstrut % flag to indicate that the master strut value
+\@IEEEeqnarrayusemasterstruttrue     % is to be used
+
+
+
+% saves the strut height and depth of the master strut
+\def\@IEEEeqnarraymasterstrutsave{\relax%
+\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
+% remove stretchability
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+% save values
+\edef\@IEEEeqnarrayTHEmasterstrutHSAVE{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEmasterstrutDSAVE{\the\dimen2}}
+
+% restores the strut height and depth of the master strut
+\def\@IEEEeqnarraymasterstrutrestore{\relax%
+\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutHSAVE\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutDSAVE\relax%
+% remove stretchability
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+% restore values
+\edef\@IEEEeqnarrayTHEmasterstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEmasterstrutdepth{\the\dimen2}}
+
+
+% globally restores the strut height and depth to the 
+% master values and sets the master strut flag to true
+\def\@IEEEeqnarraystrutreset{\relax%
+\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
+% remove stretchability
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+% restore values
+\xdef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+\xdef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+\global\@IEEEeqnarrayusemasterstruttrue}
+
+
+% if the master strut is not to be used, make the current
+% values of \@IEEEeqnarrayTHEstrutheight, \@IEEEeqnarrayTHEstrutdepth
+% and the use master strut flag, global
+% this allows user strut commands issued in the last column to be carried
+% into the isolation/strut column
+\def\@IEEEeqnarrayglobalizestrutstatus{\relax%
+\if@IEEEeqnarrayusemasterstrut\else%
+\xdef\@IEEEeqnarrayTHEstrutheight{\@IEEEeqnarrayTHEstrutheight}%
+\xdef\@IEEEeqnarrayTHEstrutdepth{\@IEEEeqnarrayTHEstrutdepth}%
+\global\@IEEEeqnarrayusemasterstrutfalse%
+\fi}
+
+
+
+% usage: \IEEEeqnarraystrutsize{height}{depth}[font size commands]
+% If called outside the lines of an IEEEeqnarray, sets the height
+% and depth of both the master and local struts. If called inside
+% an IEEEeqnarray line, sets the height and depth of the local strut
+% only and sets the flag to indicate the use of the local strut
+% values. If the height or depth is left blank, 0.7\normalbaselineskip
+% and 0.3\normalbaselineskip will be used, respectively.
+% The optional argument can be used to evaluate the lengths under
+% a different font size and styles. If none is specified, the current
+% font is used.
+% uses scratch registers \skip0, \skip2, \skip3, \dimen0, \dimen2
+\def\IEEEeqnarraystrutsize#1#2{\relax\@ifnextchar[{\@IEEEeqnarraystrutsize{#1}{#2}}{\@IEEEeqnarraystrutsize{#1}{#2}[\relax]}}
+\def\@IEEEeqnarraystrutsize#1#2[#3]{\def\@IEEEeqnarraystrutsizeARG{#1}%
+\ifx\@IEEEeqnarraystrutsizeARG\@empty%
+{\setbox0=\hbox{#3\relax\global\skip3=0.7\normalbaselineskip}}%
+\skip0=\skip3\relax%
+\else% arg one present
+{\setbox0=\hbox{#3\relax\global\skip3=#1\relax}}%
+\skip0=\skip3\relax%
+\fi% if null arg
+\def\@IEEEeqnarraystrutsizeARG{#2}%
+\ifx\@IEEEeqnarraystrutsizeARG\@empty%
+{\setbox0=\hbox{#3\relax\global\skip3=0.3\normalbaselineskip}}%
+\skip2=\skip3\relax%
+\else% arg two present
+{\setbox0=\hbox{#3\relax\global\skip3=#2\relax}}%
+\skip2=\skip3\relax%
+\fi% if null arg
+% remove stretchability, just to be safe
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+% dimen0 = height, dimen2 = depth
+\if@IEEEeqnarrayISinner% inner does not touch master strut size
+\edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+\@IEEEeqnarrayusemasterstrutfalse% do not use master
+\else% outer, have to set master strut too
+\edef\@IEEEeqnarrayTHEmasterstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEmasterstrutdepth{\the\dimen2}%
+\edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+\@IEEEeqnarrayusemasterstruttrue% use master strut
+\fi}
+
+
+% usage: \IEEEeqnarraystrutsizeadd{added height}{added depth}[font size commands]
+% If called outside the lines of an IEEEeqnarray, adds the given height
+% and depth to both the master and local struts.
+% If called inside an IEEEeqnarray line, adds the given height and depth
+% to the local strut only and sets the flag to indicate the use 
+% of the local strut values.
+% In both cases, if a height or depth is left blank, 0pt is used instead.
+% The optional argument can be used to evaluate the lengths under
+% a different font size and styles. If none is specified, the current
+% font is used.
+% uses scratch registers \skip0, \skip2, \skip3, \dimen0, \dimen2
+\def\IEEEeqnarraystrutsizeadd#1#2{\relax\@ifnextchar[{\@IEEEeqnarraystrutsizeadd{#1}{#2}}{\@IEEEeqnarraystrutsizeadd{#1}{#2}[\relax]}}
+\def\@IEEEeqnarraystrutsizeadd#1#2[#3]{\def\@IEEEeqnarraystrutsizearg{#1}%
+\ifx\@IEEEeqnarraystrutsizearg\@empty%
+\skip0=0pt\relax%
+\else% arg one present
+{\setbox0=\hbox{#3\relax\global\skip3=#1}}%
+\skip0=\skip3\relax%
+\fi% if null arg
+\def\@IEEEeqnarraystrutsizearg{#2}%
+\ifx\@IEEEeqnarraystrutsizearg\@empty%
+\skip2=0pt\relax%
+\else% arg two present
+{\setbox0=\hbox{#3\relax\global\skip3=#2}}%
+\skip2=\skip3\relax%
+\fi% if null arg
+% remove stretchability, just to be safe
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+% dimen0 = height, dimen2 = depth
+\if@IEEEeqnarrayISinner% inner does not touch master strut size
+% get local strut size
+\expandafter\skip0=\@IEEEeqnarrayTHEstrutheight\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEstrutdepth\relax%
+% add it to the user supplied values
+\advance\dimen0 by \skip0\relax%
+\advance\dimen2 by \skip2\relax%
+% update the local strut size
+\edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+\@IEEEeqnarrayusemasterstrutfalse% do not use master
+\else% outer, have to set master strut too
+% get master strut size
+\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
+% add it to the user supplied values
+\advance\dimen0 by \skip0\relax%
+\advance\dimen2 by \skip2\relax%
+% update the local and master strut sizes
+\edef\@IEEEeqnarrayTHEmasterstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEmasterstrutdepth{\the\dimen2}%
+\edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+\@IEEEeqnarrayusemasterstruttrue% use master strut
+\fi}
+
+
+% allow user a way to see the struts
+\newif\ifIEEEvisiblestruts
+\IEEEvisiblestrutsfalse
+
+% inserts an invisible strut using the master or local strut values
+% uses scratch registers \skip0, \skip2, \dimen0, \dimen2
+\def\@IEEEeqnarrayinsertstrut{\relax%
+\if@IEEEeqnarrayusemasterstrut
+% get master strut size
+\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
+\else%
+% get local strut size
+\expandafter\skip0=\@IEEEeqnarrayTHEstrutheight\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEstrutdepth\relax%
+\fi%
+% remove stretchability, probably not needed
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+% dimen0 = height, dimen2 = depth
+% allow user to see struts if desired
+\ifIEEEvisiblestruts%
+\vrule width0.2pt height\dimen0 depth\dimen2\relax%
+\else%
+\vrule width0pt height\dimen0 depth\dimen2\relax\fi}
+
+
+% creates an invisible strut, useable even outside \IEEEeqnarray
+% if \IEEEvisiblestrutstrue, the strut will be visible and 0.2pt wide. 
+% usage: \IEEEstrut[height][depth][font size commands]
+% default is \IEEEstrut[0.7\normalbaselineskip][0.3\normalbaselineskip][\relax]
+% blank arguments inherit the default values
+% uses \dimen0, \dimen2, \skip0, \skip2
+\def\IEEEstrut{\relax\@ifnextchar[{\@IEEEstrut}{\@IEEEstrut[0.7\normalbaselineskip]}}
+\def\@IEEEstrut[#1]{\relax\@ifnextchar[{\@@IEEEstrut[#1]}{\@@IEEEstrut[#1][0.3\normalbaselineskip]}}
+\def\@@IEEEstrut[#1][#2]{\relax\@ifnextchar[{\@@@IEEEstrut[#1][#2]}{\@@@IEEEstrut[#1][#2][\relax]}}
+\def\@@@IEEEstrut[#1][#2][#3]{\mbox{#3\relax%
+\def\@IEEEstrutARG{#1}%
+\ifx\@IEEEstrutARG\@empty%
+\skip0=0.7\normalbaselineskip\relax%
+\else%
+\skip0=#1\relax%
+\fi%
+\def\@IEEEstrutARG{#2}%
+\ifx\@IEEEstrutARG\@empty%
+\skip2=0.3\normalbaselineskip\relax%
+\else%
+\skip2=#2\relax%
+\fi%
+% remove stretchability, probably not needed
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+\ifIEEEvisiblestruts%
+\vrule width0.2pt height\dimen0 depth\dimen2\relax%
+\else%
+\vrule width0.0pt height\dimen0 depth\dimen2\relax\fi}}
+
+
+% enables strut mode by setting a default strut size and then zeroing the
+% \baselineskip, \lineskip, \lineskiplimit and \jot
+\def\IEEEeqnarraystrutmode{\IEEEeqnarraystrutsize{0.7\normalbaselineskip}{0.3\normalbaselineskip}[\relax]%
+\baselineskip=0pt\lineskip=0pt\lineskiplimit=0pt\jot=0pt}
+
+
+% equation and subequation forms to use to setup hyperref's \@currentHref
+\def\@IEEEtheHrefequation{equation.\theHequation}
+\def\@IEEEtheHrefsubequation{equation.\theHequation\alph{IEEEsubequation}}
+
+
+\def\IEEEeqnarray{\@IEEEeqnumpersisttrue\@IEEEsubeqnumpersistfalse\@IEEEeqnarray}
+\def\endIEEEeqnarray{\end@IEEEeqnarray}
+
+\@namedef{IEEEeqnarray*}{\@IEEEeqnumpersistfalse\@IEEEsubeqnumpersistfalse\@IEEEeqnarray}
+\@namedef{endIEEEeqnarray*}{\end@IEEEeqnarray}
+
+
+% \IEEEeqnarray is an enhanced \eqnarray. 
+% The star form defaults to not putting equation numbers at the end of each row.
+% usage: \IEEEeqnarray[decl]{cols}
+\def\@IEEEeqnarray{\relax\@ifnextchar[{\@@IEEEeqnarray}{\@@IEEEeqnarray[\relax]}}
+% We have to be careful here to normalize catcodes just before acquiring the
+% cols as that specification may contain punctuation which could be subject
+% to document catcode changes.
+\def\@@IEEEeqnarray[#1]{\begingroup\IEEEnormalcatcodes\@@@IEEEeqnarray[#1]}
+\def\@@@IEEEeqnarray[#1]#2{\endgroup
+   % default to showing the equation number or not based on whether or not
+   % the star form was involked
+   \if@IEEEeqnumpersist\global\@eqnswtrue
+   \else% not the star form
+   \global\@eqnswfalse
+   \fi% if star form
+   % provide a basic hyperref \theHequation if this has not already been setup (hyperref not loaded, or no section counter)
+   \@ifundefined{theHequation}{\def\theHequation{\arabic{equation}}}{}\relax
+   % provide dummy hyperref commands in case hyperref is not loaded
+   \providecommand{\Hy@raisedlink}[1]{}\relax
+   \providecommand{\hyper@anchorstart}[1]{}\relax
+   \providecommand{\hyper@anchorend}{}\relax
+   \providecommand{\@currentHref}{}\relax
+   \@IEEEeqnumpreadvfalse% reset eqnpreadv flag
+   \@IEEEsubeqnumpreadvfalse% reset subeqnpreadv flag
+   \@IEEEeqnarrayISinnerfalse% not yet within the lines of the halign
+   \@IEEEeqnarraystrutsize{0pt}{0pt}[\relax]% turn off struts by default
+   \@IEEEeqnarrayusemasterstruttrue% use master strut till user asks otherwise
+   \IEEEvisiblestrutsfalse% diagnostic mode defaults to off
+   % no extra space unless the user specifically requests it
+   \lineskip=0pt\relax
+   \lineskiplimit=0pt\relax
+   \baselineskip=\normalbaselineskip\relax%
+   \jot=\IEEEnormaljot\relax%
+   \mathsurround\z@\relax% no extra spacing around math
+   \@advanceIEEEeqncolcnttrue% advance the col counter for each col the user uses, 
+                             % used in \IEEEeqnarraymulticol and in the preamble build
+   %V1.8 Here we preadvance to the next equation number.
+   % If the user later wants a continued subequation, we can roll back.
+   \global\@IEEEsubeqnnumrollback=\c@IEEEsubequation%
+   \stepcounter{equation}\@IEEEeqnumpreadvtrue% advance equation counter before first line
+   \setcounter{IEEEsubequation}{0}% no subequation yet
+   \let\@IEEEcurrentlabelsave\@currentlabel% save current label as we later change it globally
+   \let\@IEEEcurrentHrefsave\@currentHref% save current href label as we later change it globally
+   \def\@currentlabel{\p@equation\theequation}% redefine the ref label
+   \def\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
+   \IEEEeqnarraydecl\relax% allow a way for the user to make global overrides
+   #1\relax% allow user to override defaults
+   \let\\\@IEEEeqnarraycr% replace newline with one that can put in eqn. numbers
+   \global\@IEEEeqncolcnt\z@% col. count = 0 for first line
+   \@IEEEbuildpreamble{#2}\relax% build the preamble and put it into \@IEEEtrantmptoksA 
+   % put in the column for the equation number
+   \ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi% col separator for those after the first
+   \toks0={##}%
+   % advance the \@IEEEeqncolcnt for the isolation col, this helps with error checking
+   \@IEEEappendtoksA{\global\advance\@IEEEeqncolcnt by 1\relax}%
+   % add the isolation column
+   \@IEEEappendtoksA{\tabskip\z@skip\bgroup\the\toks0\egroup}%
+   % advance the \@IEEEeqncolcnt for the equation number col, this helps with error checking
+   \@IEEEappendtoksA{&\global\advance\@IEEEeqncolcnt by 1\relax}%
+   % add the equation number col to the preamble
+   \@IEEEappendtoksA{\tabskip\z@skip\hb@xt@\z@\bgroup\hss\the\toks0\egroup}%
+   % note \@IEEEeqnnumcols does not count the equation col or isolation col
+   % set the starting tabskip glue as determined by the preamble build
+   \tabskip=\@IEEEBPstartglue\relax
+   % begin the display alignment
+   \@IEEEeqnarrayISinnertrue% commands are now within the lines
+   $$\everycr{}\halign to\displaywidth\bgroup
+   % "exspand" the preamble
+   \span\the\@IEEEtrantmptoksA\cr}
+
+% enter isolation/strut column (or the next column if the user did not use
+% every column), record the strut status, complete the columns, do the strut if needed,
+% restore counters (to backout any equation setup for a next line that was never used)
+% to their correct values and exit
+\def\end@IEEEeqnarray{\@IEEEeqnarrayglobalizestrutstatus&\@@IEEEeqnarraycr\egroup
+\if@IEEEsubeqnumpreadv\global\advance\c@IEEEsubequation\m@ne\fi
+\if@IEEEeqnumpreadv\global\advance\c@equation\m@ne\global\c@IEEEsubequation=\@IEEEsubeqnnumrollback\fi
+\global\let\@currentlabel\@IEEEcurrentlabelsave% restore current label
+\global\let\@currentHref\@IEEEcurrentHrefsave% restore current href label
+$$\@ignoretrue}
+
+
+% IEEEeqnarray uses a modifed \\ instead of the plain \cr to
+% end rows. This allows for things like \\*[vskip amount]
+% These "cr" macros are modified versions of those for LaTeX2e's eqnarray
+% the {\ifnum0=`} braces must be kept away from the last column to avoid
+% altering spacing of its math, so we use & to advance to the next column
+% as there is an isolation/strut column after the user's columns
+\def\@IEEEeqnarraycr{\@IEEEeqnarrayglobalizestrutstatus&% save strut status and advance to next column
+   {\ifnum0=`}\fi
+   \@ifstar{%
+      \global\@eqpen\@M\@IEEEeqnarrayYCR
+   }{%
+      \global\@eqpen\interdisplaylinepenalty \@IEEEeqnarrayYCR
+   }%
+}
+
+\def\@IEEEeqnarrayYCR{\@testopt\@IEEEeqnarrayXCR\z@skip}
+
+\def\@IEEEeqnarrayXCR[#1]{%
+   \ifnum0=`{\fi}%
+   \@@IEEEeqnarraycr
+   \noalign{\penalty\@eqpen\vskip\jot\vskip #1\relax}}%
+
+\def\@@IEEEeqnarraycr{\@IEEEtrantmptoksA={}% clear token register
+    \advance\@IEEEeqncolcnt by -1\relax% adjust col count because of the isolation column
+    \ifnum\@IEEEeqncolcnt>\@IEEEeqnnumcols\relax
+    \@IEEEclspkgerror{Too many columns within the IEEEeqnarray\MessageBreak
+                          environment}%
+    {Use fewer \string &'s or put more columns in the IEEEeqnarray column\MessageBreak 
+     specifications.}\relax%
+    \else
+    \loop% add cols if the user did not use them all
+    \ifnum\@IEEEeqncolcnt<\@IEEEeqnnumcols\relax
+    \@IEEEappendtoksA{&}%
+    \advance\@IEEEeqncolcnt by 1\relax% update the col count
+    \repeat
+    % this number of &'s will take us the the isolation column
+    \fi
+    % execute the &'s
+    \the\@IEEEtrantmptoksA%
+    % handle the strut/isolation column
+    \@IEEEeqnarrayinsertstrut% do the strut if needed
+    \@IEEEeqnarraystrutreset% reset the strut system for next line or IEEEeqnarray
+    &% and enter the equation number column
+    \if@eqnsw% only if we display something
+      \Hy@raisedlink{\hyper@anchorstart{\@currentHref}}% start a hyperref anchor
+      \global\@IEEEeqnumpreadvfalse\relax% displaying an equation number means
+      \global\@IEEEsubeqnumpreadvfalse\relax% the equation counters point to valid equations
+      % V1.8 Here we setup the counters, currentlabel and status for what would be the *next*
+      % equation line as would be the case under the current settings. However, there are two problems.
+      % One problem is that there might not ever be a next line. The second problem is that the user
+      % may later alter the meaning of a line with commands such as \IEEEyessubnumber. So, to handle
+      % these cases we have to record the current values of the (sub)equation counters and revert back
+      % to them if the next line is changed or never comes. The \if@IEEEeqnumpreadv, \if@IEEEsubeqnumpreadv
+      % and \@IEEEsubeqnnumrollback stuff tracks this.
+      % The logic to handle all this is surprisingly complex, but a nice feature of the approach here is
+      % that the equation counters and labels remain valid for what the line would be unless a
+      % \IEEEyessubnumber et al. later changes it. So, any hyperref links are always correct.
+      \ifnum\c@IEEEsubequation>0\relax% handle subequation
+         \theIEEEsubequationdis\relax
+         \if@IEEEsubeqnumpersist% setup for default type of next line
+            \stepcounter{IEEEsubequation}\global\@IEEEsubeqnumpreadvtrue\relax
+            \gdef\@currentlabel{\p@IEEEsubequation\theIEEEsubequation}\relax
+            \gdef\@currentHref{\@IEEEtheHrefsubequation}% setup hyperref label
+         \else
+             % if no subeqnum persist, go ahead and setup for a new equation number
+             \global\@IEEEsubeqnnumrollback=\c@IEEEsubequation
+             \stepcounter{equation}\global\@IEEEeqnumpreadvtrue\relax
+             \setcounter{IEEEsubequation}{0}\gdef\@currentlabel{\p@equation\theequation}\relax
+             \gdef\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
+         \fi
+      \else% display a standard equation number
+        \theequationdis\relax
+        \setcounter{IEEEsubequation}{0}\relax% not really needed
+        \if@IEEEsubeqnumpersist% setup for default type of next line
+           % subequations that follow plain equations carry the same equation number e.g, 5, 5a rather than 5, 6a
+           \stepcounter{IEEEsubequation}\global\@IEEEsubeqnumpreadvtrue\relax
+           \gdef\@currentlabel{\p@IEEEsubequation\theIEEEsubequation}\relax
+           \gdef\@currentHref{\@IEEEtheHrefsubequation}% setup hyperref label
+         \else
+             % if no subeqnum persist, go ahead and setup for a new equation number
+             \global\@IEEEsubeqnnumrollback=\c@IEEEsubequation
+             \stepcounter{equation}\global\@IEEEeqnumpreadvtrue\relax
+             \setcounter{IEEEsubequation}{0}\gdef\@currentlabel{\p@equation\theequation}\relax
+             \gdef\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
+         \fi
+      \fi%
+      \Hy@raisedlink{\hyper@anchorend}% end hyperref anchor
+    \fi% fi only if we display something
+    % reset the flags to indicate the default preferences of the display of equation numbers
+    \if@IEEEeqnumpersist\global\@eqnswtrue\else\global\@eqnswfalse\fi
+    \if@IEEEsubeqnumpersist\global\@eqnswtrue\fi% ditto for the subequation flag
+    % reset the number of columns the user actually used
+    \global\@IEEEeqncolcnt\z@\relax
+    % the real end of the line
+    \cr}
+
+
+
+
+
+% \IEEEeqnarraybox is like \IEEEeqnarray except the box form puts everything
+% inside a vtop, vbox, or vcenter box depending on the letter in the second
+% optional argument (t,b,c). Vbox is the default. Unlike \IEEEeqnarray,
+% equation numbers are not displayed and \IEEEeqnarraybox can be nested.
+% \IEEEeqnarrayboxm is for math mode (like \array) and does not put the vbox
+% within an hbox.
+% \IEEEeqnarrayboxt is for text mode (like \tabular) and puts the vbox within
+% a \hbox{$ $} construct.
+% \IEEEeqnarraybox will auto detect whether to use \IEEEeqnarrayboxm or 
+% \IEEEeqnarrayboxt depending on the math mode.
+% The third optional argument specifies the width this box is to be set to -
+% natural width is the default.
+% The * forms do not add \jot line spacing
+% usage: \IEEEeqnarraybox[decl][pos][width]{cols}
+\def\IEEEeqnarrayboxm{\@IEEEeqnarrayboxnojotfalse\@IEEEeqnarrayboxHBOXSWfalse\@IEEEeqnarraybox}
+\def\endIEEEeqnarrayboxm{\end@IEEEeqnarraybox}
+\@namedef{IEEEeqnarrayboxm*}{\@IEEEeqnarrayboxnojottrue\@IEEEeqnarrayboxHBOXSWfalse\@IEEEeqnarraybox}
+\@namedef{endIEEEeqnarrayboxm*}{\end@IEEEeqnarraybox}
+
+\def\IEEEeqnarrayboxt{\@IEEEeqnarrayboxnojotfalse\@IEEEeqnarrayboxHBOXSWtrue\@IEEEeqnarraybox}
+\def\endIEEEeqnarrayboxt{\end@IEEEeqnarraybox}
+\@namedef{IEEEeqnarrayboxt*}{\@IEEEeqnarrayboxnojottrue\@IEEEeqnarrayboxHBOXSWtrue\@IEEEeqnarraybox}
+\@namedef{endIEEEeqnarrayboxt*}{\end@IEEEeqnarraybox}
+
+\def\IEEEeqnarraybox{\@IEEEeqnarrayboxnojotfalse\ifmmode\@IEEEeqnarrayboxHBOXSWfalse\else\@IEEEeqnarrayboxHBOXSWtrue\fi%
+\@IEEEeqnarraybox}
+\def\endIEEEeqnarraybox{\end@IEEEeqnarraybox}
+
+\@namedef{IEEEeqnarraybox*}{\@IEEEeqnarrayboxnojottrue\ifmmode\@IEEEeqnarrayboxHBOXSWfalse\else\@IEEEeqnarrayboxHBOXSWtrue\fi%
+\@IEEEeqnarraybox}
+\@namedef{endIEEEeqnarraybox*}{\end@IEEEeqnarraybox}
+
+% flag to indicate if the \IEEEeqnarraybox needs to put things into an hbox{$ $} 
+% for \vcenter in non-math mode
+\newif\if@IEEEeqnarrayboxHBOXSW%
+\@IEEEeqnarrayboxHBOXSWfalse
+
+\def\@IEEEeqnarraybox{\relax\@ifnextchar[{\@@IEEEeqnarraybox}{\@@IEEEeqnarraybox[\relax]}}
+% We have to be careful here to normalize catcodes just before acquiring the
+% cols as that specification may contain punctuation which could be subject
+% to document catcode changes.
+\def\@@IEEEeqnarraybox[#1]{\relax\begingroup\IEEEnormalcatcodes\@ifnextchar[{\@@@IEEEeqnarraybox[#1]}{\@@@IEEEeqnarraybox[#1][b]}}
+\def\@@@IEEEeqnarraybox[#1][#2]{\relax\@ifnextchar[{\@@@@IEEEeqnarraybox[#1][#2]}{\@@@@IEEEeqnarraybox[#1][#2][\relax]}}
+
+% #1 = decl; #2 = t,b,c; #3 = width, #4 = col specs
+\def\@@@@IEEEeqnarraybox[#1][#2][#3]#4{\endgroup\@IEEEeqnarrayISinnerfalse % not yet within the lines of the halign
+   \@IEEEeqnarraymasterstrutsave% save current master strut values
+   \@IEEEeqnarraystrutsize{0pt}{0pt}[\relax]% turn off struts by default
+   \@IEEEeqnarrayusemasterstruttrue% use master strut till user asks otherwise
+   \IEEEvisiblestrutsfalse% diagnostic mode defaults to off
+   % no extra space unless the user specifically requests it
+   \lineskip=0pt\relax%
+   \lineskiplimit=0pt\relax%
+   \baselineskip=\normalbaselineskip\relax%
+   \jot=\IEEEnormaljot\relax%
+   \mathsurround\z@\relax% no extra spacing around math
+   % the default end glues are zero for an \IEEEeqnarraybox
+   \edef\@IEEEeqnarraycolSEPdefaultstart{\@IEEEeqnarraycolSEPzero}% default start glue
+   \edef\@IEEEeqnarraycolSEPdefaultend{\@IEEEeqnarraycolSEPzero}% default end glue
+   \edef\@IEEEeqnarraycolSEPdefaultmid{\@IEEEeqnarraycolSEPzero}% default inter-column glue
+   \@advanceIEEEeqncolcntfalse% do not advance the col counter for each col the user uses, 
+                              % used in \IEEEeqnarraymulticol and in the preamble build
+   \IEEEeqnarrayboxdecl\relax% allow a way for the user to make global overrides
+   #1\relax% allow user to override defaults
+   \let\\\@IEEEeqnarrayboxcr% replace newline with one that allows optional spacing
+   \@IEEEbuildpreamble{#4}\relax% build the preamble and put it into \@IEEEtrantmptoksA
+   % add an isolation column to the preamble to stop \\'s {} from getting into the last col
+   \ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi% col separator for those after the first
+   \toks0={##}%
+   % add the isolation column to the preamble
+   \@IEEEappendtoksA{\tabskip\z@skip\bgroup\the\toks0\egroup}% 
+   % set the starting tabskip glue as determined by the preamble build
+   \tabskip=\@IEEEBPstartglue\relax
+   % begin the alignment
+   \everycr{}%
+   % use only the very first token to determine the positioning
+   \@IEEEextracttoken{#2}\relax
+   \ifx\@IEEEextractedtokensdiscarded\@empty\else
+     \typeout{** WARNING: IEEEeqnarraybox position specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+   \fi
+   % \@IEEEextractedtoken has the first token, the rest are ignored
+   % if we need to put things into and hbox and go into math mode, do so now
+   \if@IEEEeqnarrayboxHBOXSW \leavevmode \hbox \bgroup $\fi%
+   % use the appropriate vbox type
+   \if\@IEEEextractedtoken t\relax\vtop\else\if\@IEEEextractedtoken c\relax%
+   \vcenter\else\vbox\fi\fi\bgroup%
+   \@IEEEeqnarrayISinnertrue% commands are now within the lines
+   \ifx#3\relax\halign\else\halign to #3\relax\fi%
+   \bgroup
+   % "exspand" the preamble
+   \span\the\@IEEEtrantmptoksA\cr}
+
+% carry strut status and enter the isolation/strut column, 
+% exit from math mode if needed, and exit
+\def\end@IEEEeqnarraybox{\@IEEEeqnarrayglobalizestrutstatus% carry strut status
+&% enter isolation/strut column
+\@IEEEeqnarrayinsertstrut% do strut if needed
+\@IEEEeqnarraymasterstrutrestore% restore the previous master strut values
+% reset the strut system for next IEEEeqnarray
+% (sets local strut values back to previous master strut values)
+\@IEEEeqnarraystrutreset%
+% ensure last line, exit from halign, close vbox
+\crcr\egroup\egroup%
+% exit from math mode and close hbox if needed
+\if@IEEEeqnarrayboxHBOXSW $\egroup\fi}
+
+
+
+% IEEEeqnarraybox uses a modifed \\ instead of the plain \cr to
+% end rows. This allows for things like \\[vskip amount]
+% This "cr" macros are modified versions those for LaTeX2e's eqnarray
+% For IEEEeqnarraybox, \\* is the same as \\
+% the {\ifnum0=`} braces must be kept away from the last column to avoid
+% altering spacing of its math, so we use & to advance to the isolation/strut column
+% carry strut status into isolation/strut column
+\def\@IEEEeqnarrayboxcr{\@IEEEeqnarrayglobalizestrutstatus% carry strut status
+&% enter isolation/strut column
+\@IEEEeqnarrayinsertstrut% do strut if needed
+% reset the strut system for next line or IEEEeqnarray
+\@IEEEeqnarraystrutreset%
+{\ifnum0=`}\fi%
+\@ifstar{\@IEEEeqnarrayboxYCR}{\@IEEEeqnarrayboxYCR}}
+
+% test and setup the optional argument to \\[]
+\def\@IEEEeqnarrayboxYCR{\@testopt\@IEEEeqnarrayboxXCR\z@skip}
+
+% IEEEeqnarraybox does not automatically increase line spacing by \jot
+\def\@IEEEeqnarrayboxXCR[#1]{\ifnum0=`{\fi}%
+\cr\noalign{\if@IEEEeqnarrayboxnojot\else\vskip\jot\fi\vskip#1\relax}}
+
+
+
+% usage: \@IEEEbuildpreamble{column specifiers}
+% starts the halign preamble build 
+% the assembled preamble is put in \@IEEEtrantmptoksA
+\def\@IEEEbuildpreamble#1{\@IEEEtrantmptoksA={}% clear token register
+\let\@IEEEBPcurtype=u%current column type is not yet known
+\let\@IEEEBPprevtype=s%the previous column type was the start
+\let\@IEEEBPnexttype=u%next column type is not yet known
+% ensure these are valid
+\def\@IEEEBPcurglue={0pt plus 0pt minus 0pt}%
+\def\@IEEEBPcurcolname{@IEEEdefault}% name of current column definition
+% currently acquired numerically referenced glue
+% use a name that is easier to remember
+\let\@IEEEBPcurnum=\@IEEEtrantmpcountA%
+\@IEEEBPcurnum=0%
+% tracks number of columns in the preamble
+\@IEEEeqnnumcols=0%
+% record the default end glues
+\edef\@IEEEBPstartglue{\@IEEEeqnarraycolSEPdefaultstart}%
+\edef\@IEEEBPendglue{\@IEEEeqnarraycolSEPdefaultend}%
+\edef\@IEEEedefMACRO{#1}\relax% fully expand the preamble to support macro containers
+% now parse the user's column specifications
+% \ignorespaces is used as a delimiter, need at least one trailing \relax because
+% \@@IEEEbuildpreamble looks into the future 
+\expandafter\@@IEEEbuildpreamble\@IEEEedefMACRO\ignorespaces\relax\relax}
+
+
+% usage: \@@IEEEbuildpreamble{current column}{next column}
+% parses and builds the halign preamble
+\def\@@IEEEbuildpreamble#1#2{\let\@@nextIEEEbuildpreamble=\@@IEEEbuildpreamble%
+% use only the very first token to check the end
+\@IEEEextracttokengroups{#1}\relax
+\ifx\@IEEEextractedfirsttoken\ignorespaces\let\@@nextIEEEbuildpreamble=\@@IEEEfinishpreamble\else%
+% identify current and next token type
+\@IEEEgetcoltype{#1}{\@IEEEBPcurtype}{1}% current, error on invalid
+\@IEEEgetcoltype{#2}{\@IEEEBPnexttype}{0}% next, no error on invalid next
+% if curtype is a glue, get the glue def
+\if\@IEEEBPcurtype g\@IEEEgetcurglue{#1}{\@IEEEBPcurglue}\fi%
+% if curtype is a column, get the column def and set the current column name
+\if\@IEEEBPcurtype c\@IEEEgetcurcol{#1}\fi%
+% if curtype is a numeral, acquire the user defined glue
+\if\@IEEEBPcurtype n\@IEEEprocessNcol{#1}\fi%
+% process the acquired glue 
+\if\@IEEEBPcurtype g\@IEEEprocessGcol\fi%
+% process the acquired col 
+\if\@IEEEBPcurtype c\@IEEEprocessCcol\fi%
+% ready prevtype for next col spec.
+\let\@IEEEBPprevtype=\@IEEEBPcurtype%
+% be sure and put back the future token(s) as a group
+\fi\@@nextIEEEbuildpreamble{#2}}
+
+
+% usage: \@@IEEEfinishpreamble{discarded}
+% executed just after preamble build is completed
+% warn about zero cols, and if prevtype type = u, put in end tabskip glue
+% argument is not used
+\def\@@IEEEfinishpreamble#1{\ifnum\@IEEEeqnnumcols<1\relax
+\@IEEEclspkgerror{No column specifiers declared for IEEEeqnarray}%
+{At least one column type must be declared for each IEEEeqnarray.}%
+\fi%num cols less than 1
+%if last type undefined, set default end tabskip glue
+\if\@IEEEBPprevtype u\@IEEEappendtoksA{\tabskip=\@IEEEBPendglue}\fi}
+
+
+% usage: \@IEEEgetcoltype{col specifier}{\output}{error more}
+% Identify and return the column specifier's type code in the given
+% \output macro:
+% n = number
+% g = glue (any other char in catagory 12)
+% c = letter
+% e = \ignorespaces (end of sequence)
+% u = undefined
+% error mode: 0 = no error message, 1 = error on invalid char
+\def\@IEEEgetcoltype#1#2#3{%
+% use only the very first token to determine the type
+\@IEEEextracttoken{#1}\relax
+% \@IEEEextractedtoken has the first token, the rest are discarded
+\let#2=u\relax% assume invalid until know otherwise
+\ifx\@IEEEextractedtoken\ignorespaces\let#2=e\else
+\ifcat\@IEEEextractedtoken\relax\else% screen out control sequences
+\if0\@IEEEextractedtoken\let#2=n\else
+\if1\@IEEEextractedtoken\let#2=n\else
+\if2\@IEEEextractedtoken\let#2=n\else
+\if3\@IEEEextractedtoken\let#2=n\else
+\if4\@IEEEextractedtoken\let#2=n\else
+\if5\@IEEEextractedtoken\let#2=n\else
+\if6\@IEEEextractedtoken\let#2=n\else
+\if7\@IEEEextractedtoken\let#2=n\else
+\if8\@IEEEextractedtoken\let#2=n\else
+\if9\@IEEEextractedtoken\let#2=n\else
+\ifcat,\@IEEEextractedtoken\let#2=g\relax
+\else\ifcat a\@IEEEextractedtoken\let#2=c\relax\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
+\if#2u\relax
+\if0\noexpand#3\relax\else\@IEEEclspkgerror{Invalid character in column specifications}%
+{Only letters, numerals and certain other symbols are allowed \MessageBreak
+as IEEEeqnarray column specifiers.}\fi\fi}
+
+
+% usage: \@IEEEgetcurcol{col specifier}
+% verify the letter referenced column exists
+% and return its name in \@IEEEBPcurcolname
+% if column specifier is invalid, use the default column @IEEEdefault
+\def\@IEEEgetcurcol#1{\expandafter\ifx\csname @IEEEeqnarraycolDEF#1\endcsname\@IEEEeqnarraycolisdefined%
+\def\@IEEEBPcurcolname{#1}\else% invalid column name
+\@IEEEclspkgerror{Invalid column type "#1" in column specifications.\MessageBreak
+Using a default centering column instead}%
+{You must define IEEEeqnarray column types before use.}%
+\def\@IEEEBPcurcolname{@IEEEdefault}\fi}
+
+
+% usage: \@IEEEgetcurglue{glue specifier}{\output}
+% identify the predefined (punctuation) glue value
+% and return it in the given output macro
+\def\@IEEEgetcurglue#1#2{%
+% ! = \! (neg small)  -0.16667em (-3/18 em)
+% , = \, (small)       0.16667em ( 3/18 em)
+% : = \: (med)         0.22222em ( 4/18 em)
+% ; = \; (large)       0.27778em ( 5/18 em)
+% ' = \quad            1em
+% " = \qquad           2em
+% . = 0.5\arraycolsep
+% / = \arraycolsep
+% ? = 2\arraycolsep
+% * = 1fil
+% + = \@IEEEeqnarraycolSEPcenter
+% - = \@IEEEeqnarraycolSEPzero
+% Note that all em values are referenced to the math font (textfont2) fontdimen6
+% value for 1em.
+% 
+% use only the very first token to determine the type
+\@IEEEextracttoken{#1}\relax
+\ifx\@IEEEextractedtokensdiscarded\@empty\else
+  \typeout{** WARNING: IEEEeqnarray predefined inter-column glue type specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+\fi
+% get the math font 1em value
+% LaTeX2e's NFSS2 does not preload the fonts, but \IEEEeqnarray needs
+% to gain access to the math (\textfont2) font's spacing parameters.
+% So we create a bogus box here that uses the math font to ensure
+% that \textfont2 is loaded and ready. If this is not done,
+% the \textfont2 stuff here may not work.
+% Thanks to Bernd Raichle for his 1997 post on this topic.
+{\setbox0=\hbox{$\displaystyle\relax$}}%
+% fontdimen6 has the width of 1em (a quad).
+\@IEEEtrantmpdimenA=\fontdimen6\textfont2\relax%
+% identify the glue value based on the first token
+% we discard anything after the first
+\if!\@IEEEextractedtoken\@IEEEtrantmpdimenA=-0.16667\@IEEEtrantmpdimenA\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if,\@IEEEextractedtoken\@IEEEtrantmpdimenA=0.16667\@IEEEtrantmpdimenA\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if:\@IEEEextractedtoken\@IEEEtrantmpdimenA=0.22222\@IEEEtrantmpdimenA\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if;\@IEEEextractedtoken\@IEEEtrantmpdimenA=0.27778\@IEEEtrantmpdimenA\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if'\@IEEEextractedtoken\@IEEEtrantmpdimenA=1\@IEEEtrantmpdimenA\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if"\@IEEEextractedtoken\@IEEEtrantmpdimenA=2\@IEEEtrantmpdimenA\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if.\@IEEEextractedtoken\@IEEEtrantmpdimenA=0.5\arraycolsep\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if/\@IEEEextractedtoken\edef#2{\the\arraycolsep}\else
+\if?\@IEEEextractedtoken\@IEEEtrantmpdimenA=2\arraycolsep\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if *\@IEEEextractedtoken\edef#2{0pt plus 1fil minus 0pt}\else
+\if+\@IEEEextractedtoken\edef#2{\@IEEEeqnarraycolSEPcenter}\else
+\if-\@IEEEextractedtoken\edef#2{\@IEEEeqnarraycolSEPzero}\else
+\edef#2{\@IEEEeqnarraycolSEPzero}%
+\@IEEEclspkgerror{Invalid predefined inter-column glue type "#1" in\MessageBreak
+column specifications. Using a default value of\MessageBreak
+0pt instead}%
+{Only !,:;'"./?*+ and - are valid predefined glue types in the\MessageBreak 
+IEEEeqnarray column specifications.}\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
+
+
+% usage: \@IEEEprocessNcol{digit}
+% process a numerical digit from the column specification
+% and look up the corresponding user defined glue value
+% can transform current type from n to g or a as the user defined glue is acquired
+\def\@IEEEprocessNcol#1{\if\@IEEEBPprevtype g%
+\@IEEEclspkgerror{Back-to-back inter-column glue specifiers in column\MessageBreak
+specifications. Ignoring consecutive glue specifiers\MessageBreak
+after the first}%
+{You cannot have two or more glue types next to each other\MessageBreak 
+in the IEEEeqnarray column specifications.}%
+\let\@IEEEBPcurtype=a% abort this glue, future digits will be discarded
+\@IEEEBPcurnum=0\relax%
+\else% if we previously aborted a glue
+\if\@IEEEBPprevtype a\@IEEEBPcurnum=0\let\@IEEEBPcurtype=a%maintain digit abortion
+\else%acquire this number
+% save the previous type before the numerical digits started
+\if\@IEEEBPprevtype n\else\let\@IEEEBPprevsavedtype=\@IEEEBPprevtype\fi%
+\multiply\@IEEEBPcurnum by 10\relax%
+\advance\@IEEEBPcurnum by #1\relax% add in number, \relax is needed to stop TeX's number scan
+\if\@IEEEBPnexttype n\else%close acquisition
+\expandafter\ifx\csname @IEEEeqnarraycolSEPDEF\expandafter\romannumeral\number\@IEEEBPcurnum\endcsname\@IEEEeqnarraycolisdefined%
+\edef\@IEEEBPcurglue{\csname @IEEEeqnarraycolSEP\expandafter\romannumeral\number\@IEEEBPcurnum\endcsname}%
+\else%user glue not defined
+\@IEEEclspkgerror{Invalid user defined inter-column glue type "\number\@IEEEBPcurnum" in\MessageBreak
+column specifications. Using a default value of\MessageBreak
+0pt instead}%
+{You must define all IEEEeqnarray numerical inter-column glue types via\MessageBreak
+\string\IEEEeqnarraydefcolsep \space before they are used in column specifications.}%
+\edef\@IEEEBPcurglue{\@IEEEeqnarraycolSEPzero}%
+\fi% glue defined or not
+\let\@IEEEBPcurtype=g% change the type to reflect the acquired glue
+\let\@IEEEBPprevtype=\@IEEEBPprevsavedtype% restore the prev type before this number glue
+\@IEEEBPcurnum=0\relax%ready for next acquisition
+\fi%close acquisition, get glue
+\fi%discard or acquire number
+\fi%prevtype glue or not
+}
+
+
+% process an acquired glue
+% add any acquired column/glue pair to the preamble
+\def\@IEEEprocessGcol{\if\@IEEEBPprevtype a\let\@IEEEBPcurtype=a%maintain previous glue abortions
+\else
+% if this is the start glue, save it, but do nothing else 
+% as this is not used in the preamble, but before
+\if\@IEEEBPprevtype s\edef\@IEEEBPstartglue{\@IEEEBPcurglue}%
+\else%not the start glue
+\if\@IEEEBPprevtype g%ignore if back to back glues
+\@IEEEclspkgerror{Back-to-back inter-column glue specifiers in column\MessageBreak
+specifications. Ignoring consecutive glue specifiers\MessageBreak
+after the first}%
+{You cannot have two or more glue types next to each other\MessageBreak 
+in the IEEEeqnarray column specifications.}%
+\let\@IEEEBPcurtype=a% abort this glue
+\else% not a back to back glue
+\if\@IEEEBPprevtype c\relax% if the previoustype was a col, add column/glue pair to preamble
+\ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi
+\toks0={##}%
+% make preamble advance col counter if this environment needs this
+\if@advanceIEEEeqncolcnt\@IEEEappendtoksA{\global\advance\@IEEEeqncolcnt by 1\relax}\fi
+% insert the column defintion into the preamble, being careful not to expand
+% the column definition
+\@IEEEappendtoksA{\tabskip=\@IEEEBPcurglue}%
+\@IEEEappendNOEXPANDtoksA{\begingroup\csname @IEEEeqnarraycolPRE}%
+\@IEEEappendtoksA{\@IEEEBPcurcolname}%
+\@IEEEappendNOEXPANDtoksA{\endcsname}%
+\@IEEEappendtoksA{\the\toks0}%
+\@IEEEappendNOEXPANDtoksA{\relax\relax\relax\relax\relax%
+\relax\relax\relax\relax\relax\csname @IEEEeqnarraycolPOST}%
+\@IEEEappendtoksA{\@IEEEBPcurcolname}%
+\@IEEEappendNOEXPANDtoksA{\endcsname\relax\relax\relax\relax\relax%
+\relax\relax\relax\relax\relax\endgroup}%
+\advance\@IEEEeqnnumcols by 1\relax%one more column in the preamble
+\else% error: non-start glue with no pending column
+\@IEEEclspkgerror{Inter-column glue specifier without a prior column\MessageBreak
+type in the column specifications. Ignoring this glue\MessageBreak 
+specifier}%
+{Except for the first and last positions, glue can be placed only\MessageBreak
+between column types.}%
+\let\@IEEEBPcurtype=a% abort this glue
+\fi% previous was a column
+\fi% back-to-back glues
+\fi% is start column glue
+\fi% prev type not a
+}
+
+
+% process an acquired letter referenced column and, if necessary, add it to the preamble
+\def\@IEEEprocessCcol{\if\@IEEEBPnexttype g\else
+\if\@IEEEBPnexttype n\else
+% we have a column followed by something other than a glue (or numeral glue)
+% so we must add this column to the preamble now
+\ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi%col separator for those after the first
+\if\@IEEEBPnexttype e\@IEEEappendtoksA{\tabskip=\@IEEEBPendglue\relax}\else%put in end glue
+\@IEEEappendtoksA{\tabskip=\@IEEEeqnarraycolSEPdefaultmid\relax}\fi% or default mid glue
+\toks0={##}%
+% make preamble advance col counter if this environment needs this
+\if@advanceIEEEeqncolcnt\@IEEEappendtoksA{\global\advance\@IEEEeqncolcnt by 1\relax}\fi
+% insert the column definition into the preamble, being careful not to expand
+% the column definition
+\@IEEEappendNOEXPANDtoksA{\begingroup\csname @IEEEeqnarraycolPRE}%
+\@IEEEappendtoksA{\@IEEEBPcurcolname}%
+\@IEEEappendNOEXPANDtoksA{\endcsname}%
+\@IEEEappendtoksA{\the\toks0}%
+\@IEEEappendNOEXPANDtoksA{\relax\relax\relax\relax\relax%
+\relax\relax\relax\relax\relax\csname @IEEEeqnarraycolPOST}%
+\@IEEEappendtoksA{\@IEEEBPcurcolname}%
+\@IEEEappendNOEXPANDtoksA{\endcsname\relax\relax\relax\relax\relax%
+\relax\relax\relax\relax\relax\endgroup}%
+\advance\@IEEEeqnnumcols by 1\relax%one more column in the preamble
+\fi%next type not numeral
+\fi%next type not glue
+}
+
+
+%%
+%% END OF IEEEeqnarray DEFINITIONS
+%%
+
+
+
+
+
+% set up the running headers and footers
+%
+% header and footer font and size specifications
+\def\@IEEEheaderstyle{\normalfont\scriptsize}
+\def\@IEEEfooterstyle{\normalfont\scriptsize}
+%
+% compsoc uses sans-serif headers and footers
+\ifCLASSOPTIONcompsoc
+  \def\@IEEEheaderstyle{\normalfont\sffamily\scriptsize}
+  \def\@IEEEfooterstyle{\normalfont\sffamily\scriptsize}
+\fi
+
+
+% standard page style, ps@headings
+\def\ps@headings{% default to standard twoside headers, no footers
+% will change later if the mode requires otherwise
+\def\@oddhead{\hbox{}\@IEEEheaderstyle\rightmark\hfil\thepage}\relax
+\def\@evenhead{\@IEEEheaderstyle\thepage\hfil\leftmark\hbox{}}\relax
+\let\@oddfoot\@empty
+\let\@evenfoot\@empty
+\ifCLASSOPTIONtechnote
+  % technote twoside
+  \def\@oddhead{\hbox{}\@IEEEheaderstyle\leftmark\hfil\thepage}\relax
+  \def\@evenhead{\@IEEEheaderstyle\thepage\hfil\leftmark\hbox{}}\relax
+\fi
+\ifCLASSOPTIONdraftcls
+  % draft footers
+  \def\@oddfoot{\@IEEEfooterstyle\@date\hfil DRAFT}\relax
+  \def\@evenfoot{\@IEEEfooterstyle DRAFT\hfil\@date}\relax
+\fi
+% oneside
+\if@twoside\else
+  % standard one side headers
+  \def\@oddhead{\hbox{}\@IEEEheaderstyle\leftmark\hfil\thepage}\relax
+  \let\@evenhead\@empty
+  \ifCLASSOPTIONdraftcls
+    % oneside draft footers
+    \def\@oddfoot{\@IEEEfooterstyle\@date\hfil DRAFT}\relax
+    \let\@evenfoot\@empty
+  \fi
+\fi
+% turn off headers for conferences
+\ifCLASSOPTIONconference
+  \let\@oddhead\@empty
+  \let\@evenhead\@empty
+\fi
+% turn off footers for draftclsnofoot
+\ifCLASSOPTIONdraftclsnofoot
+  \let\@oddfoot\@empty
+  \let\@evenfoot\@empty
+\fi}
+
+
+% title page style, ps@IEEEtitlepagestyle
+\def\ps@IEEEtitlepagestyle{% default title page headers, no footers
+\def\@oddhead{\hbox{}\@IEEEheaderstyle\leftmark\hfil\thepage}\relax
+\def\@evenhead{\@IEEEheaderstyle\thepage\hfil\leftmark\hbox{}}\relax
+\let\@oddfoot\@empty
+\let\@evenfoot\@empty
+% will change later if the mode requires otherwise
+\ifCLASSOPTIONdraftcls
+  % draft footers
+  \ifCLASSOPTIONdraftclsnofoot\else
+    % but only if not draftclsnofoot
+    \def\@oddfoot{\@IEEEfooterstyle\@date\hfil DRAFT}\relax
+    \def\@evenfoot{\@IEEEfooterstyle DRAFT\hfil\@date}\relax
+  \fi
+\else
+  % all nondraft mode footers
+  \if@IEEEusingpubid
+    % for title pages that are using a pubid
+    % do not repeat pubid on the title page if using a peer review cover page
+    \ifCLASSOPTIONpeerreview\else
+      % for noncompsoc papers, the pubid uses footnotesize and
+      % is at the same vertical position as where the last baseline would normally be
+      \def\@oddfoot{\hbox{}\hss\@IEEEfooterstyle\footnotesize\raisebox{\footskip}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+      \def\@evenfoot{\hbox{}\hss\@IEEEfooterstyle\footnotesize\raisebox{\footskip}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+      \ifCLASSOPTIONcompsoc
+        % for compsoc papers, the pubid is at the same vertical position as the normal footer
+        \def\@oddfoot{\hbox{}\hss\@IEEEfooterstyle\raisebox{0pt}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+        \def\@evenfoot{\hbox{}\hss\@IEEEfooterstyle\raisebox{0pt}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+      \fi
+    \fi
+  \fi
+\fi
+% turn off headers for conferences
+\ifCLASSOPTIONconference
+  \let\@oddhead\@empty
+  \let\@evenhead\@empty
+\fi}
+
+
+% peer review cover page style, ps@IEEEpeerreviewcoverpagestyle
+\def\ps@IEEEpeerreviewcoverpagestyle{% default peer review cover no headers, no footers
+\let\@oddhead\@empty
+\let\@evenhead\@empty
+\let\@oddfoot\@empty
+\let\@evenfoot\@empty
+% will change later if the mode requires otherwise
+\ifCLASSOPTIONdraftcls
+  % draft footers
+  \ifCLASSOPTIONdraftclsnofoot\else
+    % but only if not draftclsnofoot
+    \def\@oddfoot{\@IEEEfooterstyle\@date\hfil DRAFT}\relax
+    \def\@evenfoot{\@IEEEfooterstyle DRAFT\hfil\@date}\relax
+  \fi
+\else
+  % all nondraft mode footers
+  \if@IEEEusingpubid
+    % for peer review cover pages that are using a pubid
+    % for noncompsoc papers, the pubid uses footnotesize and
+    % is at the same vertical position as where the last baseline would normally be
+    \def\@oddfoot{\hbox{}\hss\@IEEEfooterstyle\footnotesize\raisebox{\footskip}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+    \def\@evenfoot{\hbox{}\hss\@IEEEfooterstyle\footnotesize\raisebox{\footskip}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+    \ifCLASSOPTIONcompsoc
+      % for compsoc papers, the pubid is at the same vertical position as the normal footer
+      \def\@oddfoot{\hbox{}\hss\@IEEEfooterstyle\raisebox{0pt}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+      \def\@evenfoot{\hbox{}\hss\@IEEEfooterstyle\raisebox{0pt}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+    \fi
+  \fi
+\fi}
+
+
+
+%% Defines the command for putting the header.
+%% Note that all the text is forced into uppercase, if you have some text
+%% that needs to be in lower case, for instance et. al., then either manually
+%% set \leftmark and \rightmark or use \MakeLowercase{et. al.} within the
+%% arguments to \markboth.
+%% V1.7b add \protect to work with Babel
+\def\markboth#1#2{\def\leftmark{\MakeUppercase{\protect#1}}%
+\def\rightmark{\MakeUppercase{\protect#2}}}
+
+\def\today{\ifcase\month\or
+    January\or February\or March\or April\or May\or June\or
+    July\or August\or September\or October\or November\or December\fi
+    \space\number\day, \number\year}
+
+
+
+
+%% CITATION AND BIBLIOGRAPHY COMMANDS
+%% 
+%% V1.6 no longer supports the older, nonstandard \shortcite and \citename setup stuff
+% 
+% 
+% Modify Latex2e \@citex to separate citations with "], ["
+\def\@citex[#1]#2{%
+  \let\@citea\@empty
+  \@cite{\@for\@citeb:=#2\do
+    {\@citea\def\@citea{], [}%
+     \edef\@citeb{\expandafter\@firstofone\@citeb\@empty}%
+     \if@filesw\immediate\write\@auxout{\string\citation{\@citeb}}\fi
+     \@ifundefined{b@\@citeb}{\mbox{\reset@font\bfseries ?}%
+       \G@refundefinedtrue
+       \@latex@warning
+         {Citation `\@citeb' on page \thepage \space undefined}}%
+       {\hbox{\csname b@\@citeb\endcsname}}}}{#1}}
+
+% V1.6 we create hooks for the optional use of Donald Arseneau's
+% cite.sty package. cite.sty is "smart" and will notice that the
+% following format controls are already defined and will not
+% redefine them. The result will be the proper sorting of the
+% citation numbers and auto detection of 3 or more entry "ranges" -
+% all in IEEE style:  [1], [2], [5]--[7], [12]
+% This also allows for an optional note, i.e., \cite[mynote]{..}.
+% If the \cite with note has more than one reference, the note will
+% be applied to the last of the listed references. It is generally
+% desired that if a note is given, only one reference is listed in
+% that \cite.
+% Thanks to Mr. Arseneau for providing the required format arguments
+% to produce the IEEE style.
+\def\citepunct{], [}
+\def\citedash{]--[}
+
+% V1.7 default to using same font for urls made by url.sty
+\AtBeginDocument{\csname url@samestyle\endcsname}
+
+% V1.6 class files should always provide these
+\def\newblock{\hskip .11em\@plus.33em\@minus.07em}
+\let\@openbib@code\@empty
+% V1.8b article.cls is now providing these too
+% we do not use \@mkboth, nor alter the page style
+\newenvironment{theindex}
+               {\if@twocolumn
+                  \@restonecolfalse
+                \else
+                  \@restonecoltrue
+                \fi
+                \twocolumn[\section*{\indexname}]%
+                \parindent\z@
+                \parskip\z@ \@plus .3\p@\relax
+                \columnseprule \z@
+                \columnsep 35\p@
+                \let\item\@idxitem}
+               {\if@restonecol\onecolumn\else\clearpage\fi}
+\newcommand\@idxitem{\par\hangindent 40\p@}
+\newcommand\subitem{\@idxitem \hspace*{20\p@}}
+\newcommand\subsubitem{\@idxitem \hspace*{30\p@}}
+\newcommand\indexspace{\par \vskip 10\p@ \@plus5\p@ \@minus3\p@\relax}
+
+
+
+% Provide support for the control entries of IEEEtran.bst V1.00 and later.
+% V1.7 optional argument allows for a different aux file to be specified in
+% order to handle multiple bibliographies. For example, with multibib.sty:
+% \newcites{sec}{Secondary Literature}
+% \bstctlcite[@auxoutsec]{BSTcontrolhak}
+\def\bstctlcite{\@ifnextchar[{\@bstctlcite}{\@bstctlcite[@auxout]}}
+\def\@bstctlcite[#1]#2{\@bsphack
+  \@for\@citeb:=#2\do{%
+    \edef\@citeb{\expandafter\@firstofone\@citeb}%
+    \if@filesw\immediate\write\csname #1\endcsname{\string\citation{\@citeb}}\fi}%
+  \@esphack}
+
+% \IEEEnoauxwrite{} allows for citations that do not add to or affect
+% the order of the existing citation list. Can be useful for \cite
+% within \thanks{}.
+\DeclareRobustCommand{\IEEEnoauxwrite}[1]{\relax
+\if@filesw
+\@fileswfalse
+#1\relax\relax\relax\relax\relax
+\@fileswtrue
+\else
+#1\relax\relax\relax\relax\relax
+\fi}
+
+% V1.6 provide a way for a user to execute a command just before 
+% a given reference number - used to insert a \newpage to balance
+% the columns on the last page
+\edef\@IEEEtriggerrefnum{0}   % the default of zero means that
+                              % the command is not executed
+\def\@IEEEtriggercmd{\newpage}
+
+% allow the user to alter the triggered command
+\long\def\IEEEtriggercmd#1{\long\def\@IEEEtriggercmd{#1}}
+
+% allow user a way to specify the reference number just before the
+% command is executed
+\def\IEEEtriggeratref#1{\@IEEEtrantmpcountA=#1%
+\edef\@IEEEtriggerrefnum{\the\@IEEEtrantmpcountA}}%
+
+% trigger command at the given reference
+\def\@IEEEbibitemprefix{\@IEEEtrantmpcountA=\@IEEEtriggerrefnum\relax%
+\advance\@IEEEtrantmpcountA by -1\relax%
+\ifnum\c@enumiv=\@IEEEtrantmpcountA\relax\@IEEEtriggercmd\relax\fi}
+
+
+\def\@biblabel#1{[#1]}
+
+% compsoc journals and conferences left align the reference numbers
+\@IEEEcompsoconly{\def\@biblabel#1{[#1]\hfill}}
+
+% controls bib item spacing
+\def\IEEEbibitemsep{0pt plus .5pt}
+
+\@IEEEcompsocconfonly{\def\IEEEbibitemsep{0.5\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}}
+
+
+\def\thebibliography#1{\section*{\refname}%
+    \addcontentsline{toc}{section}{\refname}%
+    % V1.6 add some rubber space here and provide a command trigger
+    \footnotesize\vskip 0.3\baselineskip plus 0.1\baselineskip minus 0.1\baselineskip%
+    \list{\@biblabel{\@arabic\c@enumiv}}%
+    {\settowidth\labelwidth{\@biblabel{#1}}%
+    \leftmargin\labelwidth
+    \advance\leftmargin\labelsep\relax
+    \itemsep \IEEEbibitemsep\relax
+    \usecounter{enumiv}%
+    \let\p@enumiv\@empty
+    \renewcommand\theenumiv{\@arabic\c@enumiv}}%
+    \let\@IEEElatexbibitem\bibitem%
+    \def\bibitem{\@IEEEbibitemprefix\@IEEElatexbibitem}%
+\def\newblock{\hskip .11em plus .33em minus .07em}%
+% originally:
+%   \sloppy\clubpenalty4000\widowpenalty4000%
+% by adding the \interlinepenalty here, we make it more
+% difficult, but not impossible, for LaTeX to break within a reference.
+% The IEEE almost never breaks a reference (but they do it more often with
+% technotes). You may get an underfull vbox warning around the bibliography, 
+% but the final result will be much more like what the IEEE will publish. 
+% MDS 11/2000
+\ifCLASSOPTIONtechnote\sloppy\clubpenalty4000\widowpenalty4000\interlinepenalty100%
+\else\sloppy\clubpenalty4000\widowpenalty4000\interlinepenalty500\fi%
+    \sfcode`\.=1000\relax}
+\let\endthebibliography=\endlist
+
+
+
+
+% TITLE PAGE COMMANDS
+% 
+% 
+% \IEEEmembership is used to produce the sublargesize italic font used to indicate author 
+% IEEE membership. compsoc uses a large size sans slant font
+\def\IEEEmembership#1{{\@IEEEnotcompsoconly{\sublargesize}\normalfont\@IEEEcompsoconly{\sffamily}\textit{#1}}}
+ 
+
+% \IEEEauthorrefmark{} produces a footnote type symbol to indicate author affiliation.
+% When given an argument of 1 to 9, \IEEEauthorrefmark{} follows the standard LaTeX footnote
+% symbol sequence convention. However, for arguments 10 and above, \IEEEauthorrefmark{} 
+% reverts to using lower case roman numerals, so it cannot overflow. Do note that you 
+% cannot use \footnotemark[] in place of \IEEEauthorrefmark{} within \author as the footnote
+% symbols will have been turned off to prevent \thanks from creating footnote marks.
+% \IEEEauthorrefmark{} produces a symbol that appears to LaTeX as having zero vertical
+% height - this allows for a more compact line packing, but the user must ensure that
+% the interline spacing is large enough to prevent \IEEEauthorrefmark{} from colliding
+% with the text above.
+% V1.7 make this a robust command
+% V1.8 transmag uses an arabic author affiliation symbol
+\ifCLASSOPTIONtransmag
+\DeclareRobustCommand*{\IEEEauthorrefmark}[1]{\raisebox{0pt}[0pt][0pt]{\textsuperscript{\footnotesize #1}}}
+\else
+\DeclareRobustCommand*{\IEEEauthorrefmark}[1]{\raisebox{0pt}[0pt][0pt]{\textsuperscript{\footnotesize\ensuremath{\ifcase#1\or *\or \dagger\or \ddagger\or%
+    \mathsection\or \mathparagraph\or \|\or **\or \dagger\dagger%
+    \or \ddagger\ddagger \else\textsuperscript{\expandafter\romannumeral#1}\fi}}}}
+\fi
+
+
+% FONT CONTROLS AND SPACINGS FOR CONFERENCE MODE AUTHOR NAME AND AFFILIATION BLOCKS
+% 
+% The default font styles for the author name and affiliation blocks (confmode)
+\def\@IEEEauthorblockNstyle{\normalfont\@IEEEcompsocnotconfonly{\sffamily}\sublargesize}
+\def\@IEEEauthorblockAstyle{\normalfont\@IEEEcompsocnotconfonly{\sffamily}\@IEEEcompsocconfonly{\itshape}\normalsize}
+% The default if the user does not use an author block
+\def\@IEEEauthordefaulttextstyle{\normalfont\@IEEEcompsocnotconfonly{\sffamily}\sublargesize}
+
+% adjustment spacing from title (or special paper notice) to author name blocks (confmode)
+% can be negative
+\def\@IEEEauthorblockconfadjspace{-0.25em}
+% compsoc conferences need more space here
+\@IEEEcompsocconfonly{\def\@IEEEauthorblockconfadjspace{0.75\@IEEEnormalsizeunitybaselineskip}}
+
+% spacing between name and affiliation blocks (confmode)
+% This can be negative.
+% The IEEE doesn't want any added spacing here, but I will leave these
+% controls in place in case they ever change their mind.
+% Personally, I like 0.75ex.
+%\def\@IEEEauthorblockNtopspace{0.75ex}
+%\def\@IEEEauthorblockAtopspace{0.75ex}
+\def\@IEEEauthorblockNtopspace{0.0ex}
+\def\@IEEEauthorblockAtopspace{0.0ex}
+\ifCLASSOPTIONtransmag
+% transmag uses one line of space above first affiliation block
+\def\@IEEEauthorblockAtopspace{1\@IEEEnormalsizeunitybaselineskip}
+\fi
+
+% baseline spacing within name and affiliation blocks (confmode)
+% must be positive, spacings below certain values will make 
+% the position of line of text sensitive to the contents of the
+% line above it i.e., whether or not the prior line has descenders, 
+% subscripts, etc. For this reason it is a good idea to keep
+% these above 2.6ex
+\def\@IEEEauthorblockNinterlinespace{2.6ex}
+\def\@IEEEauthorblockAinterlinespace{2.75ex}
+
+% This tracks the required strut size.
+% See the \@IEEEauthorhalign command for the actual default value used.
+\def\@IEEEauthorblockXinterlinespace{2.7ex}
+
+% variables to retain font size and style across groups
+% values given here have no effect as they will be overwritten later
+\gdef\@IEEESAVESTATEfontsize{10}
+\gdef\@IEEESAVESTATEfontbaselineskip{12}
+\gdef\@IEEESAVESTATEfontencoding{OT1}
+\gdef\@IEEESAVESTATEfontfamily{ptm}
+\gdef\@IEEESAVESTATEfontseries{m}
+\gdef\@IEEESAVESTATEfontshape{n}
+
+% saves the current font attributes
+\def\@IEEEcurfontSAVE{\global\let\@IEEESAVESTATEfontsize\f@size%
+\global\let\@IEEESAVESTATEfontbaselineskip\f@baselineskip%
+\global\let\@IEEESAVESTATEfontencoding\f@encoding%
+\global\let\@IEEESAVESTATEfontfamily\f@family%
+\global\let\@IEEESAVESTATEfontseries\f@series%
+\global\let\@IEEESAVESTATEfontshape\f@shape}
+
+% restores the saved font attributes
+\def\@IEEEcurfontRESTORE{\fontsize{\@IEEESAVESTATEfontsize}{\@IEEESAVESTATEfontbaselineskip}%
+\fontencoding{\@IEEESAVESTATEfontencoding}%
+\fontfamily{\@IEEESAVESTATEfontfamily}%
+\fontseries{\@IEEESAVESTATEfontseries}%
+\fontshape{\@IEEESAVESTATEfontshape}%
+\selectfont}
+
+
+% variable to indicate if the current block is the first block in the column
+\newif\if@IEEEprevauthorblockincol   \@IEEEprevauthorblockincolfalse
+
+
+% the command places a strut with height and depth = \@IEEEauthorblockXinterlinespace
+% we use this technique to have complete manual control over the spacing of the lines
+% within the halign environment.
+% We set the below baseline portion at 30%, the above
+% baseline portion at 70% of the total length.
+% Responds to changes in the document's \baselinestretch
+\def\@IEEEauthorstrutrule{\@IEEEtrantmpdimenA\@IEEEauthorblockXinterlinespace%
+\@IEEEtrantmpdimenA=\baselinestretch\@IEEEtrantmpdimenA%
+\rule[-0.3\@IEEEtrantmpdimenA]{0pt}{\@IEEEtrantmpdimenA}}
+
+
+% blocks to hold the authors' names and affilations. 
+% Makes formatting easy for conferences
+%
+% use real definitions in conference mode
+% name block
+\def\IEEEauthorblockN#1{\relax\@IEEEauthorblockNstyle% set the default text style
+\gdef\@IEEEauthorblockXinterlinespace{0pt}% disable strut for spacer row
+% the \expandafter hides the \cr in conditional tex, see the array.sty docs
+% for details, probably not needed here as the \cr is in a macro
+% do a spacer row if needed
+\if@IEEEprevauthorblockincol\expandafter\@IEEEauthorblockNtopspaceline\fi
+\global\@IEEEprevauthorblockincoltrue% we now have a block in this column
+%restore the correct strut value
+\gdef\@IEEEauthorblockXinterlinespace{\@IEEEauthorblockNinterlinespace}%
+% input the author names
+#1%
+% end the row if the user did not already
+\crcr}
+% spacer row for names
+\def\@IEEEauthorblockNtopspaceline{\cr\noalign{\vskip\@IEEEauthorblockNtopspace}}
+%
+% affiliation block
+\def\IEEEauthorblockA#1{\relax\@IEEEauthorblockAstyle% set the default text style
+\gdef\@IEEEauthorblockXinterlinespace{0pt}%disable strut for spacer row
+% the \expandafter hides the \cr in conditional tex, see the array.sty docs
+% for details, probably not needed here as the \cr is in a macro
+% do a spacer row if needed
+\if@IEEEprevauthorblockincol\expandafter\@IEEEauthorblockAtopspaceline\fi
+\global\@IEEEprevauthorblockincoltrue% we now have a block in this column
+%restore the correct strut value
+\gdef\@IEEEauthorblockXinterlinespace{\@IEEEauthorblockAinterlinespace}%
+% input the author affiliations
+#1%
+% end the row if the user did not already
+\crcr
+% V1.8 transmag does not use any additional affiliation spacing after the first author
+\ifCLASSOPTIONtransmag\gdef\@IEEEauthorblockAtopspace{0pt}\fi}
+
+% spacer row for affiliations
+\def\@IEEEauthorblockAtopspaceline{\cr\noalign{\vskip\@IEEEauthorblockAtopspace}}
+
+
+% allow papers to compile even if author blocks are used in modes other
+% than conference or peerreviewca. For such cases, we provide dummy blocks.
+\ifCLASSOPTIONconference
+\else
+   \ifCLASSOPTIONpeerreviewca\else
+      % not conference, peerreviewca or transmag mode
+      \ifCLASSOPTIONtransmag\else
+         \def\IEEEauthorblockN#1{#1}%
+         \def\IEEEauthorblockA#1{#1}%
+      \fi
+   \fi
+\fi
+
+
+
+% we provide our own halign so as not to have to depend on tabular
+\def\@IEEEauthorhalign{\@IEEEauthordefaulttextstyle% default text style
+   \lineskip=0pt\relax% disable line spacing
+   \lineskiplimit=0pt\relax%
+   \baselineskip=0pt\relax%
+   \@IEEEcurfontSAVE% save the current font
+   \mathsurround\z@\relax% no extra spacing around math
+   \let\\\@IEEEauthorhaligncr% replace newline with halign friendly one
+   \tabskip=0pt\relax% no column spacing
+   \everycr{}% ensure no problems here
+   \@IEEEprevauthorblockincolfalse% no author blocks yet
+   \def\@IEEEauthorblockXinterlinespace{2.7ex}% default interline space
+   \vtop\bgroup%vtop box
+   \halign\bgroup&\relax\hfil\@IEEEcurfontRESTORE\relax ##\relax
+   \hfil\@IEEEcurfontSAVE\@IEEEauthorstrutrule\cr}
+
+% ensure last line, exit from halign, close vbox
+\def\end@IEEEauthorhalign{\crcr\egroup\egroup}
+
+% handle bogus star form
+\def\@IEEEauthorhaligncr{{\ifnum0=`}\fi\@ifstar{\@@IEEEauthorhaligncr}{\@@IEEEauthorhaligncr}}
+
+% test and setup the optional argument to \\[]
+\def\@@IEEEauthorhaligncr{\@testopt\@@@IEEEauthorhaligncr\z@skip}
+
+% end the line and do the optional spacer
+\def\@@@IEEEauthorhaligncr[#1]{\ifnum0=`{\fi}\cr\noalign{\vskip#1\relax}}
+
+
+
+% flag to prevent multiple \and warning messages
+\newif\if@IEEEWARNand
+\@IEEEWARNandtrue
+
+% if in conference or peerreviewca modes, we support the use of \and as \author is a
+% tabular environment, otherwise we warn the user that \and is invalid
+% outside of conference or peerreviewca modes.
+\def\and{\relax} % provide a bogus \and that we will then override
+
+\renewcommand{\and}[1][\relax]{\if@IEEEWARNand\typeout{** WARNING: \noexpand\and is valid only
+                               when in conference or peerreviewca}\typeout{modes (line \the\inputlineno).}\fi\global\@IEEEWARNandfalse}
+
+\ifCLASSOPTIONconference%
+\renewcommand{\and}[1][\hfill]{\end{@IEEEauthorhalign}#1\begin{@IEEEauthorhalign}}%
+\fi
+\ifCLASSOPTIONpeerreviewca
+\renewcommand{\and}[1][\hfill]{\end{@IEEEauthorhalign}#1\begin{@IEEEauthorhalign}}%
+\fi
+% V1.8 transmag uses conference author format
+\ifCLASSOPTIONtransmag
+\renewcommand{\and}[1][\hfill]{\end{@IEEEauthorhalign}#1\begin{@IEEEauthorhalign}}%
+\fi
+
+% page clearing command
+% based on LaTeX2e's \cleardoublepage, but allows different page styles
+% for the inserted blank pages
+\def\@IEEEcleardoublepage#1{\clearpage\if@twoside\ifodd\c@page\else
+\hbox{}\thispagestyle{#1}\newpage\if@twocolumn\hbox{}\thispagestyle{#1}\newpage\fi\fi\fi}
+
+% V1.8b hooks to allow adjustment of space above title
+\def\IEEEtitletopspace{0.5\baselineskip}
+% an added extra amount to allow for adjustment/offset
+\def\IEEEtitletopspaceextra{0pt}
+
+% user command to invoke the title page
+\def\maketitle{\par%
+  \begingroup%
+  \normalfont%
+  \def\thefootnote{}%  the \thanks{} mark type is empty
+  \def\footnotemark{}% and kill space from \thanks within author
+  \let\@makefnmark\relax% V1.7, must *really* kill footnotemark to remove all \textsuperscript spacing as well.
+  \footnotesize%       equal spacing between thanks lines
+  \footnotesep 0.7\baselineskip%see global setting of \footnotesep for more info
+  % V1.7 disable \thanks note indention for compsoc
+  \@IEEEcompsoconly{\long\def\@makefntext##1{\parindent 1em\noindent\hbox{\@makefnmark}##1}}%
+  \normalsize%
+  \ifCLASSOPTIONpeerreview
+     \newpage\global\@topnum\z@ \@maketitle\@IEEEstatictitlevskip\@IEEEaftertitletext%
+     \thispagestyle{IEEEpeerreviewcoverpagestyle}\@thanks%
+  \else
+     \if@twocolumn%
+        \ifCLASSOPTIONtechnote%
+           \newpage\global\@topnum\z@ \@maketitle\@IEEEstatictitlevskip\@IEEEaftertitletext%
+        \else
+           \twocolumn[{\IEEEquantizevspace{\@maketitle}[\IEEEquantizedisabletitlecmds]{0pt}[-\topskip]{\baselineskip}{\@IEEENORMtitlevspace}{\@IEEEMINtitlevspace}\@IEEEaftertitletext}]%
+        \fi
+     \else
+        \newpage\global\@topnum\z@ \@maketitle\@IEEEstatictitlevskip\@IEEEaftertitletext%
+     \fi
+     \thispagestyle{IEEEtitlepagestyle}\@thanks%
+  \fi
+  % pullup page for pubid if used.
+  \if@IEEEusingpubid
+     \enlargethispage{-\@IEEEpubidpullup}%
+  \fi 
+  \endgroup
+  \setcounter{footnote}{0}\let\maketitle\relax\let\@maketitle\relax
+  \gdef\@thanks{}%
+  % v1.6b do not clear these as we will need the title again for peer review papers
+  % \gdef\@author{}\gdef\@title{}%
+  \let\thanks\relax}
+
+
+% V1.8 parbox to format \@IEEEtitleabstractindextext
+\long\def\@IEEEtitleabstractindextextbox#1{\parbox{1\textwidth}{#1}}
+% V1.8 compsoc is partial width
+\ifCLASSOPTIONcompsoc
+% comparison with proofs suggests it's in the range of 92.1-92.3%
+\long\def\@IEEEtitleabstractindextextbox#1{\parbox{0.922\textwidth}{\@IEEEcompsocnotconfonly{\rightskip\@flushglue\leftskip\z@skip}#1}}
+\fi
+
+% formats the Title, authors names, affiliations and special paper notice
+% THIS IS A CONTROLLED SPACING COMMAND! Do not allow blank lines or unintentional
+% spaces to enter the definition - use % at the end of each line
+\def\@maketitle{\newpage
+\bgroup\par\vskip\IEEEtitletopspace\vskip\IEEEtitletopspaceextra\centering%
+\ifCLASSOPTIONtechnote% technotes, V1.8a abstract and index terms are not treated differently for compsoc technotes
+   {\bfseries\large\@IEEEcompsoconly{\Large\sffamily}\@title\par}\vskip 1.3em{\lineskip .5em\@IEEEcompsoconly{\large\sffamily}\@author
+   \@IEEEspecialpapernotice\par}\relax
+\else% not a technote
+   \vskip0.2em{\Huge\ifCLASSOPTIONtransmag\bfseries\LARGE\fi\@IEEEcompsoconly{\sffamily}\@IEEEcompsocconfonly{\normalfont\normalsize\vskip 2\@IEEEnormalsizeunitybaselineskip
+   \bfseries\Large}\@IEEEcompsocnotconfonly{\vskip 0.75\@IEEEnormalsizeunitybaselineskip}\@title\par}\relax
+   \@IEEEcompsocnotconfonly{\vskip 0.5\@IEEEnormalsizeunitybaselineskip}\vskip1.0em\par%
+   % V1.6 handle \author differently if in conference mode
+   \ifCLASSOPTIONconference%
+      {\@IEEEspecialpapernotice\mbox{}\vskip\@IEEEauthorblockconfadjspace%
+       \mbox{}\hfill\begin{@IEEEauthorhalign}\@author\end{@IEEEauthorhalign}\hfill\mbox{}\par}\relax
+   \else% peerreviewca, peerreview or journal
+      \ifCLASSOPTIONpeerreviewca
+         % peerreviewca handles author names just like conference mode
+         {\@IEEEcompsoconly{\sffamily}\@IEEEspecialpapernotice\mbox{}\vskip\@IEEEauthorblockconfadjspace%
+          \mbox{}\hfill\begin{@IEEEauthorhalign}\@author\end{@IEEEauthorhalign}\hfill\mbox{}\par
+          {\@IEEEcompsoconly{\vskip 1.5em\relax
+           \@IEEEtitleabstractindextextbox{\@IEEEtitleabstractindextext}\par\noindent\hfill
+           \IEEEcompsocdiamondline\hfill\hbox{}\par}}}\relax
+      \else% journal, peerreview or transmag
+         \ifCLASSOPTIONtransmag
+            % transmag also handles author names just like conference mode
+            % it also uses \@IEEEtitleabstractindextex, but with one line less
+            % space above, and one more below
+           {\@IEEEspecialpapernotice\mbox{}\vskip\@IEEEauthorblockconfadjspace%
+            \mbox{}\hfill\begin{@IEEEauthorhalign}\@author\end{@IEEEauthorhalign}\hfill\mbox{}\par
+           {\vspace{0.5\baselineskip}\relax\@IEEEtitleabstractindextextbox{\@IEEEtitleabstractindextext}\vspace{-1\baselineskip}\par}}\relax
+         \else% journal or peerreview
+           {\lineskip.5em\@IEEEcompsoconly{\sffamily}\sublargesize\@author\@IEEEspecialpapernotice\par
+           {\@IEEEcompsoconly{\vskip 1.5em\relax
+            \@IEEEtitleabstractindextextbox{\@IEEEtitleabstractindextext}\par\noindent\hfill
+            \IEEEcompsocdiamondline\hfill\hbox{}\par}}}\relax
+         \fi
+      \fi
+   \fi
+\fi\par\addvspace{0.5\baselineskip}\egroup}
+
+
+% V1.7 Computer Society "diamond line" which follows index terms for nonconference papers
+% V1.8a full width diamond line for single column use
+\def\@IEEEcompsocdiamondlinei{\vrule depth 0pt height 0.5pt width 4cm\nobreak\hspace{7.5pt}\nobreak
+\raisebox{-3.5pt}{\fontfamily{pzd}\fontencoding{U}\fontseries{m}\fontshape{n}\fontsize{11}{12}\selectfont\char70}\nobreak
+\hspace{7.5pt}\nobreak\vrule depth 0pt height 0.5pt width 4cm\relax}
+% V1.8a narrower width diamond line for double column use
+\def\@IEEEcompsocdiamondlineii{\vrule depth 0pt height 0.5pt width 2.5cm\nobreak\hspace{7.5pt}\nobreak
+\raisebox{-3.5pt}{\fontfamily{pzd}\fontencoding{U}\fontseries{m}\fontshape{n}\fontsize{11}{12}\selectfont\char70}\nobreak
+\hspace{7.5pt}\nobreak\vrule depth 0pt height 0.5pt width 2.5cm\relax}
+% V1.8a bare core without rules to base a last resort on for very narrow linewidths
+\def\@IEEEcompsocdiamondlineiii{\mbox{}\nobreak\hspace{7.5pt}\nobreak
+\raisebox{-3.5pt}{\fontfamily{pzd}\fontencoding{U}\fontseries{m}\fontshape{n}\fontsize{11}{12}\selectfont\char70}\nobreak
+\hspace{7.5pt}\nobreak\mbox{}\relax}
+
+% V1.8a allow \IEEEcompsocdiamondline to adjust for different linewidths.
+% Use \@IEEEcompsocdiamondlinei if its width is less than 0.66\linewidth (0.487 nominal for single column)
+% if not, fall back to \@IEEEcompsocdiamondlineii if its width is less than 0.75\linewidth (0.659 nominal for double column)
+% if all else fails, try to make a custom diamondline based on the abnormally narrow linewidth
+\def\IEEEcompsocdiamondline{\settowidth{\@IEEEtrantmpdimenA}{\@IEEEcompsocdiamondlinei}\relax
+\ifdim\@IEEEtrantmpdimenA<0.66\linewidth\relax\@IEEEcompsocdiamondlinei\relax
+\else
+\settowidth{\@IEEEtrantmpdimenA}{\@IEEEcompsocdiamondlineii}\relax
+\ifdim\@IEEEtrantmpdimenA<0.75\linewidth\relax\@IEEEcompsocdiamondlineii\relax
+\else
+\settowidth{\@IEEEtrantmpdimenA}{\@IEEEcompsocdiamondlineiii}\relax
+\@IEEEtrantmpdimenB=\linewidth\relax
+\addtolength{\@IEEEtrantmpdimenB}{-1\@IEEEtrantmpdimenA}\relax
+\vrule depth 0pt height 0.5pt width 0.33\@IEEEtrantmpdimenB\@IEEEcompsocdiamondlineiii\vrule depth 0pt height 0.5pt width 0.33\@IEEEtrantmpdimenB\relax
+\fi\fi}
+
+
+% V1.7 standard LateX2e \thanks, but with \itshape under compsoc. Also make it a \long\def
+% We also need to trigger the one-shot footnote rule
+\def\@IEEEtriggeroneshotfootnoterule{\global\@IEEEenableoneshotfootnoteruletrue}
+
+
+\long\def\thanks#1{\footnotemark
+    \protected@xdef\@thanks{\@thanks
+        \protect\footnotetext[\the\c@footnote]{\@IEEEcompsoconly{\itshape
+        \protect\@IEEEtriggeroneshotfootnoterule\relax}\ignorespaces#1}}}
+\let\@thanks\@empty
+
+
+% V1.7 allow \author to contain \par's. This is needed to allow \thanks to contain \par.
+\long\def\author#1{\gdef\@author{#1}}
+
+
+% in addition to setting up IEEEitemize, we need to remove a baselineskip space above and
+% below it because \list's \pars introduce blank lines because of the footnote struts.
+\def\@IEEEsetupcompsocitemizelist{\def\labelitemi{$\bullet$}%
+\setlength{\IEEElabelindent}{0pt}\setlength{\labelsep}{1.2em}\setlength{\parskip}{0pt}%
+\setlength{\partopsep}{0pt}\setlength{\topsep}{0.5\baselineskip}\vspace{-1\baselineskip}\relax}
+
+
+% flag for fake non-compsoc \IEEEcompsocthanksitem - prevents line break on very first item
+\newif\if@IEEEbreakcompsocthanksitem \@IEEEbreakcompsocthanksitemfalse
+
+\ifCLASSOPTIONcompsoc
+% V1.7 compsoc bullet item \thanks
+% also, we need to redefine this to destroy the argument in \IEEEquantizevspace
+\long\def\IEEEcompsocitemizethanks#1{\relax\@IEEEbreakcompsocthanksitemfalse\footnotemark
+    \protected@xdef\@thanks{\@thanks
+        \protect\footnotetext[\the\c@footnote]{\itshape\protect\@IEEEtriggeroneshotfootnoterule
+        {\let\IEEEiedlistdecl\relax\protect\begin{IEEEitemize}[\protect\@IEEEsetupcompsocitemizelist]\ignorespaces#1\relax
+        \protect\end{IEEEitemize}}\protect\vspace{-1\baselineskip}}}}
+\DeclareRobustCommand*{\IEEEcompsocthanksitem}{\item}
+\else
+% non-compsoc, allow for dual compilation via rerouting to normal \thanks
+\long\def\IEEEcompsocitemizethanks#1{\thanks{#1}}
+% redirect to "pseudo-par" \hfil\break\indent after swallowing [] from \IEEEcompsocthanksitem[]
+\DeclareRobustCommand{\IEEEcompsocthanksitem}{\@ifnextchar [{\@IEEEthanksswallowoptionalarg}%
+{\@IEEEthanksswallowoptionalarg[\relax]}}
+% be sure and break only after first item, be sure and ignore spaces after optional argument
+\def\@IEEEthanksswallowoptionalarg[#1]{\relax\if@IEEEbreakcompsocthanksitem\hfil\break
+\indent\fi\@IEEEbreakcompsocthanksitemtrue\ignorespaces}
+\fi
+
+
+% V1.6b define the \IEEEpeerreviewmaketitle as needed
+\ifCLASSOPTIONpeerreview
+\def\IEEEpeerreviewmaketitle{\@IEEEcleardoublepage{empty}%
+\ifCLASSOPTIONtwocolumn
+\twocolumn[{\IEEEquantizevspace{\@IEEEpeerreviewmaketitle}[\IEEEquantizedisabletitlecmds]{0pt}[-\topskip]{\baselineskip}{\@IEEENORMtitlevspace}{\@IEEEMINtitlevspace}}]
+\else
+\newpage\@IEEEpeerreviewmaketitle\@IEEEstatictitlevskip
+\fi
+\thispagestyle{IEEEtitlepagestyle}}
+\else
+% \IEEEpeerreviewmaketitle does nothing if peer review option has not been selected
+\def\IEEEpeerreviewmaketitle{\relax}
+\fi
+
+% peerreview formats the repeated title like the title in journal papers.
+\def\@IEEEpeerreviewmaketitle{\bgroup\par\addvspace{0.5\baselineskip}\centering\@IEEEcompsoconly{\sffamily}%
+\normalfont\normalsize\vskip0.2em{\Huge\@title\par}\vskip1.0em\par
+\par\addvspace{0.5\baselineskip}\egroup}
+
+
+
+% V1.6 
+% this is a static rubber spacer between the title/authors and the main text
+% used for single column text, or when the title appears in the first column
+% of two column text (technotes). 
+\def\@IEEEstatictitlevskip{{\normalfont\normalsize
+% adjust spacing to next text
+% v1.6b handle peer review papers
+\ifCLASSOPTIONpeerreview
+% for peer review papers, the same value is used for both title pages
+% regardless of the other paper modes
+   \vskip 1\baselineskip plus 0.375\baselineskip minus 0.1875\baselineskip
+\else
+   \ifCLASSOPTIONconference% conference
+      \vskip 1\baselineskip plus 0.375\baselineskip minus 0.1875\baselineskip%
+   \else%
+      \ifCLASSOPTIONtechnote% technote
+         \vskip 1\baselineskip plus 0.375\baselineskip minus 0.1875\baselineskip%
+      \else% journal uses more space
+         \vskip 2.5\baselineskip plus 0.75\baselineskip minus 0.375\baselineskip%
+      \fi
+   \fi
+\fi}}
+
+
+% set the nominal and minimum values for the quantized title spacer
+% the quantization algorithm will not allow the spacer size to
+% become less than \@IEEEMINtitlevspace - instead it will be lengthened
+% default to journal values
+\def\@IEEENORMtitlevspace{2.5\baselineskip}
+\def\@IEEEMINtitlevspace{2\baselineskip}
+% conferences and technotes need tighter spacing
+\ifCLASSOPTIONconference% conference
+  \def\@IEEENORMtitlevspace{1\baselineskip}
+  \def\@IEEEMINtitlevspace{0.75\baselineskip}
+\fi
+\ifCLASSOPTIONtechnote% technote
+  \def\@IEEENORMtitlevspace{1\baselineskip}
+  \def\@IEEEMINtitlevspace{0.75\baselineskip}
+\fi
+
+
+% V1.8a
+\def\IEEEquantizevspace{\begingroup\@ifstar{\@IEEEquantizevspacestarformtrue\@IEEEquantizevspace}{\@IEEEquantizevspacestarformfalse\@IEEEquantizevspace}}
+% \IEEEquantizevspace[output dimen register]{object}[object decl]
+%                    {top baselineskip}
+%                    [offset][prevdepth][lineskip limit][lineskip]
+%                    {unit height}{nominal vspace}{minimum vspace}
+%
+% Calculates and creates the vspace needed to make the combined height with
+% the given object an integer multiple of the given unit height. This command
+% is more general than the older \@IEEEdynamictitlevspace it replaces.
+%
+% The star form has no effect at present, but is reserved for future use.
+%
+% If the optional argument [output dimen register] is given, the calculated
+% vspace height is stored in the given output dimen (or skip) register
+% and no other action is taken, otherwise the object followed by a vspace*
+% of the appropriate height is evaluated/output.
+%
+% The optional object decl (declarations) is code that is evaluated just
+% before the object's height is evaluated. Its intented purpose is to allow
+% for the alteration or disabling of code within the object during internal
+% height evaluation (e.g., \long\def\thanks#1{\relax} ).
+% This special code is not invoked if/when the object is rendered at the end.
+%
+% The nominal vspace is the target value of the added vspace and the minimum
+% vspace is the lower allowed limit. The vspacer will be the value that achieves
+% integral overall height, in terms of the given unit height, that is closest
+% to the nominal vspace and that is not less than the specified minimum vspace.
+%
+% The line spacing algorithm of TeX is somewhat involved and requires special
+% care with regard to the first line of a vertical list (which is indicated
+% when \prevdepth is -1000pt or less). top baselineskip specifies the
+% baselineskip or topskip used prior to the object. If the height of the
+% first line of the object is greater than the given top baselineskip, then
+% the top baselineskip is subtracted from the height of the first line and
+% that difference is considered along with the rest of the object height
+% (because the object will be shifted down by an amount = 
+% top line height - top baselineskip). Otherwise, the height of the first line
+% of the object is ignored as far as the calculations are concerned.
+% This algorithm is adequate for objects that appear at the top of a page
+% (e.g., titles) where \topskip spacing is used.
+%
+% However, as explained on page 78 of the TeXbook, interline spacing is more
+% complex when \baselineskip is being used (indicated by \prevdepth > 
+% -1000pt). The four optional parameters offset, prevdepth, lineskip limit and
+% lineskip are assumed to be equal to be 0pt, \prevdepth, \lineskiplimit and
+% \lineskip, respectively, if they are omitted.
+%
+% The prevdepth is the depth of the line before the object, the lineskip limit
+% specifies how close the top of the object can come to the bottom of the
+% previous line before \baselineskip is ignored and \lineskip is inserted
+% between the object and the line above it. Lineskip does not come into
+% play unless the first line of the object is high enough to "get too close"
+% (as specified by lineskiplimit) to the line before it. The the prevdepth,
+% lineskip limit, and lineskip optional parameters are not needed for the
+% first object/line on a page (i.e., prevdepth <= -1000pt) where the simplier
+% \topskip spacing rules are in effect.
+%
+% Offset is a manual adjustment that is added to the height calculations of
+% object irrespective of the value of \prevdepth. It is useful when the top
+% baselineskip will result in a noninteger unit height object placement even
+% if the object itself has integral height. e.g., a footnotesize baselineskip
+% is used before the object, thus an offset of, say -3pt, can be given as a
+% correction.
+
+% Common combinations of these parameters include:
+%
+% top baselineskip:   (and default values for offset, prevdepth, etc.)
+% \topskip            % for objects that appear at the top of a page
+% \maxdimen           % always ignore the height of the top line
+% 0pt                 % always consider any positive height of the top line
+%
+% for objects to appear inline in normal text:
+% top baselineskip = \baselineskip
+%
+% set prevdepth = -1000pt and top baselineskip = 0pt to consider the
+% overall height of the object without any other external skip
+% consideration
+
+\newif\if@IEEEquantizevspacestarform       % flag to indicate star form
+\newif\if@IEEEquantizevspaceuseoutdimenreg % flag to indicate output dimen register is to be used
+% Use our own private registers because the object could contain a
+% structure that uses the existing tmp scratch pad registers
+\newdimen\@IEEEquantizeheightA
+\newdimen\@IEEEquantizeheightB
+\newdimen\@IEEEquantizeheightC
+\newdimen\@IEEEquantizeprevdepth % need to save this early as can change
+\newcount\@IEEEquantizemultiple
+\newbox\@IEEEquantizeboxA
+
+
+\def\@IEEEquantizevspace{\@ifnextchar [{\@IEEEquantizevspaceuseoutdimenregtrue\@@IEEEquantizevspace}{\@IEEEquantizevspaceuseoutdimenregfalse\@@IEEEquantizevspace[]}}
+
+
+\long\def\@@IEEEquantizevspace[#1]#2{\relax
+% acquire and store
+% #1 optional output dimen register
+% #2 object
+\edef\@IEEEquantizeoutdimenreg{#1}\relax
+% allow for object specifications that contain parameters
+\@IEEEtrantmptoksA={#2}\relax
+\long\edef\@IEEEquantizeobject{\the\@IEEEtrantmptoksA}\relax
+\@ifnextchar [{\@@@IEEEquantizevspace}{\@@@IEEEquantizevspace[\relax]}}
+
+\long\def\@@@IEEEquantizevspace[#1]#2{\relax
+% acquire and store
+% [#1] optional object decl, is \relax if not given by user
+% #2 top baselineskip
+% allow for object decl specifications that have parameters
+\@IEEEtrantmptoksA={#1}\relax
+\long\edef\@IEEEquantizeobjectdecl{\the\@IEEEtrantmptoksA}\relax
+\edef\@IEEEquantizetopbaselineskip{#2}\ivIEEEquantizevspace}
+
+% acquire optional argument set and store
+% [offset][prevdepth][lineskip limit][lineskip]
+\def\ivIEEEquantizevspace{\@ifnextchar [{\@vIEEEquantizevspace}{\@vIEEEquantizevspace[0pt]}}
+\def\@vIEEEquantizevspace[#1]{\edef\@IEEEquantizeoffset{#1}\@ifnextchar [{\@viIEEEquantizevspace}{\@viIEEEquantizevspace[\prevdepth]}}
+\def\@viIEEEquantizevspace[#1]{\@IEEEquantizeprevdepth=#1\relax\@ifnextchar [{\@viiIEEEquantizevspace}{\@viiIEEEquantizevspace[\lineskiplimit]}}
+\def\@viiIEEEquantizevspace[#1]{\edef\@IEEEquantizelineskiplimit{#1}\@ifnextchar [{\@viiiIEEEquantizevspace}{\@viiiIEEEquantizevspace[\lineskip]}}
+\def\@viiiIEEEquantizevspace[#1]{\edef\@IEEEquantizelineskip{#1}\@ixIEEEquantizevspace}
+
+% main routine
+\def\@ixIEEEquantizevspace#1#2#3{\relax
+\edef\@IEEEquantizeunitheight{#1}\relax
+\edef\@IEEEquantizenomvspace{#2}\relax
+\edef\@IEEEquantizeminvspace{#3}\relax
+% \@IEEEquantizeoutdimenreg
+% \@IEEEquantizeobject
+% \@IEEEquantizeobjectdecl
+% \@IEEEquantizetopbaselineskip
+% \@IEEEquantizeoffset
+% \@IEEEquantizeprevdepth
+% \@IEEEquantizelineskiplimit
+% \@IEEEquantizelineskip
+% \@IEEEquantizeunitheight
+% \@IEEEquantizenomvspace
+% \@IEEEquantizeminvspace
+% get overall height of object
+\setbox\@IEEEquantizeboxA\vbox{\begingroup\@IEEEquantizeobjectdecl\@IEEEquantizeobject\relax\endgroup}\relax
+\@IEEEquantizeheightA\ht\@IEEEquantizeboxA\relax
+% get height of first line of object
+\setbox\@IEEEquantizeboxA\vtop{\begingroup\@IEEEquantizeobjectdecl\@IEEEquantizeobject\relax\endgroup}\relax
+\@IEEEquantizeheightB\ht\@IEEEquantizeboxA\relax
+\ifdim\@IEEEquantizeprevdepth>-1000pt\relax % prevdepth > -1000pf means full baselineskip\lineskip rules in effect
+% lineskip spacing rule takes effect if height of top line > baselineskip - prevdepth - lineskiplimit, 
+% otherwise the baselineskip rule is in effect and the height of the first line does not matter at all.
+\@IEEEquantizeheightC=\@IEEEquantizetopbaselineskip\relax
+\advance\@IEEEquantizeheightC-\@IEEEquantizeprevdepth\relax
+\advance\@IEEEquantizeheightC-\@IEEEquantizelineskiplimit\relax % this works even though \@IEEEquantizelineskiplimit is a macro because TeX allows --10pt notation
+\ifdim\@IEEEquantizeheightB>\@IEEEquantizeheightC\relax
+% lineskip spacing rule is in effect i.e., the object is going to be shifted down relative to the
+% baselineskip set position by its top line height (already a part of the total height) + prevdepth + lineskip - baselineskip
+\advance\@IEEEquantizeheightA\@IEEEquantizeprevdepth\relax
+\advance\@IEEEquantizeheightA\@IEEEquantizelineskip\relax
+\advance\@IEEEquantizeheightA-\@IEEEquantizetopbaselineskip\relax
+\else
+% height of first line <= \@IEEEquantizetopbaselineskip - \@IEEEquantizeprevdepth - \@IEEEquantizelineskiplimit
+% standard baselineskip rules are in effect, so don't consider height of first line
+\advance\@IEEEquantizeheightA-\@IEEEquantizeheightB\relax
+\fi
+%
+\else % prevdepth <= -1000pt, simplier \topskip type rules in effect
+\ifdim\@IEEEquantizeheightB>\@IEEEquantizetopbaselineskip
+% height of top line (already included in the total height) in excess of
+% baselineskip is the amount it will be downshifted
+\advance\@IEEEquantizeheightA-\@IEEEquantizetopbaselineskip\relax
+\else
+% height of first line is irrelevant, remove it
+\advance\@IEEEquantizeheightA-\@IEEEquantizeheightB\relax
+\fi
+\fi % prevdepth <= -1000pt
+%
+% adjust height for any manual offset
+\advance\@IEEEquantizeheightA\@IEEEquantizeoffset\relax
+% add in nominal spacer
+\advance\@IEEEquantizeheightA\@IEEEquantizenomvspace\relax
+% check for nonzero unitheight
+\@IEEEquantizeheightB=\@IEEEquantizeunitheight\relax
+\ifnum\@IEEEquantizeheightB=0\relax
+\@IEEEclspkgerror{IEEEquantizevspace unit height cannot be zero. Assuming 10pt.}%
+{Division by zero is not allowed.}
+\@IEEEquantizeheightB=10pt\relax
+\fi
+% get integer number of lines
+\@IEEEquantizemultiple=\@IEEEquantizeheightA\relax
+\divide\@IEEEquantizemultiple\@IEEEquantizeheightB\relax
+% set A to contain the excess height over the \@IEEEquantizemultiple of lines
+% A = height - multiple*unitheight
+\@IEEEquantizeheightC\@IEEEquantizeheightB\relax
+\multiply\@IEEEquantizeheightC\@IEEEquantizemultiple\relax
+\advance\@IEEEquantizeheightA-\@IEEEquantizeheightC\relax
+% set B to contain the height short of \@IEEEquantizemultiple+1 of lines
+% B = unitheight - A
+\advance\@IEEEquantizeheightB-\@IEEEquantizeheightA\relax
+% choose A or B based on which is closer
+\@IEEEquantizeheightC\@IEEEquantizenomvspace\relax
+\ifdim\@IEEEquantizeheightA<\@IEEEquantizeheightB\relax
+% C = nomvspace - A, go with lower
+\advance\@IEEEquantizeheightC-\@IEEEquantizeheightA\relax
+\else
+% C = nomvspace + B, go with upper
+\advance\@IEEEquantizeheightC\@IEEEquantizeheightB\relax
+\fi
+% if violate lower bound, use next integer bound
+\ifdim\@IEEEquantizeheightC<\@IEEEquantizeminvspace\relax
+% A + B = unitheight
+\advance\@IEEEquantizeheightC\@IEEEquantizeheightA\relax
+\advance\@IEEEquantizeheightC\@IEEEquantizeheightB\relax
+\fi
+% export object and spacer outside of group
+\global\let\@IEEEquantizeobjectout\@IEEEquantizeobject\relax
+\global\@IEEEquantizeheightC\@IEEEquantizeheightC\relax
+\endgroup
+\if@IEEEquantizevspaceuseoutdimenreg
+\@IEEEquantizeoutdimenreg=\@IEEEquantizeheightC\relax
+\else
+\@IEEEquantizeobjectout\relax
+\vskip\@IEEEquantizeheightC\relax
+\fi}
+
+
+% user command to disable all global assignments, possible use within object decl
+\def\IEEEquantizedisableglobal{\let\global\relax
+\let\gdef\def
+\let\xdef\edef}
+% user command to allow for the disabling of \thanks and other commands, possible use within object decl
+\def\IEEEquantizedisabletitlecmds{\long\def\thanks##1{\relax}\relax
+\long\def\IEEEcompsocitemizethanks##1{\relax}\def\newpage{\relax}}
+
+
+
+
+
+% V1.6
+% we allow the user access to the last part of the title area
+% useful in emergencies such as when a different spacing is needed
+% This text is NOT compensated for in the dynamic sizer.
+\let\@IEEEaftertitletext=\relax
+\long\def\IEEEaftertitletext#1{\def\@IEEEaftertitletext{#1}}
+
+
+% V1.7 provide a way for users to enter abstract and keywords
+% into the onecolumn title are. This text is compensated for
+% in the dynamic sizer.
+\let\@IEEEtitleabstractindextext=\relax
+\long\def\IEEEtitleabstractindextext#1{\def\@IEEEtitleabstractindextext{#1}}
+
+% V1.7 provide a way for users to get the \@IEEEtitleabstractindextext if
+% not in compsoc or transmag journal mode - this way abstract and keywords
+% can still be placed in their conventional position if not in those modes.
+\def\IEEEdisplaynontitleabstractindextext{%
+% display for all conference formats
+\ifCLASSOPTIONconference\@IEEEtitleabstractindextext\relax
+\else% non-conferences
+  % V1.8a display for all technotes
+  \ifCLASSOPTIONtechnote\@IEEEtitleabstractindextext\relax
+  % V1.8a add diamond line after abstract and index terms for compsoc technotes
+  \@IEEEcompsoconly{\noindent\hfill\IEEEcompsocdiamondline\hfill\hbox{}\par}\relax
+  \else % non-conferences and non-technotes
+    \ifCLASSOPTIONcompsoc% display if not compsoc and not transmag
+    \else
+      \ifCLASSOPTIONtransmag
+      \else% not compsoc journal nor transmag journal
+         \@IEEEtitleabstractindextext\relax
+      \fi
+    \fi
+  \fi
+\fi}
+
+
+% command to allow alteration of baselinestretch, but only if the current
+% baselineskip is unity. Used to tweak the compsoc abstract and keywords line spacing.
+\def\@IEEEtweakunitybaselinestretch#1{{\def\baselinestretch{1}\selectfont
+\global\@tempskipa\baselineskip}\ifnum\@tempskipa=\baselineskip%
+\def\baselinestretch{#1}\selectfont\fi\relax}
+
+
+% abstract and keywords are in \small, except 
+% for 9pt docs in which they are in \footnotesize
+% Because 9pt docs use an 8pt footnotesize, \small
+% becomes a rather awkward 8.5pt
+\def\@IEEEabskeysecsize{\small}
+\ifx\CLASSOPTIONpt\@IEEEptsizenine
+ \def\@IEEEabskeysecsize{\footnotesize}
+\fi
+
+% compsoc journals use \footnotesize, compsoc conferences use normalsize
+\@IEEEcompsoconly{\def\@IEEEabskeysecsize{\footnotesize}}
+\@IEEEcompsocconfonly{\def\@IEEEabskeysecsize{\small}}
+
+
+% V1.6 have abstract and keywords strip leading spaces, pars and newlines
+% so that spacing is more tightly controlled.
+\def\abstract{\normalfont
+    \if@twocolumn
+      \@IEEEabskeysecsize\bfseries\textit{\abstractname}---\relax
+    \else
+      \bgroup\par\addvspace{0.5\baselineskip}\centering\vspace{-1.78ex}\@IEEEabskeysecsize\textbf{\abstractname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize
+    \fi\@IEEEgobbleleadPARNLSP}
+% V1.6 The IEEE wants only 1 pica from end of abstract to introduction heading when in 
+% conference mode (the heading already has this much above it)
+\def\endabstract{\relax\ifCLASSOPTIONconference\vspace{0ex}\else\vspace{1.34ex}\fi\par\if@twocolumn\else\endquotation\fi
+    \normalfont\normalsize}
+
+\def\IEEEkeywords{\normalfont
+    \if@twocolumn
+      \@IEEEabskeysecsize\bfseries\textit{\IEEEkeywordsname}---\relax
+    \else
+      \bgroup\par\addvspace{0.5\baselineskip}\centering\@IEEEabskeysecsize\textbf{\IEEEkeywordsname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize
+    \fi\@IEEEgobbleleadPARNLSP}
+\def\endIEEEkeywords{\relax\ifCLASSOPTIONtechnote\vspace{1.34ex}\else\vspace{0.67ex}\fi
+    \par\if@twocolumn\else\endquotation\fi%
+    \normalfont\normalsize}
+
+% V1.7 compsoc keywords index terms
+\ifCLASSOPTIONcompsoc
+  \ifCLASSOPTIONconference% compsoc conference
+\def\abstract{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\bfseries
+    \if@twocolumn
+      \@IEEEabskeysecsize\noindent\textit{\abstractname}---\relax
+    \else
+      \bgroup\par\addvspace{0.5\baselineskip}\centering\vspace{-1.78ex}\@IEEEabskeysecsize\textbf{\abstractname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize%
+    \fi\@IEEEgobbleleadPARNLSP}
+\def\IEEEkeywords{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\bfseries
+    \if@twocolumn
+      \@IEEEabskeysecsize\vskip 0.5\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip\noindent
+      \textit{\IEEEkeywordsname}---\relax
+    \else
+      \bgroup\par\addvspace{0.5\baselineskip}\centering\@IEEEabskeysecsize\textbf{\IEEEkeywordsname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize%
+    \fi\@IEEEgobbleleadPARNLSP}
+  \else% compsoc not conference
+\def\abstract{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\sffamily
+    \if@twocolumn
+      \@IEEEabskeysecsize\noindent\textbf{\abstractname}---\relax
+    \else
+      \bgroup\par\addvspace{0.5\baselineskip}\centering\vspace{-1.78ex}\@IEEEabskeysecsize\textbf{\abstractname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize%
+    \fi\@IEEEgobbleleadPARNLSP}
+\def\IEEEkeywords{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\sffamily
+    \if@twocolumn
+      \@IEEEabskeysecsize\vskip 0.5\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip\noindent
+      \textbf{\IEEEkeywordsname}---\relax
+    \else
+      \bgroup\par\addvspace{0.5\baselineskip}\centering\@IEEEabskeysecsize\textbf{\IEEEkeywordsname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize%
+    \fi\@IEEEgobbleleadPARNLSP}
+  \fi
+\fi
+
+% V1.8 transmag keywords index terms
+% no abstract name, use indentation
+\ifCLASSOPTIONtransmag
+\def\abstract{\normalfont\parindent 1em\relax
+    \if@twocolumn
+      \@IEEEabskeysecsize\bfseries\indent
+    \else
+      \bgroup\par\addvspace{0.5\baselineskip}\centering\vspace{-1.78ex}\@IEEEabskeysecsize
+      \textbf{\abstractname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize
+    \fi\@IEEEgobbleleadPARNLSP}
+
+\def\IEEEkeywords{\normalfont\parindent 1em\relax
+    \if@twocolumn
+      \@IEEEabskeysecsize\vspace{1\baselineskip}\bfseries\indent\textit{\IEEEkeywordsname}---\relax
+    \else
+      \bgroup\par\vspace{1\baselineskip}\centering\@IEEEabskeysecsize
+      \textbf{\IEEEkeywordsname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize
+    \fi\@IEEEgobbleleadPARNLSP}
+\fi
+
+
+
+% gobbles all leading \, \\ and \par, upon finding first token that
+% is not a \ , \\ or a \par, it ceases and returns that token
+% 
+% used to strip leading \, \\ and \par from the input
+% so that such things in the beginning of an environment will not
+% affect the formatting of the text
+\long\def\@IEEEgobbleleadPARNLSP#1{\let\@IEEEswallowthistoken=0%
+\let\@IEEEgobbleleadPARNLSPtoken#1%
+\let\@IEEEgobbleleadPARtoken=\par%
+\let\@IEEEgobbleleadNLtoken=\\%
+\let\@IEEEgobbleleadSPtoken=\ %
+\def\@IEEEgobbleleadSPMACRO{\ }%
+\ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadPARtoken%
+\let\@IEEEswallowthistoken=1%
+\fi%
+\ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadNLtoken%
+\let\@IEEEswallowthistoken=1%
+\fi%
+\ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadSPtoken%
+\let\@IEEEswallowthistoken=1%
+\fi%
+% a control space will come in as a macro
+% when it is the last one on a line
+\ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadSPMACRO%
+\let\@IEEEswallowthistoken=1%
+\fi%
+% if we have to swallow this token, do so and taste the next one
+% else spit it out and stop gobbling
+\ifx\@IEEEswallowthistoken 1\let\@IEEEnextgobbleleadPARNLSP=\@IEEEgobbleleadPARNLSP\else%
+\let\@IEEEnextgobbleleadPARNLSP=#1\fi%
+\@IEEEnextgobbleleadPARNLSP}%
+
+
+
+
+% TITLING OF SECTIONS
+\def\@IEEEsectpunct{:\ \,}  % Punctuation after run-in section heading  (headings which are
+                            % part of the paragraphs), need little bit more than a single space
+                            % spacing from section number to title
+% compsoc conferences use regular period/space punctuation
+\ifCLASSOPTIONcompsoc
+\ifCLASSOPTIONconference
+\def\@IEEEsectpunct{.\ }
+\fi\fi
+
+
+\def\@seccntformat#1{\csname the#1dis\endcsname\hskip 0.5em\relax}
+
+\ifCLASSOPTIONcompsoc
+% compsoc journals need extra spacing
+\ifCLASSOPTIONconference\else
+\def\@seccntformat#1{\csname the#1dis\endcsname\hskip 1em\relax}
+\fi\fi
+
+%v1.7 put {} after #6 to allow for some types of user font control
+%and use \@@par rather than \par
+\def\@sect#1#2#3#4#5#6[#7]#8{%
+  \ifnum #2>\c@secnumdepth
+     \let\@svsec\@empty
+  \else
+     \refstepcounter{#1}%
+     % load section label and spacer into \@svsec
+     \protected@edef\@svsec{\@seccntformat{#1}\relax}%
+  \fi%
+  \@tempskipa #5\relax
+  \ifdim \@tempskipa>\z@% tempskipa determines whether is treated as a high
+     \begingroup #6{\relax% or low level heading
+      \noindent % subsections are NOT indented
+       % print top level headings. \@svsec is label, #8 is heading title
+       % The IEEE does not block indent the section title text, it flows like normal
+       {\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\@@par}}%
+     \endgroup
+     \addcontentsline{toc}{#1}{\ifnum #2>\c@secnumdepth\relax\else
+               \protect\numberline{\csname the#1\endcsname}\fi#7}%
+  \else % printout low level headings
+     % svsechd seems to swallow the trailing space, protect it with \mbox{}
+     % got rid of sectionmark stuff
+     \def\@svsechd{#6{\hskip #3\relax\@svsec #8\@IEEEsectpunct\mbox{}}%
+     \addcontentsline{toc}{#1}{\ifnum #2>\c@secnumdepth\relax\else
+               \protect\numberline{\csname the#1\endcsname}\fi#7}}%
+  \fi%skip down
+  \@xsect{#5}}
+
+
+% section* handler
+%v1.7 put {} after #4 to allow for some types of user font control
+%and use \@@par rather than \par
+\def\@ssect#1#2#3#4#5{\@tempskipa #3\relax
+  \ifdim \@tempskipa>\z@
+     %\begingroup #4\@hangfrom{\hskip #1}{\interlinepenalty \@M #5\par}\endgroup
+     % The IEEE does not block indent the section title text, it flows like normal
+     \begingroup \noindent #4{\relax{\hskip #1}{\interlinepenalty \@M #5\@@par}}\endgroup
+  % svsechd swallows the trailing space, protect it with \mbox{}
+  \else \def\@svsechd{#4{\hskip #1\relax #5\@IEEEsectpunct\mbox{}}}\fi
+  \@xsect{#3}}
+
+
+%% SECTION heading spacing and font
+%%
+% arguments are: #1 - sectiontype name
+% (for \@sect)   #2 - section level
+%                #3 - section heading indent
+%                #4 - top separation (absolute value used, neg indicates not to indent main text)
+%                     If negative, make stretch parts negative too!
+%                #5 - (absolute value used) positive: bottom separation after heading,
+%                      negative: amount to indent main text after heading
+%                Both #4 and #5 negative means to indent main text and use negative top separation
+%                #6 - font control
+% You've got to have \normalfont\normalsize in the font specs below to prevent
+% trouble when you do something like:
+% \section{Note}{\ttfamily TT-TEXT} is known to ... 
+% The IEEE sometimes REALLY stretches the area before a section
+% heading by up to about 0.5in. However, it may not be a good
+% idea to let LaTeX have quite this much rubber.
+\ifCLASSOPTIONconference%
+% The IEEE wants section heading spacing to decrease for conference mode
+\def\section{\@startsection{section}{1}{\z@}{1.5ex plus 1.5ex minus 0.5ex}%
+{0.7ex plus 1ex minus 0ex}{\normalfont\normalsize\centering\scshape}}%
+\def\subsection{\@startsection{subsection}{2}{\z@}{1.5ex plus 1.5ex minus 0.5ex}%
+{0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\itshape}}%
+\else % for journals
+\def\section{\@startsection{section}{1}{\z@}{3.0ex plus 1.5ex minus 1.5ex}% V1.6 3.0ex from 3.5ex
+{0.7ex plus 1ex minus 0ex}{\normalfont\normalsize\centering\scshape}}%
+\def\subsection{\@startsection{subsection}{2}{\z@}{3.5ex plus 1.5ex minus 1.5ex}%
+{0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\itshape}}%
+\fi
+
+% for both journals and conferences
+% decided to put in a little rubber above the section, might help somebody
+\def\subsubsection{\@startsection{subsubsection}{3}{\parindent}{0ex plus 0.1ex minus 0.1ex}%
+{0ex}{\normalfont\normalsize\itshape}}%
+\def\paragraph{\@startsection{paragraph}{4}{2\parindent}{0ex plus 0.1ex minus 0.1ex}%
+{0ex}{\normalfont\normalsize\itshape}}%
+
+
+% compsoc
+\ifCLASSOPTIONcompsoc
+\ifCLASSOPTIONconference
+% compsoc conference
+\def\section{\@startsection{section}{1}{\z@}{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}%
+{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}{\normalfont\large\bfseries}}%
+\def\subsection{\@startsection{subsection}{2}{\z@}{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}%
+{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}{\normalfont\sublargesize\bfseries}}%
+\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}%
+{0ex}{\normalfont\normalsize\bfseries}}%
+\def\paragraph{\@startsection{paragraph}{4}{2\parindent}{0ex plus 0.1ex minus 0.1ex}%
+{0ex}{\normalfont\normalsize}}%
+\else% compsoc journals
+% use negative top separation as compsoc journals do not indent paragraphs after section titles
+\def\section{\@startsection{section}{1}{\z@}{-3.5ex plus -2ex minus -1.5ex}%
+{0.7ex plus 1ex minus 0ex}{\normalfont\sublargesize\sffamily\bfseries\scshape}}%
+% Note that subsection and smaller may not be correct for the Computer Society,
+% I have to look up an example.
+\def\subsection{\@startsection{subsection}{2}{\z@}{-3.5ex plus -1.5ex minus -1.5ex}%
+{0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\sffamily\bfseries}}%
+\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-2.5ex plus -1ex minus -1ex}%
+{0.5ex plus 0.5ex minus 0ex}{\normalfont\normalsize\sffamily\itshape}}%
+\def\paragraph{\@startsection{paragraph}{4}{2\parindent}{-0ex plus -0.1ex minus -0.1ex}%
+{0ex}{\normalfont\normalsize}}%
+\fi\fi
+
+% transmag
+\ifCLASSOPTIONtransmag
+\def\subsection{\@startsection{subsection}{2}{0.75\parindent}{3.5ex plus 1.5ex minus 1.5ex}%
+{0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\itshape}}%
+\def\subsubsection{\@startsection{subsubsection}{3}{1.25\parindent}{0.1ex plus 0.1ex minus 0.1ex}%
+{0.1ex}{\normalfont\normalsize\itshape}}%
+\fi
+
+
+% V1.8a provide for a raised line Introduction section for use with Computer
+% Society papers. We have to remove any spacing glue after the section
+% heading and then remove the blank line for the new paragraph after it.
+% LaTeX's section handler alters \everypar and we need to propogate those
+% changes outside of the \parbox lest there be spacing problems at the top
+% of the next section.
+\def\IEEEraisesectionheading#1{\noindent\raisebox{1.5\baselineskip}[0pt][0pt]{\parbox[b]{\columnwidth}{#1\unskip\global\everypar=\everypar}}\vspace{-1\baselineskip}\vspace{-\parskip}\par}
+
+
+
+%% ENVIRONMENTS
+% "box" symbols at end of proofs
+\def\IEEEQEDclosed{\mbox{\rule[0pt]{1.3ex}{1.3ex}}} % for a filled box
+% V1.6 some journals use an open box instead that will just fit around a closed one
+\def\IEEEQEDopen{{\setlength{\fboxsep}{0pt}\setlength{\fboxrule}{0.2pt}\fbox{\rule[0pt]{0pt}{1.3ex}\rule[0pt]{1.3ex}{0pt}}}}
+\ifCLASSOPTIONcompsoc
+\def\IEEEQED{\IEEEQEDopen}   % default to open for compsoc
+\else
+\def\IEEEQED{\IEEEQEDclosed} % otherwise default to closed
+\fi
+
+%V1.8 flag to indicate if QED symbol is to be shown
+\newif\if@IEEEQEDshow  \@IEEEQEDshowtrue
+\def\IEEEproofindentspace{2\parindent}% V1.8 allow user to change indentation amount if desired
+% v1.7 name change to avoid namespace collision with amsthm. Also add support
+% for an optional argument.
+\def\IEEEproof{\@ifnextchar[{\@IEEEproof}{\@IEEEproof[\IEEEproofname]}}
+\def\@IEEEproof[#1]{\@IEEEQEDshowtrue\par\noindent\hspace{\IEEEproofindentspace}{\itshape #1: }}
+\def\endIEEEproof{\if@IEEEQEDshow\hspace*{\fill}\nobreakspace\IEEEQED\fi\par}
+% qedhere for equation environments, similar to AMS \qedhere
+\def\IEEEQEDhereeqn{\global\@IEEEQEDshowfalse\eqno\let\eqno\relax\let\leqno\relax
+                    \let\veqno\relax\hbox{\IEEEQED}}
+% IEEE style qedhere for IEEEeqnarray and other environments
+\def\IEEEQEDhere{\global\@IEEEQEDshowfalse\IEEEQED}
+% command to disable QED at end of IEEEproof
+\def\IEEEQEDoff{\global\@IEEEQEDshowfalse}
+
+
+%\itemindent is set to \z@ by list, so define new temporary variable
+\newdimen\@IEEEtmpitemindent
+
+\ifCLASSOPTIONcompsoc
+% V1.8a compsoc uses bold theorem titles, a period instead of a colon, vertical spacing, and hanging indentation
+% V1.8 allow long theorem names to break across lines.
+% Thanks to Miquel Payaro for reporting this.
+\def\@begintheorem#1#2{\@IEEEtmpitemindent\itemindent\relax
+    \topsep 0.2\@IEEEnormalsizeunitybaselineskip plus 0.26\@IEEEnormalsizeunitybaselineskip minus 0.05\@IEEEnormalsizeunitybaselineskip
+    \rmfamily\trivlist\hangindent\parindent%
+    \item[]\textit{\bfseries\noindent #1\ #2.} \itemindent\@IEEEtmpitemindent\relax}
+\def\@opargbegintheorem#1#2#3{\@IEEEtmpitemindent\itemindent\relax
+\topsep 0.2\@IEEEnormalsizeunitybaselineskip plus 0.26\@IEEEnormalsizeunitybaselineskip minus 0.05\@IEEEnormalsizeunitybaselineskip
+\rmfamily\trivlist\hangindent\parindent%
+% V1.6 The IEEE is back to using () around theorem names which are also in italics
+% Thanks to Christian Peel for reporting this.
+    \item[]\textit{\bfseries\noindent #1\ #2\ (#3).} \itemindent\@IEEEtmpitemindent\relax}
+% V1.7 remove bogus \unskip that caused equations in theorems to collide with
+% lines below.
+\def\@endtheorem{\endtrivlist\vskip 0.25\@IEEEnormalsizeunitybaselineskip plus 0.26\@IEEEnormalsizeunitybaselineskip minus 0.05\@IEEEnormalsizeunitybaselineskip}
+\else
+%
+% noncompsoc
+% 
+% V1.8 allow long theorem names to break across lines.
+% Thanks to Miquel Payaro for reporting this.
+\def\@begintheorem#1#2{\@IEEEtmpitemindent\itemindent\relax\topsep 0pt\rmfamily\trivlist%
+    \item[]\textit{\indent #1\ #2:} \itemindent\@IEEEtmpitemindent\relax}
+\def\@opargbegintheorem#1#2#3{\@IEEEtmpitemindent\itemindent\relax\topsep 0pt\rmfamily \trivlist%
+% V1.6 The IEEE is back to using () around theorem names which are also in italics
+% Thanks to Christian Peel for reporting this.
+    \item[]\textit{\indent #1\ #2\ (#3):} \itemindent\@IEEEtmpitemindent\relax}
+% V1.7 remove bogus \unskip that caused equations in theorems to collide with
+% lines below.
+\def\@endtheorem{\endtrivlist}
+\fi
+
+
+
+% V1.6
+% display command for the section the theorem is in - so that \thesection
+% is not used as this will be in Roman numerals when we want arabic.
+% LaTeX2e uses \def\@thmcounter#1{\noexpand\arabic{#1}} for the theorem number
+% (second part) display and \def\@thmcountersep{.} as a separator.
+% V1.7 intercept calls to the section counter and reroute to \@IEEEthmcounterinsection
+% to allow \appendix(ices} to override as needed.
+%
+% special handler for sections, allows appendix(ices) to override
+\gdef\@IEEEthmcounterinsection#1{\arabic{#1}}
+% string macro
+\edef\@IEEEstringsection{section}
+
+% redefine the #1#2[#3] form of newtheorem to use a hook to \@IEEEthmcounterinsection
+% if section in_counter is used
+\def\@xnthm#1#2[#3]{%
+  \expandafter\@ifdefinable\csname #1\endcsname
+    {\@definecounter{#1}\@newctr{#1}[#3]%
+     \edef\@IEEEstringtmp{#3}
+     \ifx\@IEEEstringtmp\@IEEEstringsection
+     \expandafter\xdef\csname the#1\endcsname{%
+     \noexpand\@IEEEthmcounterinsection{#3}\@thmcountersep
+          \@thmcounter{#1}}%
+     \else
+     \expandafter\xdef\csname the#1\endcsname{%
+       \expandafter\noexpand\csname the#3\endcsname \@thmcountersep
+          \@thmcounter{#1}}%
+     \fi
+     \global\@namedef{#1}{\@thm{#1}{#2}}%
+     \global\@namedef{end#1}{\@endtheorem}}}
+
+
+
+%% SET UP THE DEFAULT PAGESTYLE
+\pagestyle{headings}
+\pagenumbering{arabic}
+
+% normally the page counter starts at 1
+\setcounter{page}{1}
+% however, for peerreview the cover sheet is page 0 or page -1
+% (for duplex printing)
+\ifCLASSOPTIONpeerreview
+   \if@twoside
+      \setcounter{page}{-1}
+   \else
+      \setcounter{page}{0}
+   \fi
+\fi
+
+% standard book class behavior - let bottom line float up and down as
+% needed when single sided
+\ifCLASSOPTIONtwoside\else\raggedbottom\fi
+% if two column - turn on twocolumn, allow word spacings to stretch more and
+% enforce a rigid position for the last lines
+\ifCLASSOPTIONtwocolumn
+% the peer review option delays invoking twocolumn
+   \ifCLASSOPTIONpeerreview\else
+      \twocolumn
+   \fi
+\sloppy 
+\flushbottom
+\fi
+
+
+
+
+% \APPENDIX and \APPENDICES definitions
+
+% This is the \@ifmtarg command from the LaTeX ifmtarg package
+% by Peter Wilson (CUA) and Donald Arseneau
+% \@ifmtarg is used to determine if an argument to a command
+% is present or not.
+% For instance:
+% \@ifmtarg{#1}{\typeout{empty}}{\typeout{has something}}
+% \@ifmtarg is used with our redefined \section command if
+% \appendices is invoked.
+% The command \section will behave slightly differently depending
+% on whether the user specifies a title: 
+% \section{My appendix title}
+% or not:
+% \section{}
+% This way, we can eliminate the blank lines where the title
+% would be, and the unneeded : after Appendix in the table of
+% contents 
+\begingroup
+\catcode`\Q=3
+\long\gdef\@ifmtarg#1{\@xifmtarg#1QQ\@secondoftwo\@firstoftwo\@nil}
+\long\gdef\@xifmtarg#1#2Q#3#4#5\@nil{#4}
+\endgroup
+% end of \@ifmtarg defs
+
+
+% V1.7
+% command that allows the one time saving of the original definition
+% of section to \@IEEEappendixsavesection for \appendix or \appendices 
+% we don't save \section here as it may be redefined later by other
+% packages (hyperref.sty, etc.)
+\def\@IEEEsaveoriginalsectiononce{\let\@IEEEappendixsavesection\section
+\let\@IEEEsaveoriginalsectiononce\relax}
+
+% neat trick to grab and process the argument from \section{argument}
+% we process differently if the user invoked \section{} with no
+% argument (title)
+% note we reroute the call to the old \section*
+\def\@IEEEprocessthesectionargument#1{%
+\@ifmtarg{#1}{%
+\@IEEEappendixsavesection*{\appendixname\nobreakspace\thesectiondis}%
+\addcontentsline{toc}{section}{\appendixname\nobreakspace\thesection}}{%
+\@IEEEappendixsavesection*{\appendixname\nobreakspace\thesectiondis\\* #1}%
+\addcontentsline{toc}{section}{\appendixname\nobreakspace\thesection: #1}}}
+
+% we use this if the user calls \section{} after
+% \appendix-- which has no meaning. So, we ignore the
+% command and its argument. Then, warn the user.
+\def\@IEEEdestroythesectionargument#1{\typeout{** WARNING: Ignoring useless
+\protect\section\space in Appendix (line \the\inputlineno).}}
+
+
+% remember \thesection forms will be displayed in \ref calls
+% and in the Table of Contents.
+% The \sectiondis form is used in the actual heading itself
+
+% appendix command for one single appendix
+% normally has no heading. However, if you want a 
+% heading, you can do so via the optional argument:
+% \appendix[Optional Heading]
+\def\appendix{\relax}
+\renewcommand{\appendix}[1][]{\@IEEEsaveoriginalsectiononce\par
+    % v1.6 keep hyperref's identifiers unique
+    \gdef\theHsection{Appendix.A}%
+    % v1.6 adjust hyperref's string name for the section
+    \xdef\Hy@chapapp{appendix}%
+    \setcounter{section}{0}%
+    \setcounter{subsection}{0}%
+    \setcounter{subsubsection}{0}%
+    \setcounter{paragraph}{0}%
+    \gdef\thesection{A}%
+    \gdef\thesectiondis{}% 
+    \gdef\thesubsection{\Alph{subsection}}%
+    \gdef\@IEEEthmcounterinsection##1{A}
+    \refstepcounter{section}% update the \ref counter
+    \@ifmtarg{#1}{\@IEEEappendixsavesection*{\appendixname}%
+                  \addcontentsline{toc}{section}{\appendixname}}{%
+             \@IEEEappendixsavesection*{\appendixname\nobreakspace\\* #1}%
+             \addcontentsline{toc}{section}{\appendixname: #1}}%
+    % redefine \section command for appendix
+    % leave \section* as is
+    \def\section{\@ifstar{\@IEEEappendixsavesection*}{%
+                    \@IEEEdestroythesectionargument}}% throw out the argument
+                                                     % of the normal form
+}
+
+
+
+% appendices command for multiple appendices
+% user then calls \section with an argument (possibly empty) to
+% declare the individual appendices
+\def\appendices{\@IEEEsaveoriginalsectiononce\par
+    % v1.6 keep hyperref's identifiers unique
+    \gdef\theHsection{Appendix.\Alph{section}}%
+    % v1.6 adjust hyperref's string name for the section
+    \xdef\Hy@chapapp{appendix}%
+    \setcounter{section}{-1}% we want \refstepcounter to use section 0
+    \setcounter{subsection}{0}%
+    \setcounter{subsubsection}{0}%
+    \setcounter{paragraph}{0}%
+    \ifCLASSOPTIONromanappendices%
+    \gdef\thesection{\Roman{section}}%
+    \gdef\thesectiondis{\Roman{section}}%
+    \@IEEEcompsocconfonly{\gdef\thesectiondis{\Roman{section}.}}%
+    \gdef\@IEEEthmcounterinsection##1{A\arabic{##1}}
+    \else%
+    \gdef\thesection{\Alph{section}}%
+    \gdef\thesectiondis{\Alph{section}}%
+    \@IEEEcompsocconfonly{\gdef\thesectiondis{\Alph{section}.}}%
+    \gdef\@IEEEthmcounterinsection##1{\Alph{##1}}
+    \fi%
+    \refstepcounter{section}% update the \ref counter
+    \setcounter{section}{0}% NEXT \section will be the FIRST appendix
+    % redefine \section command for appendices
+    % leave \section* as is
+    \def\section{\@ifstar{\@IEEEappendixsavesection*}{% process the *-form
+                    \refstepcounter{section}% or is a new section so,
+                    \@IEEEprocessthesectionargument}}% process the argument 
+                                                 % of the normal form
+}
+
+
+
+% V1.7 compoc uses nonbold drop cap and small caps word style
+\ifCLASSOPTIONcompsoc
+  \def\IEEEPARstartFONTSTYLE{\mdseries}
+  \def\IEEEPARstartWORDFONTSTYLE{\scshape}
+  \def\IEEEPARstartWORDCAPSTYLE{\relax}
+\fi
+%
+%
+% \IEEEPARstart
+% Definition for the big two line drop cap letter at the beginning of the
+% first paragraph of journal papers. The first argument is the first letter
+% of the first word, the second argument is the remaining letters of the
+% first word which will be rendered in upper case.
+% In V1.6 this has been completely rewritten to:
+% 
+% 1. no longer have problems when the user begins an environment
+%    within the paragraph that uses \IEEEPARstart.
+% 2. auto-detect and use the current font family
+% 3. revise handling of the space at the end of the first word so that
+%    interword glue will now work as normal.
+% 4. produce correctly aligned edges for the (two) indented lines.
+% 
+% We generalize things via control macros - playing with these is fun too.
+% 
+% V1.7 added more control macros to make it easy for IEEEtrantools.sty users
+% to change the font style.
+% 
+% the number of lines that are indented to clear it
+% may need to increase if using decenders
+\providecommand{\IEEEPARstartDROPLINES}{2}
+% minimum number of lines left on a page to allow a \@IEEEPARstart
+% Does not take into consideration rubber shrink, so it tends to
+% be overly cautious
+\providecommand{\IEEEPARstartMINPAGELINES}{2}
+% V1.7 the height of the drop cap is adjusted to match the height of this text
+% in the current font (when \IEEEPARstart is called).
+\providecommand{\IEEEPARstartHEIGHTTEXT}{T}
+% the depth the letter is lowered below the baseline
+% the height (and size) of the letter is determined by the sum
+% of this value and the height of the \IEEEPARstartHEIGHTTEXT in the current
+% font. It is a good idea to set this value in terms of the baselineskip
+% so that it can respond to changes therein.
+\providecommand{\IEEEPARstartDROPDEPTH}{1.1\baselineskip}
+% V1.7 the font the drop cap will be rendered in,
+% can take zero or one argument.
+\providecommand{\IEEEPARstartFONTSTYLE}{\bfseries}
+% V1.7 any additional, non-font related commands needed to modify
+% the drop cap letter, can take zero or one argument.
+\providecommand{\IEEEPARstartCAPSTYLE}{\MakeUppercase}
+% V1.7 the font that will be used to render the rest of the word,
+% can take zero or one argument.
+\providecommand{\IEEEPARstartWORDFONTSTYLE}{\relax}
+% V1.7 any additional, non-font related commands needed to modify
+% the rest of the word, can take zero or one argument.
+\providecommand{\IEEEPARstartWORDCAPSTYLE}{\MakeUppercase}
+% This is the horizontal separation distance from the drop letter to the main text.
+% Lengths that depend on the font (e.g., ex, em, etc.) will be referenced
+% to the font that is active when \IEEEPARstart is called. 
+\providecommand{\IEEEPARstartSEP}{0.15em}
+% V1.7 horizontal offset applied to the left of the drop cap.
+\providecommand{\IEEEPARstartHOFFSET}{0em}
+% V1.7 Italic correction command applied at the end of the drop cap.
+\providecommand{\IEEEPARstartITLCORRECT}{\/}
+
+% width of the letter output, set globally. Can be used in \IEEEPARstartSEP
+% or \IEEEPARstartHOFFSET, but not the height lengths.
+\newdimen\IEEEPARstartletwidth
+\IEEEPARstartletwidth 0pt\relax
+
+% definition of \IEEEPARstart
+% THIS IS A CONTROLLED SPACING AREA, DO NOT ALLOW SPACES WITHIN THESE LINES
+% 
+% The token \@IEEEPARstartfont will be globally defined after the first use
+% of \IEEEPARstart and will be a font command which creates the big letter
+% The first argument is the first letter of the first word and the second
+% argument is the rest of the first word(s).
+\def\IEEEPARstart#1#2{\par{%
+% if this page does not have enough space, break it and lets start
+% on a new one
+\@IEEEtranneedspace{\IEEEPARstartMINPAGELINES\baselineskip}{\relax}%
+% V1.7 move this up here in case user uses \textbf for \IEEEPARstartFONTSTYLE
+% which uses command \leavevmode which causes an unwanted \indent to be issued
+\noindent
+% calculate the desired height of the big letter
+% it extends from the top of \IEEEPARstartHEIGHTTEXT in the current font
+% down to \IEEEPARstartDROPDEPTH below the current baseline
+\settoheight{\@IEEEtrantmpdimenA}{\IEEEPARstartHEIGHTTEXT}%
+\addtolength{\@IEEEtrantmpdimenA}{\IEEEPARstartDROPDEPTH}%
+% extract the name of the current font in bold
+% and place it in \@IEEEPARstartFONTNAME
+\def\@IEEEPARstartGETFIRSTWORD##1 ##2\relax{##1}%
+{\IEEEPARstartFONTSTYLE{\selectfont\edef\@IEEEPARstartFONTNAMESPACE{\fontname\font\space}%
+\xdef\@IEEEPARstartFONTNAME{\expandafter\@IEEEPARstartGETFIRSTWORD\@IEEEPARstartFONTNAMESPACE\relax}}}%
+% define a font based on this name with a point size equal to the desired
+% height of the drop letter
+\font\@IEEEPARstartsubfont\@IEEEPARstartFONTNAME\space at \@IEEEtrantmpdimenA\relax%
+% save this value as a counter (integer) value (sp points)
+\@IEEEtrantmpcountA=\@IEEEtrantmpdimenA%
+% now get the height of the actual letter produced by this font size
+\settoheight{\@IEEEtrantmpdimenB}{\@IEEEPARstartsubfont\IEEEPARstartCAPSTYLE{#1}}%
+% If something bogus happens like the first argument is empty or the
+% current font is strange, do not allow a zero height.
+\ifdim\@IEEEtrantmpdimenB=0pt\relax%
+\typeout{** WARNING: IEEEPARstart drop letter has zero height! (line \the\inputlineno)}%
+\typeout{ Forcing the drop letter font size to 10pt.}%
+\@IEEEtrantmpdimenB=10pt%
+\fi%
+% and store it as a counter
+\@IEEEtrantmpcountB=\@IEEEtrantmpdimenB%
+% Since a font size doesn't exactly correspond to the height of the capital
+% letters in that font, the actual height of the letter, \@IEEEtrantmpcountB,
+% will be less than that desired, \@IEEEtrantmpcountA
+% we need to raise the font size, \@IEEEtrantmpdimenA 
+% by \@IEEEtrantmpcountA / \@IEEEtrantmpcountB
+% But, TeX doesn't have floating point division, so we have to use integer
+% division. Hence the use of the counters.
+% We need to reduce the denominator so that the loss of the remainder will
+% have minimal affect on the accuracy of the result
+\divide\@IEEEtrantmpcountB by 200%
+\divide\@IEEEtrantmpcountA by \@IEEEtrantmpcountB%
+% Then reequalize things when we use TeX's ability to multiply by
+% floating point values
+\@IEEEtrantmpdimenB=0.005\@IEEEtrantmpdimenA%
+\multiply\@IEEEtrantmpdimenB by \@IEEEtrantmpcountA%
+% \@IEEEPARstartfont is globaly set to the calculated font of the big letter
+% We need to carry this out of the local calculation area to to create the
+% big letter.
+\global\font\@IEEEPARstartfont\@IEEEPARstartFONTNAME\space at \@IEEEtrantmpdimenB%
+% Now set \@IEEEtrantmpdimenA to the width of the big letter
+% We need to carry this out of the local calculation area to set the
+% hanging indent
+\settowidth{\global\@IEEEtrantmpdimenA}{\@IEEEPARstartfont
+\IEEEPARstartCAPSTYLE{#1\IEEEPARstartITLCORRECT}}}%
+% end of the isolated calculation environment
+\global\IEEEPARstartletwidth\@IEEEtrantmpdimenA\relax%
+% add in the extra clearance we want
+\advance\@IEEEtrantmpdimenA by \IEEEPARstartSEP\relax%
+% add in the optional offset
+\advance\@IEEEtrantmpdimenA by \IEEEPARstartHOFFSET\relax%
+% V1.7 don't allow negative offsets to produce negative hanging indents
+\@IEEEtrantmpdimenB\@IEEEtrantmpdimenA
+\ifnum\@IEEEtrantmpdimenB < 0 \@IEEEtrantmpdimenB 0pt\fi
+% \@IEEEtrantmpdimenA has the width of the big letter plus the
+% separation space and \@IEEEPARstartfont is the font we need to use
+% Now, we make the letter and issue the hanging indent command
+% The letter is placed in a box of zero width and height so that other
+% text won't be displaced by it.
+\hangindent\@IEEEtrantmpdimenB\hangafter=-\IEEEPARstartDROPLINES%
+\makebox[0pt][l]{\hspace{-\@IEEEtrantmpdimenA}%
+\raisebox{-\IEEEPARstartDROPDEPTH}[0pt][0pt]{\hspace{\IEEEPARstartHOFFSET}%
+\@IEEEPARstartfont\IEEEPARstartCAPSTYLE{#1\IEEEPARstartITLCORRECT}%
+\hspace{\IEEEPARstartSEP}}}%
+{\IEEEPARstartWORDFONTSTYLE{\IEEEPARstartWORDCAPSTYLE{\selectfont#2}}}}
+
+
+
+
+% determines if the space remaining on a given page is equal to or greater
+% than the specified space of argument one
+% if not, execute argument two (only if the remaining space is greater than zero)
+% and issue a \newpage
+% 
+% example: \@IEEEtranneedspace{2in}{\vfill}
+% 
+% Does not take into consideration rubber shrinkage, so it tends to
+% be overly cautious
+% Based on an example posted by Donald Arseneau
+% Note this macro uses \@IEEEtrantmpdimenB internally for calculations,
+% so DO NOT PASS \@IEEEtrantmpdimenB to this routine
+% if you need a dimen register, import with \@IEEEtrantmpdimenA instead
+\def\@IEEEtranneedspace#1#2{\penalty-100\begingroup%shield temp variable
+\@IEEEtrantmpdimenB\pagegoal\advance\@IEEEtrantmpdimenB-\pagetotal% space left
+\ifdim #1>\@IEEEtrantmpdimenB\relax% not enough space left
+\ifdim\@IEEEtrantmpdimenB>\z@\relax #2\fi%
+\newpage%
+\fi\endgroup}
+
+
+
+% IEEEbiography ENVIRONMENT
+% Allows user to enter biography leaving place for picture (adapts to font size)
+% As of V1.5, a new optional argument allows you to have a real graphic!
+% V1.5 and later also fixes the "colliding biographies" which could happen when a 
+% biography's text was shorter than the space for the photo.
+% MDS 7/2001
+% V1.6 prevent multiple biographies from making multiple TOC entries
+\newif\if@IEEEbiographyTOCentrynotmade
+\global\@IEEEbiographyTOCentrynotmadetrue
+
+% biography counter so hyperref can jump directly to the biographies
+% and not just the previous section
+\newcounter{IEEEbiography}
+\setcounter{IEEEbiography}{0}
+
+% photo area size
+\def\@IEEEBIOphotowidth{1.0in}    % width of the biography photo area
+\def\@IEEEBIOphotodepth{1.25in}   % depth (height) of the biography photo area
+% area cleared for photo
+\def\@IEEEBIOhangwidth{1.14in}    % width cleared for the biography photo area
+\def\@IEEEBIOhangdepth{1.25in}    % depth cleared for the biography photo area
+                                  % actual depth will be a multiple of 
+                                  % \baselineskip, rounded up
+\def\@IEEEBIOskipN{4\baselineskip}% nominal value of the vskip above the biography
+
+\newenvironment{IEEEbiography}[2][]{\normalfont\@IEEEcompsoconly{\sffamily}\footnotesize%
+\unitlength 1in\parskip=0pt\par\parindent 1em\interlinepenalty500%
+% we need enough space to support the hanging indent
+% the nominal value of the spacer
+% and one extra line for good measure
+\@IEEEtrantmpdimenA=\@IEEEBIOhangdepth%
+\advance\@IEEEtrantmpdimenA by \@IEEEBIOskipN%
+\advance\@IEEEtrantmpdimenA by 1\baselineskip%
+% if this page does not have enough space, break it and lets start
+% with a new one
+\@IEEEtranneedspace{\@IEEEtrantmpdimenA}{\relax}%
+% nominal spacer can strech, not shrink use 1fil so user can out stretch with \vfill
+\vskip \@IEEEBIOskipN plus 1fil minus 0\baselineskip%
+% the default box for where the photo goes
+\def\@IEEEtempbiographybox{{\setlength{\fboxsep}{0pt}\framebox{%
+\begin{minipage}[b][\@IEEEBIOphotodepth][c]{\@IEEEBIOphotowidth}\centering PLACE\\ PHOTO\\ HERE \end{minipage}}}}%
+%
+% detect if the optional argument was supplied, this requires the
+% \@ifmtarg command as defined in the appendix section above
+% and if so, override the default box with what they want
+\@ifmtarg{#1}{\relax}{\def\@IEEEtempbiographybox{\mbox{\begin{minipage}[b][\@IEEEBIOphotodepth][c]{\@IEEEBIOphotowidth}%
+\centering%
+#1%
+\end{minipage}}}}% end if optional argument supplied
+% Make an entry into the table of contents only if we have not done so before
+\if@IEEEbiographyTOCentrynotmade%
+% link labels to the biography counter so hyperref will jump
+% to the biography, not the previous section
+\setcounter{IEEEbiography}{-1}%
+\refstepcounter{IEEEbiography}%
+\addcontentsline{toc}{section}{Biographies}%
+\global\@IEEEbiographyTOCentrynotmadefalse%
+\fi%
+% one more biography
+\refstepcounter{IEEEbiography}%
+% Make an entry for this name into the table of contents 
+\addcontentsline{toc}{subsection}{#2}%
+% V1.6 properly handle if a new paragraph should occur while the
+% hanging indent is still active. Do this by redefining \par so
+% that it will not start a new paragraph. (But it will appear to the
+% user as if it did.) Also, strip any leading pars, newlines, or spaces.
+\let\@IEEEBIOORGparCMD=\par% save the original \par command
+\edef\par{\hfil\break\indent}% the new \par will not be a "real" \par
+\settoheight{\@IEEEtrantmpdimenA}{\@IEEEtempbiographybox}% get height of biography box
+\@IEEEtrantmpdimenB=\@IEEEBIOhangdepth%
+\@IEEEtrantmpcountA=\@IEEEtrantmpdimenB% countA has the hang depth
+\divide\@IEEEtrantmpcountA by \baselineskip%  calculates lines needed to produce the hang depth
+\advance\@IEEEtrantmpcountA by 1% ensure we overestimate
+% set the hanging indent
+\hangindent\@IEEEBIOhangwidth%
+\hangafter-\@IEEEtrantmpcountA%
+% reference the top of the photo area to the top of a capital T
+\settoheight{\@IEEEtrantmpdimenB}{\mbox{T}}%
+% set the photo box, give it zero width and height so as not to disturb anything
+\noindent\makebox[0pt][l]{\hspace{-\@IEEEBIOhangwidth}\raisebox{\@IEEEtrantmpdimenB}[0pt][0pt]{%
+\raisebox{-\@IEEEBIOphotodepth}[0pt][0pt]{\@IEEEtempbiographybox}}}%
+% now place the author name and begin the bio text
+\noindent\textbf{#2\ }\@IEEEgobbleleadPARNLSP}{\relax\let\par=\@IEEEBIOORGparCMD\par%
+% 7/2001 V1.5 detect when the biography text is shorter than the photo area
+% and pad the unused area - preventing a collision from the next biography entry
+% MDS
+\ifnum \prevgraf <\@IEEEtrantmpcountA\relax% detect when the biography text is shorter than the photo
+    \advance\@IEEEtrantmpcountA by -\prevgraf% calculate how many lines we need to pad
+    \advance\@IEEEtrantmpcountA by -1\relax% we compensate for the fact that we indented an extra line
+    \@IEEEtrantmpdimenA=\baselineskip% calculate the length of the padding
+    \multiply\@IEEEtrantmpdimenA by \@IEEEtrantmpcountA%
+    \noindent\rule{0pt}{\@IEEEtrantmpdimenA}% insert an invisible support strut
+\fi%
+\par\normalfont}
+
+
+
+% V1.6
+% added biography without a photo environment
+\newenvironment{IEEEbiographynophoto}[1]{%
+% Make an entry into the table of contents only if we have not done so before
+\if@IEEEbiographyTOCentrynotmade%
+% link labels to the biography counter so hyperref will jump
+% to the biography, not the previous section
+\setcounter{IEEEbiography}{-1}%
+\refstepcounter{IEEEbiography}%
+\addcontentsline{toc}{section}{Biographies}%
+\global\@IEEEbiographyTOCentrynotmadefalse%
+\fi%
+% one more biography
+\refstepcounter{IEEEbiography}%
+% Make an entry for this name into the table of contents 
+\addcontentsline{toc}{subsection}{#1}%
+\normalfont\@IEEEcompsoconly{\sffamily}\footnotesize\interlinepenalty500%
+\vskip 4\baselineskip plus 1fil minus 0\baselineskip%
+\parskip=0pt\par%
+\noindent\textbf{#1\ }\@IEEEgobbleleadPARNLSP}{\relax\par\normalfont}
+
+
+% provide the user with some old font commands
+% got this from article.cls
+\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
+\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
+\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
+\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
+\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
+\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
+\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
+\DeclareRobustCommand*\cal{\@fontswitch\relax\mathcal}
+\DeclareRobustCommand*\mit{\@fontswitch\relax\mathnormal}
+
+
+% SPECIAL PAPER NOTICE COMMANDS
+% 
+% holds the special notice text
+\def\@IEEEspecialpapernotice{\relax}
+ 
+% for special papers, like invited papers, the user can do:
+% \IEEEspecialpapernotice{(Invited Paper)} before \maketitle
+\def\IEEEspecialpapernotice#1{\ifCLASSOPTIONconference%
+\def\@IEEEspecialpapernotice{{\sublargesize\textit{#1}\vspace*{1em}}}%
+\else%
+\def\@IEEEspecialpapernotice{{\\*[1.5ex]\sublargesize\textit{#1}}\vspace*{-2ex}}%
+\fi}
+
+
+
+
+% PUBLISHER ID COMMANDS
+% to insert a publisher's ID footer
+% V1.6 \IEEEpubid has been changed so that the change in page size and style
+% occurs in \maketitle. \IEEEpubid must now be issued prior to \maketitle
+% use \IEEEpubidadjcol as before - in the second column of the title page
+% These changes allow \maketitle to take the reduced page height into
+% consideration when dynamically setting the space between the author 
+% names and the maintext.
+%
+% the amount the main text is pulled up to make room for the
+% publisher's ID footer
+% The IEEE uses about 1.3\baselineskip for journals, 
+% dynamic title spacing will clean up the fraction
+\def\@IEEEpubidpullup{1.3\baselineskip}
+\ifCLASSOPTIONtechnote
+% for technotes it must be an integer of baselineskip as there can be no
+% dynamic title spacing for two column mode technotes (the title is in the
+% in first column) and we should maintain an integer number of lines in the
+% second column
+% There are some examples (such as older issues of "Transactions on
+% Information Theory") in which the IEEE really pulls the text off the ID for
+% technotes - about 0.55in (or 4\baselineskip). We'll use 2\baselineskip
+% and call it even.
+\def\@IEEEpubidpullup{2\baselineskip}
+\fi
+
+% V1.7 compsoc does not use a pullup
+\ifCLASSOPTIONcompsoc
+\def\@IEEEpubidpullup{0pt}
+\fi
+
+% holds the ID text
+\def\@IEEEpubid{\relax}
+
+% flag so \maketitle can tell if \IEEEpubid was called
+\newif\if@IEEEusingpubid
+\global\@IEEEusingpubidfalse
+% issue this command in the page to have the ID at the bottom
+% V1.6 use before \maketitle
+\def\IEEEpubid#1{\def\@IEEEpubid{#1}\global\@IEEEusingpubidtrue}
+
+
+% command which will pull up (shorten) the column it is executed in
+% to make room for the publisher ID. Place in the second column of
+% the title page when using \IEEEpubid
+% Is smart enough not to do anything when in single column text or
+% if the user hasn't called \IEEEpubid
+% currently needed in for the second column of a page with the
+% publisher ID. If not needed in future releases, please provide this
+% command and define it as \relax for backward compatibility
+% v1.6b do not allow command to operate if the peer review option has been 
+% selected because \IEEEpubidadjcol will not be on the cover page.
+% V1.7 do nothing if compsoc
+\def\IEEEpubidadjcol{\ifCLASSOPTIONcompsoc\else\ifCLASSOPTIONpeerreview\else
+\if@twocolumn\if@IEEEusingpubid\enlargethispage{-\@IEEEpubidpullup}\fi\fi\fi\fi}
+
+% Special thanks to Peter Wilson, Daniel Luecking, and the other
+% gurus at comp.text.tex, for helping me to understand how best to
+% implement the IEEEpubid command in LaTeX.
+
+
+
+%% Lockout some commands under various conditions
+
+% general purpose bit bucket
+\newsavebox{\@IEEEtranrubishbin}
+
+% flags to prevent multiple warning messages
+\newif\if@IEEEWARNthanks
+\newif\if@IEEEWARNIEEEPARstart
+\newif\if@IEEEWARNIEEEbiography
+\newif\if@IEEEWARNIEEEbiographynophoto
+\newif\if@IEEEWARNIEEEpubid
+\newif\if@IEEEWARNIEEEpubidadjcol
+\newif\if@IEEEWARNIEEEmembership
+\newif\if@IEEEWARNIEEEaftertitletext
+\@IEEEWARNthankstrue
+\@IEEEWARNIEEEPARstarttrue
+\@IEEEWARNIEEEbiographytrue
+\@IEEEWARNIEEEbiographynophototrue
+\@IEEEWARNIEEEpubidtrue
+\@IEEEWARNIEEEpubidadjcoltrue
+\@IEEEWARNIEEEmembershiptrue
+\@IEEEWARNIEEEaftertitletexttrue
+
+
+%% Lockout some commands when in various modes, but allow them to be restored if needed
+%%
+% save commands which might be locked out
+% so that the user can later restore them if needed
+\let\@IEEESAVECMDthanks\thanks
+\let\@IEEESAVECMDIEEEPARstart\IEEEPARstart
+\let\@IEEESAVECMDIEEEbiography\IEEEbiography
+\let\@IEEESAVECMDendIEEEbiography\endIEEEbiography
+\let\@IEEESAVECMDIEEEbiographynophoto\IEEEbiographynophoto
+\let\@IEEESAVECMDendIEEEbiographynophoto\endIEEEbiographynophoto
+\let\@IEEESAVECMDIEEEpubid\IEEEpubid
+\let\@IEEESAVECMDIEEEpubidadjcol\IEEEpubidadjcol
+\let\@IEEESAVECMDIEEEmembership\IEEEmembership
+\let\@IEEESAVECMDIEEEaftertitletext\IEEEaftertitletext
+
+
+% disable \IEEEPARstart when in draft mode
+% This may have originally been done because the pre-V1.6 drop letter
+% algorithm had problems with a non-unity baselinestretch
+% At any rate, it seems too formal to have a drop letter in a draft
+% paper.
+\ifCLASSOPTIONdraftcls
+\def\IEEEPARstart#1#2{#1#2\if@IEEEWARNIEEEPARstart\typeout{** ATTENTION: \noexpand\IEEEPARstart
+ is disabled in draft mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEPARstartfalse}
+\fi
+% and for technotes
+\ifCLASSOPTIONtechnote
+\def\IEEEPARstart#1#2{#1#2\if@IEEEWARNIEEEPARstart\typeout{** WARNING: \noexpand\IEEEPARstart
+ is locked out for technotes (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEPARstartfalse}
+\fi
+
+
+% lockout unneeded commands when in conference mode
+\ifCLASSOPTIONconference
+% when locked out, \thanks, \IEEEbiography, \IEEEbiographynophoto, \IEEEpubid,
+% \IEEEmembership and \IEEEaftertitletext will all swallow their given text. 
+% \IEEEPARstart will output a normal character instead
+% warn the user about these commands only once to prevent the console screen
+% from filling up with redundant messages
+\def\thanks#1{\if@IEEEWARNthanks\typeout{** WARNING: \noexpand\thanks
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNthanksfalse}
+\def\IEEEPARstart#1#2{#1#2\if@IEEEWARNIEEEPARstart\typeout{** WARNING: \noexpand\IEEEPARstart
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEPARstartfalse}
+
+
+% LaTeX treats environments and commands with optional arguments differently.
+% the actual ("internal") command is stored as \\commandname 
+% (accessed via \csname\string\commandname\endcsname )
+% the "external" command \commandname is a macro with code to determine
+% whether or not the optional argument is presented and to provide the 
+% default if it is absent. So, in order to save and restore such a command
+% we would have to save and restore \\commandname as well. But, if LaTeX
+% ever changes the way it names the internal names, the trick would break.
+% Instead let us just define a new environment so that the internal
+% name can be left undisturbed.
+\newenvironment{@IEEEbogusbiography}[2][]{\if@IEEEWARNIEEEbiography\typeout{** WARNING: \noexpand\IEEEbiography
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEbiographyfalse%
+\setbox\@IEEEtranrubishbin\vbox\bgroup}{\egroup\relax}
+% and make biography point to our bogus biography
+\let\IEEEbiography=\@IEEEbogusbiography
+\let\endIEEEbiography=\end@IEEEbogusbiography
+
+\renewenvironment{IEEEbiographynophoto}[1]{\if@IEEEWARNIEEEbiographynophoto\typeout{** WARNING: \noexpand\IEEEbiographynophoto
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEbiographynophotofalse%
+\setbox\@IEEEtranrubishbin\vbox\bgroup}{\egroup\relax}
+
+\def\IEEEpubid#1{\if@IEEEWARNIEEEpubid\typeout{** WARNING: \noexpand\IEEEpubid 
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEpubidfalse}
+\def\IEEEpubidadjcol{\if@IEEEWARNIEEEpubidadjcol\typeout{** WARNING: \noexpand\IEEEpubidadjcol
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEpubidadjcolfalse}
+\def\IEEEmembership#1{\if@IEEEWARNIEEEmembership\typeout{** WARNING: \noexpand\IEEEmembership
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEmembershipfalse}
+\def\IEEEaftertitletext#1{\if@IEEEWARNIEEEaftertitletext\typeout{** WARNING: \noexpand\IEEEaftertitletext
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEaftertitletextfalse}
+\fi
+
+
+% provide a way to restore the commands that are locked out
+\def\IEEEoverridecommandlockouts{%
+\typeout{** ATTENTION: Overriding command lockouts (line \the\inputlineno).}%
+\let\thanks\@IEEESAVECMDthanks%
+\let\IEEEPARstart\@IEEESAVECMDIEEEPARstart%
+\let\IEEEbiography\@IEEESAVECMDIEEEbiography%
+\let\endIEEEbiography\@IEEESAVECMDendIEEEbiography%
+\let\IEEEbiographynophoto\@IEEESAVECMDIEEEbiographynophoto%
+\let\endIEEEbiographynophoto\@IEEESAVECMDendIEEEbiographynophoto%
+\let\IEEEpubid\@IEEESAVECMDIEEEpubid%
+\let\IEEEpubidadjcol\@IEEESAVECMDIEEEpubidadjcol%
+\let\IEEEmembership\@IEEESAVECMDIEEEmembership%
+\let\IEEEaftertitletext\@IEEESAVECMDIEEEaftertitletext}
+
+
+
+% need a backslash character for typeout output
+{\catcode`\|=0 \catcode`\\=12
+|xdef|@IEEEbackslash{\}}
+
+
+% hook to allow easy disabling of all legacy warnings
+\def\@IEEElegacywarn#1#2{\typeout{** ATTENTION: \@IEEEbackslash #1 is deprecated (line \the\inputlineno).
+Use \@IEEEbackslash #2 instead.}}
+
+
+% provide some legacy IEEEtran commands
+\def\IEEEcompsoctitleabstractindextext{\@IEEElegacywarn{IEEEcompsoctitleabstractindextext}{IEEEtitleabstractindextext}\IEEEtitleabstractindextext}
+\def\IEEEdisplaynotcompsoctitleabstractindextext{\@IEEElegacywarn{IEEEdisplaynotcompsoctitleabstractindextext}{IEEEdisplaynontitleabstractindextext}\IEEEdisplaynontitleabstractindextext}
+% provide some legacy IEEEtran environments
+
+
+% V1.8a no more support for these legacy commands
+%\def\authorblockA{\@IEEElegacywarn{authorblockA}{IEEEauthorblockA}\IEEEauthorblockA}
+%\def\authorblockN{\@IEEElegacywarn{authorblockN}{IEEEauthorblockN}\IEEEauthorblockN}
+%\def\authorrefmark{\@IEEElegacywarn{authorrefmark}{IEEEauthorrefmark}\IEEEauthorrefmark}
+%\def\PARstart{\@IEEElegacywarn{PARstart}{IEEEPARstart}\IEEEPARstart}
+%\def\pubid{\@IEEElegacywarn{pubid}{IEEEpubid}\IEEEpubid}
+%\def\pubidadjcol{\@IEEElegacywarn{pubidadjcol}{IEEEpubidadjcol}\IEEEpubidadjcol}
+%\def\specialpapernotice{\@IEEElegacywarn{specialpapernotice}{IEEEspecialpapernotice}\IEEEspecialpapernotice}
+% and environments
+%\def\keywords{\@IEEElegacywarn{keywords}{IEEEkeywords}\IEEEkeywords}
+%\def\endkeywords{\endIEEEkeywords}
+% V1.8 no more support for legacy IED list commands
+%\let\labelindent\IEEElabelindent
+%\def\calcleftmargin{\@IEEElegacywarn{calcleftmargin}{IEEEcalcleftmargin}\IEEEcalcleftmargin}
+%\def\setlabelwidth{\@IEEElegacywarn{setlabelwidth}{IEEEsetlabelwidth}\IEEEsetlabelwidth}
+%\def\usemathlabelsep{\@IEEElegacywarn{usemathlabelsep}{IEEEusemathlabelsep}\IEEEusemathlabelsep}
+%\def\iedlabeljustifyc{\@IEEElegacywarn{iedlabeljustifyc}{IEEEiedlabeljustifyc}\IEEEiedlabeljustifyc}
+%\def\iedlabeljustifyl{\@IEEElegacywarn{iedlabeljustifyl}{IEEEiedlabeljustifyl}\IEEEiedlabeljustifyl}
+%\def\iedlabeljustifyr{\@IEEElegacywarn{iedlabeljustifyr}{IEEEiedlabeljustifyr}\IEEEiedlabeljustifyr}
+% V1.8 no more support for QED and proof stuff
+%\def\QED{\@IEEElegacywarn{QED}{IEEEQED}\IEEEQED}
+%\def\QEDclosed{\@IEEElegacywarn{QEDclosed}{IEEEQEDclosed}\IEEEQEDclosed}
+%\def\QEDopen{\@IEEElegacywarn{QEDopen}{IEEEQEDopen}\IEEEQEDopen}
+%\AtBeginDocument{\def\proof{\@IEEElegacywarn{proof}{IEEEproof}\IEEEproof}\def\endproof{\endIEEEproof}}
+% V1.8 no longer support biography or biographynophoto
+%\def\biography{\@IEEElegacywarn{biography}{IEEEbiography}\IEEEbiography}
+%\def\biographynophoto{\@IEEElegacywarn{biographynophoto}{IEEEbiographynophoto}\IEEEbiographynophoto}
+%\def\endbiography{\endIEEEbiography}
+%\def\endbiographynophoto{\endIEEEbiographynophoto}
+% V1.7 and later no longer supports \overrideIEEEmargins
+%\def\overrideIEEEmargins{%
+%\typeout{** WARNING: \string\overrideIEEEmargins \space no longer supported (line \the\inputlineno).}%
+%\typeout{** Use the \string\CLASSINPUTinnersidemargin, \string\CLASSINPUToutersidemargin \space controls instead.}}
+
+\endinput
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%% End of IEEEtran.cls  %%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% That's all folks!
+
diff --git a/article/IEEEtran_HOWTO.pdf b/article/IEEEtran_HOWTO.pdf
new file mode 100644
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
new file mode 100644
index 0000000..07b0739
--- /dev/null
+++ b/article/cutting-planes-article.tex
@@ -0,0 +1,288 @@
+\documentclass[conference]{IEEEtran}
+\IEEEoverridecommandlockouts
+% The preceding line is only needed to identify funding in the first footnote. If that is unneeded, please comment it out.
+\usepackage{cite}
+\usepackage{amsmath,amssymb,amsfonts}
+\usepackage{algorithmic}
+\usepackage{graphicx}
+\usepackage{textcomp}
+\usepackage{xcolor}
+\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
+    T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
+\begin{document}
+
+\title{Conference Planes Title*\\
+{\footnotesize \textsuperscript{*}Note: Sub-titles are not captured in Xplore and
+should not be used}
+\thanks{Identify applicable funding agency here. If none, delete this.}
+}
+
+\author{\IEEEauthorblockN{1\textsuperscript{st} Given Name Surname}
+    \IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
+        \textit{name of organization (of Aff.)}\\
+        City, Country \\
+        email address or ORCID}
+    \and
+    \IEEEauthorblockN{2\textsuperscript{nd} Given Name Surname}
+    \IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
+        \textit{name of organization (of Aff.)}\\
+        City, Country \\
+        email address or ORCID}
+    \and
+    \IEEEauthorblockN{3\textsuperscript{rd} Given Name Surname}
+    \IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
+        \textit{name of organization (of Aff.)}\\
+        City, Country \\
+        email address or ORCID}
+    \and
+    \IEEEauthorblockN{4\textsuperscript{th} Given Name Surname}
+    \IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
+        \textit{name of organization (of Aff.)}\\
+        City, Country \\
+        email address or ORCID}
+    \and
+    \IEEEauthorblockN{5\textsuperscript{th} Given Name Surname}
+    \IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
+        \textit{name of organization (of Aff.)}\\
+        City, Country \\
+        email address or ORCID}
+    \and
+    \IEEEauthorblockN{6\textsuperscript{th} Given Name Surname}
+    \IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
+        \textit{name of organization (of Aff.)}\\
+        City, Country \\
+        email address or ORCID}
+}
+
+\maketitle
+
+\begin{abstract}
+    This document is a model and instructions for \LaTeX.
+    This and the IEEEtran.cls file define the components of your paper [title, text, heads, etc.]. *CRITICAL: Do Not Use Symbols, Special Characters, Footnotes,
+    or Math in Paper Title or Abstract.
+\end{abstract}
+
+\begin{IEEEkeywords}
+    component, formatting, style, styling, insert
+\end{IEEEkeywords}
+
+\section{Introduction}
+This document is a model and instructions for \LaTeX.
+Please observe the conference page limits.
+
+\section{Ease of Use}
+
+\subsection{Maintaining the Integrity of the Specifications}
+
+The IEEEtran class file is used to format your paper and style the text. All margins,
+column widths, line spaces, and text fonts are prescribed; please do not
+alter them. You may note peculiarities. For example, the head margin
+measures proportionately more than is customary. This measurement
+and others are deliberate, using specifications that anticipate your paper
+as one part of the entire proceedings, and not as an independent document.
+Please do not revise any of the current designations.
+
+\section{Prepare Your Paper Before Styling}
+Before you begin to format your paper, first write and save the content as a
+separate text file. Complete all content and organizational editing before
+formatting. Please note sections \ref{AA}--\ref{SCM} below for more information on
+proofreading, spelling and grammar.
+
+Keep your text and graphic files separate until after the text has been
+formatted and styled. Do not number text heads---{\LaTeX} will do that
+for you.
+
+\subsection{Abbreviations and Acronyms}\label{AA}
+Define abbreviations and acronyms the first time they are used in the text,
+even after they have been defined in the abstract. Abbreviations such as
+IEEE, SI, MKS, CGS, ac, dc, and rms do not have to be defined. Do not use
+abbreviations in the title or heads unless they are unavoidable.
+
+\subsection{Units}
+\begin{itemize}
+    \item Use either SI (MKS) or CGS as primary units. (SI units are encouraged.) English units may be used as secondary units (in parentheses). An exception would be the use of English units as identifiers in trade, such as ``3.5-inch disk drive''.
+    \item Avoid combining SI and CGS units, such as current in amperes and magnetic field in oersteds. This often leads to confusion because equations do not balance dimensionally. If you must use mixed units, clearly state the units for each quantity that you use in an equation.
+    \item Do not mix complete spellings and abbreviations of units: ``Wb/m\textsuperscript{2}'' or ``webers per square meter'', not ``webers/m\textsuperscript{2}''. Spell out units when they appear in text: ``. . . a few henries'', not ``. . . a few H''.
+    \item Use a zero before decimal points: ``0.25'', not ``.25''. Use ``cm\textsuperscript{3}'', not ``cc''.)
+\end{itemize}
+
+\subsection{Equations}
+Number equations consecutively. To make your
+equations more compact, you may use the solidus (~/~), the exp function, or
+appropriate exponents. Italicize Roman symbols for quantities and variables,
+but not Greek symbols. Use a long dash rather than a hyphen for a minus
+sign. Punctuate equations with commas or periods when they are part of a
+sentence, as in:
+\begin{equation}
+    a+b=\gamma\label{eq}
+\end{equation}
+
+Be sure that the
+symbols in your equation have been defined before or immediately following
+the equation. Use ``\eqref{eq}'', not ``Eq.~\eqref{eq}'' or ``equation \eqref{eq}'', except at
+the beginning of a sentence: ``Equation \eqref{eq} is . . .''
+
+\subsection{\LaTeX-Specific Advice}
+
+Please use ``soft'' (e.g., \verb|\eqref{Eq}|) cross references instead
+of ``hard'' references (e.g., \verb|(1)|). That will make it possible
+to combine sections, add equations, or change the order of figures or
+citations without having to go through the file line by line.
+
+Please don't use the \verb|{eqnarray}| equation environment. Use
+\verb|{align}| or \verb|{IEEEeqnarray}| instead. The \verb|{eqnarray}|
+environment leaves unsightly spaces around relation symbols.
+
+Please note that the \verb|{subequations}| environment in {\LaTeX}
+will increment the main equation counter even when there are no
+equation numbers displayed. If you forget that, you might write an
+article in which the equation numbers skip from (17) to (20), causing
+the copy editors to wonder if you've discovered a new method of
+counting.
+
+    {\BibTeX} does not work by magic. It doesn't get the bibliographic
+data from thin air but from .bib files. If you use {\BibTeX} to produce a
+bibliography you must send the .bib files.
+
+    {\LaTeX} can't read your mind. If you assign the same label to a
+subsubsection and a table, you might find that Table I has been cross
+referenced as Table IV-B3.
+
+{\LaTeX} does not have precognitive abilities. If you put a
+\verb|\label| command before the command that updates the counter it's
+supposed to be using, the label will pick up the last counter to be
+cross referenced instead. In particular, a \verb|\label| command
+should not go before the caption of a figure or a table.
+
+Do not use \verb|\nonumber| inside the \verb|{array}| environment. It
+will not stop equation numbers inside \verb|{array}| (there won't be
+any anyway) and it might stop a wanted equation number in the
+surrounding equation.
+
+\subsection{Some Common Mistakes}\label{SCM}
+\begin{itemize}
+    \item The word ``data'' is plural, not singular.
+    \item The subscript for the permeability of vacuum $\mu_{0}$, and other common scientific constants, is zero with subscript formatting, not a lowercase letter ``o''.
+    \item In American English, commas, semicolons, periods, question and exclamation marks are located within quotation marks only when a complete thought or name is cited, such as a title or full quotation. When quotation marks are used, instead of a bold or italic typeface, to highlight a word or phrase, punctuation should appear outside of the quotation marks. A parenthetical phrase or statement at the end of a sentence is punctuated outside of the closing parenthesis (like this). (A parenthetical sentence is punctuated within the parentheses.)
+    \item A graph within a graph is an ``inset'', not an ``insert''. The word alternatively is preferred to the word ``alternately'' (unless you really mean something that alternates).
+    \item Do not use the word ``essentially'' to mean ``approximately'' or ``effectively''.
+    \item In your paper title, if the words ``that uses'' can accurately replace the word ``using'', capitalize the ``u''; if not, keep using lower-cased.
+    \item Be aware of the different meanings of the homophones ``affect'' and ``effect'', ``complement'' and ``compliment'', ``discreet'' and ``discrete'', ``principal'' and ``principle''.
+    \item Do not confuse ``imply'' and ``infer''.
+    \item The prefix ``non'' is not a word; it should be joined to the word it modifies, usually without a hyphen.
+    \item There is no period after the ``et'' in the Latin abbreviation ``et al.''.
+    \item The abbreviation ``i.e.'' means ``that is'', and the abbreviation ``e.g.'' means ``for example''.
+\end{itemize}
+An excellent style manual for science writers is \cite{b7}.
+
+\subsection{Authors and Affiliations}
+\textbf{The class file is designed for, but not limited to, six authors.} A
+minimum of one author is required for all conference articles. Author names
+should be listed starting from left to right and then moving down to the
+next line. This is the author sequence that will be used in future citations
+and by indexing services. Names should not be listed in columns nor group by
+affiliation. Please keep your affiliations as succinct as possible (for
+example, do not differentiate among departments of the same organization).
+
+\subsection{Identify the Headings}
+Headings, or heads, are organizational devices that guide the reader through
+your paper. There are two types: component heads and text heads.
+
+Component heads identify the different components of your paper and are not
+topically subordinate to each other. Examples include Acknowledgments and
+References and, for these, the correct style to use is ``Heading 5''. Use
+``figure caption'' for your Figure captions, and ``table head'' for your
+table title. Run-in heads, such as ``Abstract'', will require you to apply a
+style (in this case, italic) in addition to the style provided by the drop
+down menu to differentiate the head from the text.
+
+Text heads organize the topics on a relational, hierarchical basis. For
+example, the paper title is the primary text head because all subsequent
+material relates and elaborates on this one topic. If there are two or more
+sub-topics, the next level head (uppercase Roman numerals) should be used
+and, conversely, if there are not at least two sub-topics, then no subheads
+should be introduced.
+
+\subsection{Figures and Tables}
+\paragraph{Positioning Figures and Tables} Place figures and tables at the top and
+bottom of columns. Avoid placing them in the middle of columns. Large
+figures and tables may span across both columns. Figure captions should be
+below the figures; table heads should appear above the tables. Insert
+figures and tables after they are cited in the text. Use the abbreviation
+``Fig.~\ref{fig}'', even at the beginning of a sentence.
+
+\begin{table}[htbp]
+    \caption{Table Type Styles}
+    \begin{center}
+        \begin{tabular}{|c|c|c|c|}
+            \hline
+            \textbf{Table} & \multicolumn{3}{|c|}{\textbf{Table Column Head}}                                                         \\
+            \cline{2-4}
+            \textbf{Head}  & \textbf{\textit{Table column subhead}}           & \textbf{\textit{Subhead}} & \textbf{\textit{Subhead}} \\
+            \hline
+            copy           & More table copy$^{\mathrm{a}}$                   &                           &                           \\
+            \hline
+            \multicolumn{4}{l}{$^{\mathrm{a}}$Sample of a Table footnote.}
+        \end{tabular}
+        \label{tab1}
+    \end{center}
+\end{table}
+
+\begin{figure}[htbp]
+    \centerline{\includegraphics{fig1.png}}
+    \caption{Example of a figure caption.}
+    \label{fig}
+\end{figure}
+
+Figure Labels: Use 8 point Times New Roman for Figure labels. Use words
+rather than symbols or abbreviations when writing Figure axis labels to
+avoid confusing the reader. As an example, write the quantity
+``Magnetization'', or ``Magnetization, M'', not just ``M''. If including
+units in the label, present them within parentheses. Do not label axes only
+with units. In the example, write ``Magnetization (A/m)'' or ``Magnetization
+\{A[m(1)]\}'', not just ``A/m''. Do not label axes with a ratio of
+quantities and units. For example, write ``Temperature (K)'', not
+``Temperature/K''.
+
+\section*{Acknowledgment}
+
+The preferred spelling of the word ``acknowledgment'' in America is without
+an ``e'' after the ``g''. Avoid the stilted expression ``one of us (R. B.
+G.) thanks $\ldots$''. Instead, try ``R. B. G. thanks$\ldots$''. Put sponsor
+acknowledgments in the unnumbered footnote on the first page.
+
+\section*{References}
+
+Please number citations consecutively within brackets \cite{b1}. The
+sentence punctuation follows the bracket \cite{b2}. Refer simply to the reference
+number, as in \cite{b3}---do not use ``Ref. \cite{b3}'' or ``reference \cite{b3}'' except at
+the beginning of a sentence: ``Reference \cite{b3} was the first $\ldots$''
+
+Number footnotes separately in superscripts. Place the actual footnote at
+the bottom of the column in which it was cited. Do not put footnotes in the
+abstract or reference list. Use letters for table footnotes.
+
+Unless there are six authors or more give all authors' names; do not use
+``et al.''. Papers that have not been published, even if they have been
+submitted for publication, should be cited as ``unpublished'' \cite{b4}. Papers
+that have been accepted for publication should be cited as ``in press'' \cite{b5}.
+Capitalize only the first word in a paper title, except for proper nouns and
+element symbols.
+
+For papers published in translation journals, please give the English
+citation first, followed by the original foreign-language citation \cite{b6}.
+
+\begin{thebibliography}{00}
+    \bibitem{b1} G. Eason, B. Noble, and I. N. Sneddon, ``On certain integrals of Lipschitz-Hankel type involving products of Bessel functions,'' Phil. Trans. Roy. Soc. London, vol. A247, pp. 529--551, April 1955.
+    \bibitem{b2} J. Clerk Maxwell, A Treatise on Electricity and Magnetism, 3rd ed., vol. 2. Oxford: Clarendon, 1892, pp.68--73.
+    \bibitem{b3} I. S. Jacobs and C. P. Bean, ``Fine particles, thin films and exchange anisotropy,'' in Magnetism, vol. III, G. T. Rado and H. Suhl, Eds. New York: Academic, 1963, pp. 271--350.
+    \bibitem{b4} K. Elissa, ``Title of paper if known,'' unpublished.
+    \bibitem{b5} R. Nicole, ``Title of paper with only first word capitalized,'' J. Name Stand. Abbrev., in press.
+    \bibitem{b6} Y. Yorozu, M. Hirano, K. Oka, and Y. Tagawa, ``Electron spectroscopy studies on magneto-optical media and plastic substrate interface,'' IEEE Transl. J. Magn. Japan, vol. 2, pp. 740--741, August 1987 [Digests 9th Annual Conf. Magnetics Japan, p. 301, 1982].
+    \bibitem{b7} M. Young, The Technical Writer's Handbook. Mill Valley, CA: University Science, 1989.
+\end{thebibliography}
+\vspace{12pt}
+\color{red}
+IEEE conference templates contain guidance text for composing and formatting conference papers. Please ensure that all template text is removed from your conference paper prior to submission to the conference. Failure to remove the template text from your paper may result in your paper not being published.
+
+\end{document}
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diff --git a/docs/poc.tex b/docs/poc.tex
deleted file mode 100644
index a5efd09..0000000
--- a/docs/poc.tex
+++ /dev/null
@@ -1,48 +0,0 @@
-\documentclass[a4paper]{article}
-\usepackage{listings}
-\usepackage{color}
-
-% \lstset{style=bw}
-
-\begin{document}
-
-\section{Introduction}
-
-\[
-    \sum_i{a} = b + c
-\]
-
-%Custom Colors
-\definecolor{plainText}{RGB}{74,131,31}
-\definecolor{comments}{RGB}{74,131,31}
-\definecolor{strings}{RGB}{180,54,34}
-\definecolor{numbers}{RGB}{44,45,211}
-\definecolor{keywords}{RGB}{160,49,158}
-\definecolor{preprocessorStatements}{RGB}{109,75,48}
-\definecolor{classNames}{RGB}{101,63,165}
-
-
-\lstset{frame=tb,
-    language=C++,
-    aboveskip=3mm,
-    belowskip=3mm,
-    showstringspaces=false,
-    columns=flexible,
-    basicstyle={\small\ttfamily},
-    numbers=left,
-    numberstyle=\tiny\color{plainText},
-    keywordstyle=\color{keywords},
-    commentstyle=\color{comments},
-    stringstyle=\color{strings},
-    breaklines=true,
-    breakatwhitespace=true
-    tabsize=4
-}
-
-\lstset{language=python}
-\begin{lstlisting}
-    def foo:
-        print "oi"
-\end{lstlisting}
-
-\end{document}
\ No newline at end of file

From 91c126b59db7cba9da39c3af3a83e49b9317d836 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 26 Jun 2024 14:05:12 -0300
Subject: [PATCH 64/93] =?UTF-8?q?Definindo=20se=C3=A7=C3=B5es=20do=20artig?=
 =?UTF-8?q?o?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 article/cutting-planes-article.pdf | Bin 134996 -> 52217 bytes
 article/cutting-planes-article.tex | 275 +++--------------------------
 2 files changed, 23 insertions(+), 252 deletions(-)

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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 07b0739..2f5b508 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -11,278 +11,49 @@
     T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
 \begin{document}
 
-\title{Conference Planes Title*\\
-{\footnotesize \textsuperscript{*}Note: Sub-titles are not captured in Xplore and
-should not be used}
-\thanks{Identify applicable funding agency here. If none, delete this.}
+\title{Modelando e verificando o cálculo formal de Planos de Corte com Lean 4\\
+{\footnotesize Projeto Orientadado em Computação I - Pesquisa Científica}
 }
 
-\author{\IEEEauthorblockN{1\textsuperscript{st} Given Name Surname}
-    \IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
-        \textit{name of organization (of Aff.)}\\
-        City, Country \\
-        email address or ORCID}
-    \and
-    \IEEEauthorblockN{2\textsuperscript{nd} Given Name Surname}
-    \IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
-        \textit{name of organization (of Aff.)}\\
-        City, Country \\
-        email address or ORCID}
-    \and
-    \IEEEauthorblockN{3\textsuperscript{rd} Given Name Surname}
-    \IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
-        \textit{name of organization (of Aff.)}\\
-        City, Country \\
-        email address or ORCID}
-    \and
-    \IEEEauthorblockN{4\textsuperscript{th} Given Name Surname}
-    \IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
-        \textit{name of organization (of Aff.)}\\
-        City, Country \\
-        email address or ORCID}
-    \and
-    \IEEEauthorblockN{5\textsuperscript{th} Given Name Surname}
-    \IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
-        \textit{name of organization (of Aff.)}\\
-        City, Country \\
-        email address or ORCID}
-    \and
-    \IEEEauthorblockN{6\textsuperscript{th} Given Name Surname}
-    \IEEEauthorblockA{\textit{dept. name of organization (of Aff.)} \\
-        \textit{name of organization (of Aff.)}\\
-        City, Country \\
-        email address or ORCID}
+\author{\IEEEauthorblockN{Bernardo Borges}
+    \IEEEauthorblockA{\textit{Departamento de Ciência da Computação} \\
+        \textit{Universidade Federal de Minas Gerais}\\
+        Belo Horizonte, Brasil\\
+        bernardoborges@dcc.ufmg.br}
 }
 
 \maketitle
 
 \begin{abstract}
-    This document is a model and instructions for \LaTeX.
-    This and the IEEEtran.cls file define the components of your paper [title, text, heads, etc.]. *CRITICAL: Do Not Use Symbols, Special Characters, Footnotes,
-    or Math in Paper Title or Abstract.
+    Este documento ?
 \end{abstract}
 
 \begin{IEEEkeywords}
-    component, formatting, style, styling, insert
+    lean, cutting planes, formal methods, pseudo boolean reasoning
 \end{IEEEkeywords}
 
-\section{Introduction}
-This document is a model and instructions for \LaTeX.
-Please observe the conference page limits.
+\section{Introdução}
+Capítulo introdutório: caracterização do problema, motivação, objetivos, breve descrição da estrutura do trabalho.
 
-\section{Ease of Use}
+\section{Referencial Teórico}
+Capítulo referencial: identificação de trabalhos correlatos, referencial teórico.
 
-\subsection{Maintaining the Integrity of the Specifications}
+\subsection{Pseudo-Booleans}
+O que são PBs?
 
-The IEEEtran class file is used to format your paper and style the text. All margins,
-column widths, line spaces, and text fonts are prescribed; please do not
-alter them. You may note peculiarities. For example, the head margin
-measures proportionately more than is customary. This measurement
-and others are deliberate, using specifications that anticipate your paper
-as one part of the entire proceedings, and not as an independent document.
-Please do not revise any of the current designations.
+\section{Contribuições}
+Capítulo de contribuição: descrição organizada das atividades conduzidas pelo
+aluno.
 
-\section{Prepare Your Paper Before Styling}
-Before you begin to format your paper, first write and save the content as a
-separate text file. Complete all content and organizational editing before
-formatting. Please note sections \ref{AA}--\ref{SCM} below for more information on
-proofreading, spelling and grammar.
+\section*{Conclusões}
+Capítulo de fechamento: conclusões e relação de trabalhos futuros.
 
-Keep your text and graphic files separate until after the text has been
-formatted and styled. Do not number text heads---{\LaTeX} will do that
-for you.
+\section*{Referencias Bibliográficas}
 
-\subsection{Abbreviations and Acronyms}\label{AA}
-Define abbreviations and acronyms the first time they are used in the text,
-even after they have been defined in the abstract. Abbreviations such as
-IEEE, SI, MKS, CGS, ac, dc, and rms do not have to be defined. Do not use
-abbreviations in the title or heads unless they are unavoidable.
-
-\subsection{Units}
-\begin{itemize}
-    \item Use either SI (MKS) or CGS as primary units. (SI units are encouraged.) English units may be used as secondary units (in parentheses). An exception would be the use of English units as identifiers in trade, such as ``3.5-inch disk drive''.
-    \item Avoid combining SI and CGS units, such as current in amperes and magnetic field in oersteds. This often leads to confusion because equations do not balance dimensionally. If you must use mixed units, clearly state the units for each quantity that you use in an equation.
-    \item Do not mix complete spellings and abbreviations of units: ``Wb/m\textsuperscript{2}'' or ``webers per square meter'', not ``webers/m\textsuperscript{2}''. Spell out units when they appear in text: ``. . . a few henries'', not ``. . . a few H''.
-    \item Use a zero before decimal points: ``0.25'', not ``.25''. Use ``cm\textsuperscript{3}'', not ``cc''.)
-\end{itemize}
-
-\subsection{Equations}
-Number equations consecutively. To make your
-equations more compact, you may use the solidus (~/~), the exp function, or
-appropriate exponents. Italicize Roman symbols for quantities and variables,
-but not Greek symbols. Use a long dash rather than a hyphen for a minus
-sign. Punctuate equations with commas or periods when they are part of a
-sentence, as in:
-\begin{equation}
-    a+b=\gamma\label{eq}
-\end{equation}
-
-Be sure that the
-symbols in your equation have been defined before or immediately following
-the equation. Use ``\eqref{eq}'', not ``Eq.~\eqref{eq}'' or ``equation \eqref{eq}'', except at
-the beginning of a sentence: ``Equation \eqref{eq} is . . .''
-
-\subsection{\LaTeX-Specific Advice}
-
-Please use ``soft'' (e.g., \verb|\eqref{Eq}|) cross references instead
-of ``hard'' references (e.g., \verb|(1)|). That will make it possible
-to combine sections, add equations, or change the order of figures or
-citations without having to go through the file line by line.
-
-Please don't use the \verb|{eqnarray}| equation environment. Use
-\verb|{align}| or \verb|{IEEEeqnarray}| instead. The \verb|{eqnarray}|
-environment leaves unsightly spaces around relation symbols.
-
-Please note that the \verb|{subequations}| environment in {\LaTeX}
-will increment the main equation counter even when there are no
-equation numbers displayed. If you forget that, you might write an
-article in which the equation numbers skip from (17) to (20), causing
-the copy editors to wonder if you've discovered a new method of
-counting.
-
-    {\BibTeX} does not work by magic. It doesn't get the bibliographic
-data from thin air but from .bib files. If you use {\BibTeX} to produce a
-bibliography you must send the .bib files.
-
-    {\LaTeX} can't read your mind. If you assign the same label to a
-subsubsection and a table, you might find that Table I has been cross
-referenced as Table IV-B3.
-
-{\LaTeX} does not have precognitive abilities. If you put a
-\verb|\label| command before the command that updates the counter it's
-supposed to be using, the label will pick up the last counter to be
-cross referenced instead. In particular, a \verb|\label| command
-should not go before the caption of a figure or a table.
-
-Do not use \verb|\nonumber| inside the \verb|{array}| environment. It
-will not stop equation numbers inside \verb|{array}| (there won't be
-any anyway) and it might stop a wanted equation number in the
-surrounding equation.
-
-\subsection{Some Common Mistakes}\label{SCM}
-\begin{itemize}
-    \item The word ``data'' is plural, not singular.
-    \item The subscript for the permeability of vacuum $\mu_{0}$, and other common scientific constants, is zero with subscript formatting, not a lowercase letter ``o''.
-    \item In American English, commas, semicolons, periods, question and exclamation marks are located within quotation marks only when a complete thought or name is cited, such as a title or full quotation. When quotation marks are used, instead of a bold or italic typeface, to highlight a word or phrase, punctuation should appear outside of the quotation marks. A parenthetical phrase or statement at the end of a sentence is punctuated outside of the closing parenthesis (like this). (A parenthetical sentence is punctuated within the parentheses.)
-    \item A graph within a graph is an ``inset'', not an ``insert''. The word alternatively is preferred to the word ``alternately'' (unless you really mean something that alternates).
-    \item Do not use the word ``essentially'' to mean ``approximately'' or ``effectively''.
-    \item In your paper title, if the words ``that uses'' can accurately replace the word ``using'', capitalize the ``u''; if not, keep using lower-cased.
-    \item Be aware of the different meanings of the homophones ``affect'' and ``effect'', ``complement'' and ``compliment'', ``discreet'' and ``discrete'', ``principal'' and ``principle''.
-    \item Do not confuse ``imply'' and ``infer''.
-    \item The prefix ``non'' is not a word; it should be joined to the word it modifies, usually without a hyphen.
-    \item There is no period after the ``et'' in the Latin abbreviation ``et al.''.
-    \item The abbreviation ``i.e.'' means ``that is'', and the abbreviation ``e.g.'' means ``for example''.
-\end{itemize}
-An excellent style manual for science writers is \cite{b7}.
-
-\subsection{Authors and Affiliations}
-\textbf{The class file is designed for, but not limited to, six authors.} A
-minimum of one author is required for all conference articles. Author names
-should be listed starting from left to right and then moving down to the
-next line. This is the author sequence that will be used in future citations
-and by indexing services. Names should not be listed in columns nor group by
-affiliation. Please keep your affiliations as succinct as possible (for
-example, do not differentiate among departments of the same organization).
-
-\subsection{Identify the Headings}
-Headings, or heads, are organizational devices that guide the reader through
-your paper. There are two types: component heads and text heads.
-
-Component heads identify the different components of your paper and are not
-topically subordinate to each other. Examples include Acknowledgments and
-References and, for these, the correct style to use is ``Heading 5''. Use
-``figure caption'' for your Figure captions, and ``table head'' for your
-table title. Run-in heads, such as ``Abstract'', will require you to apply a
-style (in this case, italic) in addition to the style provided by the drop
-down menu to differentiate the head from the text.
-
-Text heads organize the topics on a relational, hierarchical basis. For
-example, the paper title is the primary text head because all subsequent
-material relates and elaborates on this one topic. If there are two or more
-sub-topics, the next level head (uppercase Roman numerals) should be used
-and, conversely, if there are not at least two sub-topics, then no subheads
-should be introduced.
-
-\subsection{Figures and Tables}
-\paragraph{Positioning Figures and Tables} Place figures and tables at the top and
-bottom of columns. Avoid placing them in the middle of columns. Large
-figures and tables may span across both columns. Figure captions should be
-below the figures; table heads should appear above the tables. Insert
-figures and tables after they are cited in the text. Use the abbreviation
-``Fig.~\ref{fig}'', even at the beginning of a sentence.
-
-\begin{table}[htbp]
-    \caption{Table Type Styles}
-    \begin{center}
-        \begin{tabular}{|c|c|c|c|}
-            \hline
-            \textbf{Table} & \multicolumn{3}{|c|}{\textbf{Table Column Head}}                                                         \\
-            \cline{2-4}
-            \textbf{Head}  & \textbf{\textit{Table column subhead}}           & \textbf{\textit{Subhead}} & \textbf{\textit{Subhead}} \\
-            \hline
-            copy           & More table copy$^{\mathrm{a}}$                   &                           &                           \\
-            \hline
-            \multicolumn{4}{l}{$^{\mathrm{a}}$Sample of a Table footnote.}
-        \end{tabular}
-        \label{tab1}
-    \end{center}
-\end{table}
-
-\begin{figure}[htbp]
-    \centerline{\includegraphics{fig1.png}}
-    \caption{Example of a figure caption.}
-    \label{fig}
-\end{figure}
-
-Figure Labels: Use 8 point Times New Roman for Figure labels. Use words
-rather than symbols or abbreviations when writing Figure axis labels to
-avoid confusing the reader. As an example, write the quantity
-``Magnetization'', or ``Magnetization, M'', not just ``M''. If including
-units in the label, present them within parentheses. Do not label axes only
-with units. In the example, write ``Magnetization (A/m)'' or ``Magnetization
-\{A[m(1)]\}'', not just ``A/m''. Do not label axes with a ratio of
-quantities and units. For example, write ``Temperature (K)'', not
-``Temperature/K''.
-
-\section*{Acknowledgment}
-
-The preferred spelling of the word ``acknowledgment'' in America is without
-an ``e'' after the ``g''. Avoid the stilted expression ``one of us (R. B.
-G.) thanks $\ldots$''. Instead, try ``R. B. G. thanks$\ldots$''. Put sponsor
-acknowledgments in the unnumbered footnote on the first page.
-
-\section*{References}
-
-Please number citations consecutively within brackets \cite{b1}. The
-sentence punctuation follows the bracket \cite{b2}. Refer simply to the reference
-number, as in \cite{b3}---do not use ``Ref. \cite{b3}'' or ``reference \cite{b3}'' except at
-the beginning of a sentence: ``Reference \cite{b3} was the first $\ldots$''
-
-Number footnotes separately in superscripts. Place the actual footnote at
-the bottom of the column in which it was cited. Do not put footnotes in the
-abstract or reference list. Use letters for table footnotes.
-
-Unless there are six authors or more give all authors' names; do not use
-``et al.''. Papers that have not been published, even if they have been
-submitted for publication, should be cited as ``unpublished'' \cite{b4}. Papers
-that have been accepted for publication should be cited as ``in press'' \cite{b5}.
-Capitalize only the first word in a paper title, except for proper nouns and
-element symbols.
-
-For papers published in translation journals, please give the English
-citation first, followed by the original foreign-language citation \cite{b6}.
+Referências bibliográficas contidas no texto.
 
 \begin{thebibliography}{00}
-    \bibitem{b1} G. Eason, B. Noble, and I. N. Sneddon, ``On certain integrals of Lipschitz-Hankel type involving products of Bessel functions,'' Phil. Trans. Roy. Soc. London, vol. A247, pp. 529--551, April 1955.
-    \bibitem{b2} J. Clerk Maxwell, A Treatise on Electricity and Magnetism, 3rd ed., vol. 2. Oxford: Clarendon, 1892, pp.68--73.
-    \bibitem{b3} I. S. Jacobs and C. P. Bean, ``Fine particles, thin films and exchange anisotropy,'' in Magnetism, vol. III, G. T. Rado and H. Suhl, Eds. New York: Academic, 1963, pp. 271--350.
-    \bibitem{b4} K. Elissa, ``Title of paper if known,'' unpublished.
-    \bibitem{b5} R. Nicole, ``Title of paper with only first word capitalized,'' J. Name Stand. Abbrev., in press.
-    \bibitem{b6} Y. Yorozu, M. Hirano, K. Oka, and Y. Tagawa, ``Electron spectroscopy studies on magneto-optical media and plastic substrate interface,'' IEEE Transl. J. Magn. Japan, vol. 2, pp. 740--741, August 1987 [Digests 9th Annual Conf. Magnetics Japan, p. 301, 1982].
-    \bibitem{b7} M. Young, The Technical Writer's Handbook. Mill Valley, CA: University Science, 1989.
+    \bibitem{b1} J. Nordström, ``A Unified Proof System for Discrete Combinatorial Problems'', November 2023.
 \end{thebibliography}
 \vspace{12pt}
-\color{red}
-IEEE conference templates contain guidance text for composing and formatting conference papers. Please ensure that all template text is removed from your conference paper prior to submission to the conference. Failure to remove the template text from your paper may result in your paper not being published.
-
 \end{document}

From 0847e1c4cba72f692a493f528e3dbba120cccae8 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Wed, 26 Jun 2024 14:49:29 -0300
Subject: [PATCH 65/93] Article Introduction

---
 article/IEEEtran.cls               |   2 +-
 article/cutting-planes-article.pdf | Bin 52217 -> 55637 bytes
 article/cutting-planes-article.tex |  35 +++++++++++++++++++++++------
 article/fig1.png                   | Bin 4535 -> 0 bytes
 4 files changed, 29 insertions(+), 8 deletions(-)
 delete mode 100644 article/fig1.png

diff --git a/article/IEEEtran.cls b/article/IEEEtran.cls
index 905a10c..b0cf599 100644
--- a/article/IEEEtran.cls
+++ b/article/IEEEtran.cls
@@ -2602,7 +2602,7 @@
 \def\contentsname{Contents}
 \def\listfigurename{List of Figures}
 \def\listtablename{List of Tables}
-\def\refname{References}
+\def\refname{Referências Bibliográficas}
 \def\indexname{Index}
 \def\figurename{Fig.}
 \def\tablename{TABLE}
diff --git a/article/cutting-planes-article.pdf b/article/cutting-planes-article.pdf
index 58db7c51da112da9674ab5cb3ccd9b59e824d9a9..2df99966b898b020818c1a3e08167570bfde01d6 100644
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 2f5b508..71647dd 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -25,7 +25,9 @@
 \maketitle
 
 \begin{abstract}
-    Este documento ?
+    Esta pesquisa envolve a definição e verificação da lógica pseudo-booleana de
+    Planos de Corte utilizando o provador de teoremas Lean 4
+
 \end{abstract}
 
 \begin{IEEEkeywords}
@@ -33,7 +35,26 @@
 \end{IEEEkeywords}
 
 \section{Introdução}
-Capítulo introdutório: caracterização do problema, motivação, objetivos, breve descrição da estrutura do trabalho.
+O problema da satisfabilidade booleana (SAT) é muito importante para a Ciência da Computação, sendo o
+primeiro demonstrado ser da class NP-Completo. Ele é o problema de decidir para dada expressão booleana, é
+possível escolher valores para as variáveis de forma que a expressão como um todo seja verdadeira.
+Este problema é extensivamente pesquisado na área de métodos formais\cite{SatLive} e existem inclusive competições para
+definir qual é o melhor solucionador do mundo \cite{SatComp}.
+
+Um formato comumente utilizado para representar essas expressões é a Forma Normal Conjuntiva (CNF), que
+consiste da conjunção de cláusulas, em que cada cláusula é a disjunção de variáveis ou negação de variáveis.
+% Formula CNF aqui
+Entretando, introduzimos uma outra representação para expressões, chamados Pseudo-Booleanos, funções estudadas
+desde os anos 1960 na área de pesquisa operacional.
+% Formula PB aqui
+Este formato é exponencialmente mais compacto que o CNF, o que motiva seu uso\cite{PBSolve}.
+
+% TODO: https://youtu.be/xoAov5RLcis?feature=shared&t=165
+
+
+O cálculo formal de Planos de Corte é uma técnica matemática utilizada na área de otimização,
+particularmente em Programação Linear que
+% Capítulo introdutório: caracterização do problema, motivação, objetivos, breve descrição da estrutura do trabalho.
 
 \section{Referencial Teórico}
 Capítulo referencial: identificação de trabalhos correlatos, referencial teórico.
@@ -48,12 +69,12 @@ \section{Contribuições}
 \section*{Conclusões}
 Capítulo de fechamento: conclusões e relação de trabalhos futuros.
 
-\section*{Referencias Bibliográficas}
-
-Referências bibliográficas contidas no texto.
-
 \begin{thebibliography}{00}
-    \bibitem{b1} J. Nordström, ``A Unified Proof System for Discrete Combinatorial Problems'', November 2023.
+    \bibitem{SatLive}   ``Sat Live! keep up to date with research on the satisfiability problem'', access at http://www.satlive.org/
+    \bibitem{SatComp}   ``The International SAT Competition Web Page'', access at https://satcompetition.github.io/.
+    \bibitem{PBSolve}   J. Nordström, ``Pseudo-Boolean Solving and Optimization'', February 2021.
+    \bibitem{CutPlane}  J. Nordström, ``A Unified Proof System for Discrete Combinatorial Problems'', November 2023.
+
 \end{thebibliography}
 \vspace{12pt}
 \end{document}
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From cace8b677898629bf4db0359eb6c339f690c4363 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 27 Jun 2024 13:52:22 -0300
Subject: [PATCH 66/93] Divided Theoretical Subsections

---
 article/cutting-planes-article.pdf | Bin 55637 -> 114775 bytes
 article/cutting-planes-article.tex |  75 +++++++++++++++++++++--------
 2 files changed, 55 insertions(+), 20 deletions(-)

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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 71647dd..153e019 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -35,32 +35,67 @@
 \end{IEEEkeywords}
 
 \section{Introdução}
+
+% TODO
+% O cálculo formal de Planos de Corte é uma técnica matemática utilizada na área de otimização,
+% particularmente em Programação Linear.  
+% Capítulo introdutório: caracterização do problema, motivação, objetivos, breve descrição da estrutura do trabalho.
+% * problema de utilizar pseudo booleanos para representacao eficiente de problemas SAT
+% * motivacao: Ter certeza que solucoes que utilizam os pseudo-booleans estao corretas
+% * objetivos: Usar metodos formais para validar a logica pseudo-booleana e provas que a utilizem
+% * estrutura do trabalho: biblioteca de codigo em Lean 4 que implementa as regras da logica pseudo-booleana e prova sua
+%   corretude. Alem disso, mostrar exemplos de uso dessa biblioteca com o "Toy Example" presente na apresentacao de Jakob.
+%   Criacao de uma documentacao em formato mdbook, que estara disponivel para pesquisadores e matematicos que desejem usar
+%   a biblioteca.
+Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+
+\section{Referencial Teórico}
+% Capítulo referencial: identificação de trabalhos correlatos, referencial teórico.
+
+\subsection{Satisfiabilidade}
 O problema da satisfabilidade booleana (SAT) é muito importante para a Ciência da Computação, sendo o
-primeiro demonstrado ser da class NP-Completo. Ele é o problema de decidir para dada expressão booleana, é
-possível escolher valores para as variáveis de forma que a expressão como um todo seja verdadeira.
+primeiro demonstrado ser da class NP-Completo. Ele é o problema de decidir para dada expressão booleana,
+se é possível escolher valores para as variáveis de forma que a expressão como um todo seja verdadeira.
 Este problema é extensivamente pesquisado na área de métodos formais\cite{SatLive} e existem inclusive competições para
 definir qual é o melhor solucionador do mundo \cite{SatComp}.
 
 Um formato comumente utilizado para representar essas expressões é a Forma Normal Conjuntiva (CNF), que
 consiste da conjunção de cláusulas, em que cada cláusula é a disjunção de variáveis ou negação de variáveis.
-% Formula CNF aqui
-Entretando, introduzimos uma outra representação para expressões, chamados Pseudo-Booleanos, funções estudadas
-desde os anos 1960 na área de pesquisa operacional.
-% Formula PB aqui
-Este formato é exponencialmente mais compacto que o CNF, o que motiva seu uso\cite{PBSolve}.
-
-% TODO: https://youtu.be/xoAov5RLcis?feature=shared&t=165
+% Formula CNF
+\begin{equation}
+    C_1 \land C_2 \land \dots \land C_n
+\end{equation}
+onde,
+\begin{equation}
+    C_i = a_1^i \lor a_2^i \lor \dots \lor a_k^i
+\end{equation}
+e para cada $j$
+\begin{equation}
+    a_j^i \in \{ x_1,x_2,\dots,x_m,\neg x_1,\neg x_2,\dots, \neg x_m \}
+\end{equation}
 
+\subsection{Pseudo-Booleans}
+Neste trabalho, introduzimos uma outra representação para expressões, chamados Pseudo-Booleanos, funções estudadas
+desde os anos 1960 na área de pesquisa operacional, em programação inteira. Este formato consiste de:
+% Formula PB
+\begin{equation}
+    \sum_i{a_i l_i} \ge A
+\end{equation}
+onde,
+\begin{equation}
+    \begin{gathered}
+        A, a_i \in \mathbb{N} \\
+        l_i \in \{ x_i, \overline x_i \}, \qquad x_i + \overline x_i = 1
+    \end{gathered}
+\end{equation}
 
-O cálculo formal de Planos de Corte é uma técnica matemática utilizada na área de otimização,
-particularmente em Programação Linear que
-% Capítulo introdutório: caracterização do problema, motivação, objetivos, breve descrição da estrutura do trabalho.
+Este formato é exponencialmente mais compacto que o CNF, o que motiva seu uso\cite{PBSolve}.
 
-\section{Referencial Teórico}
-Capítulo referencial: identificação de trabalhos correlatos, referencial teórico.
+\subsection{Pseudo-Boolean Reasoning}
+A lógica formal de Cutting Planes introduz 2 axiomas e 4 regras de inferência, nomeadamente
+\textbf{Adição}, \textbf{Multiplicação}, \textbf{Divisão} e \textbf{Saturação},
+que permitem derivar novas inequações a partir de outras\cite{CutPlane}.
 
-\subsection{Pseudo-Booleans}
-O que são PBs?
 
 \section{Contribuições}
 Capítulo de contribuição: descrição organizada das atividades conduzidas pelo
@@ -70,10 +105,10 @@ \section*{Conclusões}
 Capítulo de fechamento: conclusões e relação de trabalhos futuros.
 
 \begin{thebibliography}{00}
-    \bibitem{SatLive}   ``Sat Live! keep up to date with research on the satisfiability problem'', access at http://www.satlive.org/
-    \bibitem{SatComp}   ``The International SAT Competition Web Page'', access at https://satcompetition.github.io/.
-    \bibitem{PBSolve}   J. Nordström, ``Pseudo-Boolean Solving and Optimization'', February 2021.
-    \bibitem{CutPlane}  J. Nordström, ``A Unified Proof System for Discrete Combinatorial Problems'', November 2023.
+    \bibitem{SatLive} ``Sat Live! keep up to date with research on the satisfiability problem'', access at http://www.satlive.org/.
+    \bibitem{SatComp} ``The International SAT Competition Web Page'', access at https://satcompetition.github.io/.
+    \bibitem{PBSolve} J. Nordström, ``Pseudo-Boolean Solving and Optimization'', February 2021.
+    \bibitem{CutPlane}J. Nordström, ``A Unified Proof System for Discrete Combinatorial Problems'', November 2023.
 
 \end{thebibliography}
 \vspace{12pt}

From 9ad8a988430b2f3fa6a77c1f9d6f41dd0b9672d5 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 27 Jun 2024 14:22:03 -0300
Subject: [PATCH 67/93] Lean Prover in Theoretical References

---
 article/cutting-planes-article.pdf | Bin 114775 -> 117933 bytes
 article/cutting-planes-article.tex |  28 +++++++++++++++++++++-------
 2 files changed, 21 insertions(+), 7 deletions(-)

diff --git a/article/cutting-planes-article.pdf b/article/cutting-planes-article.pdf
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 153e019..bc55af6 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -96,19 +96,33 @@ \subsection{Pseudo-Boolean Reasoning}
 \textbf{Adição}, \textbf{Multiplicação}, \textbf{Divisão} e \textbf{Saturação},
 que permitem derivar novas inequações a partir de outras\cite{CutPlane}.
 
+\subsection{Lean Theorem Prover}
+Lean é uma linguagem de programação e provador de teoremas criado por Leonardo de Moura em 2013\cite{LeanProver}, com
+influência de ML, Coq e Haskell.
+A sua versão mais atual de 2021, Lean 4\cite{Lean4}, se tornou proeminente entre matemáticos, por permitir a
+\textbf{formalização} e \textbf{verificação} de teoremas, auxiliando o trabalho teórico.
+Em 2021, uma equipe de pesquisadores usou o Lean para verificar a correção de uma prova de Peter Scholze na área
+de matemática condensada\cite{LTE}, o que atraiu atenção por formalizar um resultado na vanguarda da pesquisa matemática.
+Em 2023, Terence Tao usou o Lean para formalizar uma prova da conjectura Polinomial de Freiman-Ruzsa\cite{PFR},
+um resultado publicado por Tao e colaboradores no mesmo ano.
 
 \section{Contribuições}
-Capítulo de contribuição: descrição organizada das atividades conduzidas pelo
-aluno.
+% Capítulo de contribuição: descrição organizada das atividades conduzidas pelo aluno.
+Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
 
 \section*{Conclusões}
-Capítulo de fechamento: conclusões e relação de trabalhos futuros.
+% Capítulo de fechamento: conclusões e relação de trabalhos futuros.
+Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
 
 \begin{thebibliography}{00}
-    \bibitem{SatLive} ``Sat Live! keep up to date with research on the satisfiability problem'', access at http://www.satlive.org/.
-    \bibitem{SatComp} ``The International SAT Competition Web Page'', access at https://satcompetition.github.io/.
-    \bibitem{PBSolve} J. Nordström, ``Pseudo-Boolean Solving and Optimization'', February 2021.
-    \bibitem{CutPlane}J. Nordström, ``A Unified Proof System for Discrete Combinatorial Problems'', November 2023.
+    \bibitem{SatLive}       D. Le Berre, ``Sat Live! keep up to date with research on the satisfiability problem'', Acesso em http://www.satlive.org/.
+    \bibitem{SatComp}       M. Heule, M. Jävisalo, M. Suda, ``The International SAT Competition Web Page'', Acesso em https://satcompetition.github.io/.
+    \bibitem{PBSolve}       J. Nordström, ``Pseudo-Boolean Solving and Optimization'', Fevereiro de 2021.
+    \bibitem{CutPlane}      J. Nordström, ``A Unified Proof System for Discrete Combinatorial Problems'', Novembro de 2023.
+    \bibitem{LeanProver}    L. de Moura, S. Kong, J. Avigad, F. van Doorn, J. von Raumer ``The Lean Theorem Prover'', 25th International Conference on Automated Deduction (CADE-25), Berlin, Germany, 2015. Acesso em https://lean-lang.org/papers/system.pdf.
+    \bibitem{Lean4}         L. de Moura, S. Ullrich ``The Lean 4 Theorem Prover and Programming Language'', 28th International Conference on Automated Deduction (CADE-28), Pittsburgh, USA, 2021. Acesso em https://lean-lang.org/papers/lean4.pdf.
+    \bibitem{LTE}           J. Commelin, P. Scholze ``Liquid Tensor Experiment''. Acesso em https://math.commelin.net/files/LTE.pdf
+    \bibitem{PFR}           T. Tao, ``The Polynomial Freiman-Ruzsa Conjecture'', Novembro de 2023. Acesso em https://teorth.github.io/pfr/.
 
 \end{thebibliography}
 \vspace{12pt}

From 3e488cd058d52b937b885f1a095c5c6a81bb5447 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 27 Jun 2024 14:33:52 -0300
Subject: [PATCH 68/93] Subdivision of Contributions

---
 article/cutting-planes-article.pdf | Bin 117933 -> 119496 bytes
 article/cutting-planes-article.tex |  29 ++++++++++++++++++++++++++++-
 2 files changed, 28 insertions(+), 1 deletion(-)

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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index bc55af6..51c4f78 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -108,6 +108,33 @@ \subsection{Lean Theorem Prover}
 
 \section{Contribuições}
 % Capítulo de contribuição: descrição organizada das atividades conduzidas pelo aluno.
+A contribuição deste trabalho está na criação de código em Lean 4 para lidar com a lógica de cutting planes.
+Primeiramente, as inequações foram formalmente definidas em Lean, utilizando teoremas e definições de sua biblioteca
+matemática mathlib4\cite{mathlib4}. Assim as quatro regras foram definidas e formalizadas, e em seguida um exemplo de
+raciocínio, apresentado por Jakob Nordström foi formalizado nessa biblioteca.
+
+% * Definição da inequação pseudo-booleana
+\subsection{Definindo Inequações Pseudo-Booleanas}
+Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+
+% * Prova da Regra Multiplication
+\subsection{Provando a Regra Multiplicação}
+Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+
+% * Prova da Regra Saturation
+\subsection{Provando a Regra Saturação}
+Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+
+% * Prova da Regra Division
+\subsection{Provando a Regra Divisão}
+Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+
+% * Prova da Regra Addition
+\subsection{Provando a Regra Adição}
+Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+
+% * Implementação do Toy Example
+\subsection{Implementação do Exemplo}
 Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
 
 \section*{Conclusões}
@@ -123,7 +150,7 @@ \section*{Conclusões}
     \bibitem{Lean4}         L. de Moura, S. Ullrich ``The Lean 4 Theorem Prover and Programming Language'', 28th International Conference on Automated Deduction (CADE-28), Pittsburgh, USA, 2021. Acesso em https://lean-lang.org/papers/lean4.pdf.
     \bibitem{LTE}           J. Commelin, P. Scholze ``Liquid Tensor Experiment''. Acesso em https://math.commelin.net/files/LTE.pdf
     \bibitem{PFR}           T. Tao, ``The Polynomial Freiman-Ruzsa Conjecture'', Novembro de 2023. Acesso em https://teorth.github.io/pfr/.
-
+    \bibitem{mathlib4}      The mathlib Community. 2020. ``The lean mathematical library''. In CPP 2020. 367-381. https://doi.org/10.1145/3372885.3373824
 \end{thebibliography}
 \vspace{12pt}
 \end{document}

From 94dfcb23cd8d97ca8cdd6752066b9303c88a3c39 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 27 Jun 2024 14:51:07 -0300
Subject: [PATCH 69/93] Rename subsections

---
 article/cutting-planes-article.pdf | Bin 119496 -> 119502 bytes
 article/cutting-planes-article.tex |  12 ++++++------
 2 files changed, 6 insertions(+), 6 deletions(-)

diff --git a/article/cutting-planes-article.pdf b/article/cutting-planes-article.pdf
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 51c4f78..264a9af 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -114,23 +114,23 @@ \section{Contribuições}
 raciocínio, apresentado por Jakob Nordström foi formalizado nessa biblioteca.
 
 % * Definição da inequação pseudo-booleana
-\subsection{Definindo Inequações Pseudo-Booleanas}
-Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+\subsection{Definição de Inequações Pseudo-Booleanas}
+Teste
 
 % * Prova da Regra Multiplication
-\subsection{Provando a Regra Multiplicação}
+\subsection{Prova da Regra Multiplicação}
 Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
 
 % * Prova da Regra Saturation
-\subsection{Provando a Regra Saturação}
+\subsection{Prova da Regra Saturação}
 Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
 
 % * Prova da Regra Division
-\subsection{Provando a Regra Divisão}
+\subsection{Prova da Regra Divisão}
 Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
 
 % * Prova da Regra Addition
-\subsection{Provando a Regra Adição}
+\subsection{Prova da Regra Adição}
 Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
 
 % * Implementação do Toy Example

From 2f3fc2c1359ea7cf2c0a211d50dd17a6822483ab Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 27 Jun 2024 20:21:34 -0300
Subject: [PATCH 70/93] Add `minted` package to article (Missing unicode
 supports)

---
 article/cutting-planes-article.pdf | Bin 119502 -> 153320 bytes
 article/cutting-planes-article.tex |  26 ++++++++++++++++++++++++++
 2 files changed, 26 insertions(+)

diff --git a/article/cutting-planes-article.pdf b/article/cutting-planes-article.pdf
index 12e5df88ee92546ae08ade50d43714178ce6e1b6..a5d305c44c4f832ad47b2f578bf5964ef4aba1fa 100644
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 264a9af..4c7a6c7 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -7,6 +7,7 @@
 \usepackage{graphicx}
 \usepackage{textcomp}
 \usepackage{xcolor}
+\usepackage{minted}
 \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
     T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
 \begin{document}
@@ -152,5 +153,30 @@ \section*{Conclusões}
     \bibitem{PFR}           T. Tao, ``The Polynomial Freiman-Ruzsa Conjecture'', Novembro de 2023. Acesso em https://teorth.github.io/pfr/.
     \bibitem{mathlib4}      The mathlib Community. 2020. ``The lean mathematical library''. In CPP 2020. 367-381. https://doi.org/10.1145/3372885.3373824
 \end{thebibliography}
+
 \vspace{12pt}
+
+\newpage
+
+\section{Apêndice}
+
+\subsection{Definição de Pseudo-Boolean}
+
+\begin{minted}{lean4}
+
+open FinVec BigOperators
+
+-- TODO: Add support for unicode chars in LaTeX
+abbrev Coeff (n : .) := Fin n → (. . .)
+
+def PBSum (cs : Coeff n) (xs : Fin n → Fin 2) :=
+  . i, let (p,n) := cs i;
+    if xs i = 1 then p else n
+
+def PBIneq (cs : Coeff n) (xs : Fin n → Fin 2) (const : .) :=
+  PBSum cs xs . const
+
+\end{minted}
+
+
 \end{document}

From d05a8abcf8664c75bb7a0f3e320c9a7930a10501 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Fri, 28 Jun 2024 13:28:59 -0300
Subject: [PATCH 71/93] Commented out `minted` for faster compilation
 (temporary)

---
 article/cutting-planes-article.pdf | Bin 153320 -> 119843 bytes
 article/cutting-planes-article.tex |  24 +++++++++++++-----------
 2 files changed, 13 insertions(+), 11 deletions(-)

diff --git a/article/cutting-planes-article.pdf b/article/cutting-planes-article.pdf
index a5d305c44c4f832ad47b2f578bf5964ef4aba1fa..45bf32d83cf31f1eef832ba6161880a8d9d083db 100644
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 4c7a6c7..e302c60 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -7,7 +7,8 @@
 \usepackage{graphicx}
 \usepackage{textcomp}
 \usepackage{xcolor}
-\usepackage{minted}
+% \usepackage{minted}
+
 \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
     T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
 \begin{document}
@@ -162,21 +163,22 @@ \section{Apêndice}
 
 \subsection{Definição de Pseudo-Boolean}
 
-\begin{minted}{lean4}
+% \begin{minted}{lean4}
+
+% open FinVec BigOperators
 
-open FinVec BigOperators
+% -- TODO: Add support for unicode chars in LaTeX
+% abbrev Coeff (n : .) := Fin n → (. . .)
 
--- TODO: Add support for unicode chars in LaTeX
-abbrev Coeff (n : .) := Fin n → (. . .)
 
-def PBSum (cs : Coeff n) (xs : Fin n → Fin 2) :=
-  . i, let (p,n) := cs i;
-    if xs i = 1 then p else n
+% def PBSum (cs : Coeff n) (xs : Fin n → Fin 2) :=
+%   . i, let (p,n) := cs i;
+%     if xs i = 1 then p else n
 
-def PBIneq (cs : Coeff n) (xs : Fin n → Fin 2) (const : .) :=
-  PBSum cs xs . const
+% def PBIneq (cs : Coeff n) (xs : Fin n → Fin 2) (const : .) :=
+%   PBSum cs xs . const
 
-\end{minted}
+% \end{minted}
 
 
 \end{document}

From e0c7511318df23adb04e3a71660459c8f11187f8 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Fri, 28 Jun 2024 14:40:44 -0300
Subject: [PATCH 72/93] Added Unicode support

---
 article/cutting-planes-article.pdf | Bin 119843 -> 160257 bytes
 article/cutting-planes-article.tex |  91 +++++++++++++++++++----------
 2 files changed, 61 insertions(+), 30 deletions(-)

diff --git a/article/cutting-planes-article.pdf b/article/cutting-planes-article.pdf
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index e302c60..b485223 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -7,7 +7,9 @@
 \usepackage{graphicx}
 \usepackage{textcomp}
 \usepackage{xcolor}
-% \usepackage{minted}
+\usepackage[mathletters]{ucs}
+\usepackage[utf8x]{inputenc}
+\usepackage{minted}
 
 \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
     T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
@@ -61,36 +63,12 @@ \subsection{Satisfiabilidade}
 Este problema é extensivamente pesquisado na área de métodos formais\cite{SatLive} e existem inclusive competições para
 definir qual é o melhor solucionador do mundo \cite{SatComp}.
 
-Um formato comumente utilizado para representar essas expressões é a Forma Normal Conjuntiva (CNF), que
-consiste da conjunção de cláusulas, em que cada cláusula é a disjunção de variáveis ou negação de variáveis.
-% Formula CNF
-\begin{equation}
-    C_1 \land C_2 \land \dots \land C_n
-\end{equation}
-onde,
-\begin{equation}
-    C_i = a_1^i \lor a_2^i \lor \dots \lor a_k^i
-\end{equation}
-e para cada $j$
-\begin{equation}
-    a_j^i \in \{ x_1,x_2,\dots,x_m,\neg x_1,\neg x_2,\dots, \neg x_m \}
-\end{equation}
-
 \subsection{Pseudo-Booleans}
+Um formato comumente utilizado para representar expressões booleanas é a Forma Normal Conjuntiva (CNF), que
+consiste da conjunção de cláusulas, em que cada cláusula é a disjunção de variáveis ou negação de variáveis\cite{CNF}.
 Neste trabalho, introduzimos uma outra representação para expressões, chamados Pseudo-Booleanos, funções estudadas
-desde os anos 1960 na área de pesquisa operacional, em programação inteira. Este formato consiste de:
-% Formula PB
-\begin{equation}
-    \sum_i{a_i l_i} \ge A
-\end{equation}
-onde,
-\begin{equation}
-    \begin{gathered}
-        A, a_i \in \mathbb{N} \\
-        l_i \in \{ x_i, \overline x_i \}, \qquad x_i + \overline x_i = 1
-    \end{gathered}
-\end{equation}
-
+desde os anos 1960 na área de pesquisa operacional, em programação inteira. Este formato consiste de um somatório
+do produto de um coeficiente por um literal, que é maior ou igual a uma constante natural.
 Este formato é exponencialmente mais compacto que o CNF, o que motiva seu uso\cite{PBSolve}.
 
 \subsection{Pseudo-Boolean Reasoning}
@@ -117,7 +95,59 @@ \section{Contribuições}
 
 % * Definição da inequação pseudo-booleana
 \subsection{Definição de Inequações Pseudo-Booleanas}
-Teste
+Uma expressão booleana na forma CNF consiste de:
+\begin{equation}
+    C_1 \land C_2 \land \dots \land C_n
+\end{equation}
+onde,
+\begin{equation}
+    C_i = a_1^i \lor a_2^i \lor \dots \lor a_k^i
+\end{equation}
+e para cada $j$
+\begin{equation}
+    a_j^i \in \{ x_1,x_2,\dots,x_m,\neg x_1,\neg x_2,\dots, \neg x_m \}
+\end{equation}
+
+Um exemplo desse formato é a formula:
+\begin{equation}
+    (x_1 \lor x_2) \land (x_1 \lor \neg x_3) \land (x_2 \lor \neg x_3)
+\end{equation}
+Onde $x_1$, $x_2$ e $x_3$ são nossas variáveis booleanas e a atribuição $x_1=T$, $x_2=T$, $x_3=T$ satisfaz essa fórmula.
+
+O formato pseudo-booleano consiste de:
+\begin{equation}
+    \sum_i{a_i l_i} \ge A
+\end{equation}
+onde,
+\begin{equation}
+    \begin{gathered}
+        A, a_i \in \mathbb{N} \\
+        l_i \in \{ x_i, \overline x_i \}, \qquad x_i + \overline x_i = 1
+    \end{gathered}
+\end{equation}
+
+A mesma fórmula acima nesse formato é expressa por:
+\begin{equation}
+    x_1 + x_2 + \overline x_3 \ge 2
+\end{equation}
+
+A implementação em Lean 4 consiste de definir o tipo para coeficientes \textit{Coeff}, onde usamos a estrutura
+\textit{FinVec}, que permite termos uma lista com $n$ elementos, onde cada elemento da lista será um par
+de dois números naturais, o tipo $\mathbb{N}$ em Lean:
+\begin{minted}[fontsize=\small]{lean4}
+abbrev Coeff (n : ℕ) := Fin n → (ℕ × ℕ)
+\end{minted}
+
+Esse par consiste do coeficiente de $x_i$ no primeiro elemento e o coeficiente de $\overline x_i$ no segundo elemento.
+
+Com essa definição, definimos o \textit{PBSum}, que, com os coeficientes $cs$ e os valores 0-1 $xs$,
+utiliza o \textit{BigOperator} de somatório ($\sum$) para somar os elementos da lista iterando no índice $i$:
+\begin{minted}[fontsize=\small]{lean4}
+def PBSum (cs : Coeff n) (xs : Fin n → Fin 2) :=
+  ∑ i, let (p,n) := cs i;
+    if xs i = 1 then p else n
+\end{minted}
+
 
 % * Prova da Regra Multiplication
 \subsection{Prova da Regra Multiplicação}
@@ -146,6 +176,7 @@ \section*{Conclusões}
 \begin{thebibliography}{00}
     \bibitem{SatLive}       D. Le Berre, ``Sat Live! keep up to date with research on the satisfiability problem'', Acesso em http://www.satlive.org/.
     \bibitem{SatComp}       M. Heule, M. Jävisalo, M. Suda, ``The International SAT Competition Web Page'', Acesso em https://satcompetition.github.io/.
+    \bibitem{CNF}           ``Conjunctive normal form'', Encyclopedia of Mathematics, EMS Press, 2001.
     \bibitem{PBSolve}       J. Nordström, ``Pseudo-Boolean Solving and Optimization'', Fevereiro de 2021.
     \bibitem{CutPlane}      J. Nordström, ``A Unified Proof System for Discrete Combinatorial Problems'', Novembro de 2023.
     \bibitem{LeanProver}    L. de Moura, S. Kong, J. Avigad, F. van Doorn, J. von Raumer ``The Lean Theorem Prover'', 25th International Conference on Automated Deduction (CADE-25), Berlin, Germany, 2015. Acesso em https://lean-lang.org/papers/system.pdf.

From 70f3c6c4e565fd96e00a5a8e55e1158ca78dcd1f Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Fri, 28 Jun 2024 14:52:42 -0300
Subject: [PATCH 73/93] Finalize definition of PBs

---
 article/cutting-planes-article.pdf | Bin 160257 -> 172515 bytes
 article/cutting-planes-article.tex |  28 +++++++++++++++++++++++++++-
 2 files changed, 27 insertions(+), 1 deletion(-)

diff --git a/article/cutting-planes-article.pdf b/article/cutting-planes-article.pdf
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index b485223..684353d 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -138,7 +138,8 @@ \subsection{Definição de Inequações Pseudo-Booleanas}
 abbrev Coeff (n : ℕ) := Fin n → (ℕ × ℕ)
 \end{minted}
 
-Esse par consiste do coeficiente de $x_i$ no primeiro elemento e o coeficiente de $\overline x_i$ no segundo elemento.
+Esse par consiste do coeficiente de $x_i$ no primeiro elemento e
+o coeficiente de $\overline x_i$ no segundo elemento.
 
 Com essa definição, definimos o \textit{PBSum}, que, com os coeficientes $cs$ e os valores 0-1 $xs$,
 utiliza o \textit{BigOperator} de somatório ($\sum$) para somar os elementos da lista iterando no índice $i$:
@@ -148,6 +149,31 @@ \subsection{Definição de Inequações Pseudo-Booleanas}
     if xs i = 1 then p else n
 \end{minted}
 
+E com esse somatório podemos agora criar o \textit{PBIneq} que é a verificação que o \textit{PBSum}
+é maior ou igual à constante $const$:
+\begin{minted}[fontsize=\small]{lean4}
+def PBIneq (cs : Coeff n) (xs : Fin n → Fin 2)
+    (const : ℕ) :=
+  PBSum cs xs ≥ const
+\end{minted}
+
+Para criar uma expressão desse tipo basta fornecer as listas de coeficientes, pseudo-booleanos e a constante,
+e em seguida provar que a propriedade vale:
+\begin{minted}[fontsize=\small]{lean4}
+example : PBIneq ![(1,0),(2,0)] ![0,1] 2 := by
+  -- Change goal to 1 * 0 + 2 * 1 ≥ 2
+  reduce               
+  -- Prove 1 * 0 + 2 * 1 ≥ 2
+  exact Nat.le_refl 2  
+  done
+\end{minted}
+
+O exemplo acima prova a expressão
+\begin{equation}
+    x_1 + 2 x_2 \ge 2
+\end{equation}
+quando $x_1 = 0$ e $x_2 = 1$.
+
 
 % * Prova da Regra Multiplication
 \subsection{Prova da Regra Multiplicação}

From 413493ef193a089d92995fa4f269e564da9820d3 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Fri, 28 Jun 2024 15:52:22 -0300
Subject: [PATCH 74/93] Finish Multiplication Proof article

---
 article/cutting-planes-article.pdf | Bin 172515 -> 174766 bytes
 article/cutting-planes-article.tex |  22 +++++++++++++++++++++-
 2 files changed, 21 insertions(+), 1 deletion(-)

diff --git a/article/cutting-planes-article.pdf b/article/cutting-planes-article.pdf
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 684353d..0be8661 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -177,7 +177,27 @@ \subsection{Definição de Inequações Pseudo-Booleanas}
 
 % * Prova da Regra Multiplication
 \subsection{Prova da Regra Multiplicação}
-Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+A primeira regra a ser formalizada é a multiplicação, que diz que dada uma inequação pseudo-booleana, podemos obter
+outra inequação válida ao multiplicar os coeficientes por um escalar natural $c \in \mathbb{N}^+ $, ao mesmo tempo que multiplicamos
+a constante pelo mesmo valor.
+
+\begin{equation}
+    \frac
+    {\sum_i{a_i l_i} \ge A}
+    {\sum_i{c a_i l_i} \ge c A}
+\end{equation}
+
+O teorema \textit{Multiplication} implementa esse comportamento em Lean:
+\begin{minted}[fontsize=\small]{lean4}
+theorem Multiplication
+  {xs : Fin n → Fin 2}
+  {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
+  (c : ℕ)
+  : PBIneq (c • as) xs (c • A)
+\end{minted}
+
+A prova completa se encontra no Apêndice A. Como a muliplicação por escalares já está definida na \textit{mathlib4} para
+os \textit{FinVec}s pelo teorema \texttt{Finset.sum\_nsmul} da biblioteca.
 
 % * Prova da Regra Saturation
 \subsection{Prova da Regra Saturação}

From 831f7102f8860cb060deef29a481f0ea88610144 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Fri, 28 Jun 2024 16:36:55 -0300
Subject: [PATCH 75/93] Proof of Saturation article

---
 article/cutting-planes-article.pdf | Bin 174766 -> 188555 bytes
 article/cutting-planes-article.tex |  44 ++++++++++++++++++++++++++++-
 2 files changed, 43 insertions(+), 1 deletion(-)

diff --git a/article/cutting-planes-article.pdf b/article/cutting-planes-article.pdf
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 0be8661..966c974 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -196,12 +196,54 @@ \subsection{Prova da Regra Multiplicação}
   : PBIneq (c • as) xs (c • A)
 \end{minted}
 
+% TODO: Criar apêndice A com código completo
 A prova completa se encontra no Apêndice A. Como a muliplicação por escalares já está definida na \textit{mathlib4} para
 os \textit{FinVec}s pelo teorema \texttt{Finset.sum\_nsmul} da biblioteca.
 
 % * Prova da Regra Saturation
 \subsection{Prova da Regra Saturação}
-Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+A regra da Saturação permite substituir os coeficientes de uma inequação pelo \textit{mínimo} desse número com a constante $A$:
+\begin{equation}
+    \frac
+    {\sum_i{a_i l_i} \ge A}
+    {\sum_i{ \min(a_i,A)\cdot l_i} \ge A}
+\end{equation}
+
+O teorema \textit{Saturation} implementa esse comportamento ao aplicar \textit{map} na lista de coeficientes com a função
+\texttt{mapBoth (min A)}, que transforma ambos elementos do par no mínimo entre eles e a constante $A$:
+\begin{minted}[fontsize=\small]{lean4}
+theorem Saturation
+  {xs : Fin n → Fin 2}
+  {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
+  : PBIneq (map (mapBoth (min A)) as) xs A
+\end{minted}
+
+% TODO: Criar apêndice B com código completo
+Este teorema (Apêndice B) envolveu mais passos, pois não havia o mesmo suporte nativo da \textit{mathlib4}.
+O lema que provamos, chamado \texttt{le\_sum\_min\_of\_le\_sum} é o caso mais simples, onde trabalhamos com uma
+lista de naturais e desejamos mostrar que a relação menor-ou-igual se mantém ao aplicar o mínimo de $A$:
+\begin{minted}[fontsize=\small]{lean4}
+lemma le_sum_min_of_le_sum {n A : ℕ}
+  {as : Fin n → ℕ}
+  (h : A ≤ ∑i, as i)
+  : A ≤ ∑i, min A (as i)
+\end{minted}
+
+Em alto nível, podemos provar isso por casos:
+\begin{enumerate}
+    \item Todos os elementos de $as$ são menores-ou-iguais a $A$.
+          Nesse caso $\min(A,as_i) = as_i$, para todo $i$, logo temos a mesma lista.
+          % os elementos,
+          Então a afirmação vale pela hipótese.
+    \item Caso contrário, existe ao menos um índice $k$ da lista $as$, onde $as_k > A$.
+          Podemos dividir o somatório em $\sum_{i\neq k}{\min(A,as_i) + \min(A,as_k)}$, separando esse índice em específico.
+          Como $as_k > A$, substituímos $\min(A,as_k)$ por $A$.
+          Como queremos mostrar que $A \le \sum_{i\neq k}{\min(A,as_i) + A}$,
+          terminamos a prova com o teorema \texttt{Nat.le\_add\_left A}, que diz
+          $A \le B + A$, para qualquer $B \in \mathbb{N}$.
+
+\end{enumerate}
+
 
 % * Prova da Regra Division
 \subsection{Prova da Regra Divisão}

From e0db857f5895719f8e51c878b8504e9faa541233 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Fri, 28 Jun 2024 17:58:32 -0300
Subject: [PATCH 76/93] Proof of Division article

---
 article/cutting-planes-article.pdf | Bin 188555 -> 209408 bytes
 article/cutting-planes-article.tex |  55 +++++++++++++++++++++++++----
 2 files changed, 49 insertions(+), 6 deletions(-)

diff --git a/article/cutting-planes-article.pdf b/article/cutting-planes-article.pdf
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 966c974..c4dbc23 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -168,11 +168,7 @@ \subsection{Definição de Inequações Pseudo-Booleanas}
   done
 \end{minted}
 
-O exemplo acima prova a expressão
-\begin{equation}
-    x_1 + 2 x_2 \ge 2
-\end{equation}
-quando $x_1 = 0$ e $x_2 = 1$.
+O exemplo acima prova a expressão $x_1 + 2 x_2 \ge 2$, quando $x_1 = 0$ e $x_2 = 1$.
 
 
 % * Prova da Regra Multiplication
@@ -247,7 +243,54 @@ \subsection{Prova da Regra Saturação}
 
 % * Prova da Regra Division
 \subsection{Prova da Regra Divisão}
-Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+A regra Divisão nos permite fazer o caminho inverso da Multiplicação, dividindo por um escalar $c \in \mathbb{N}^+$, com a diferença que divisões não exatas serão arrendondadas
+para cima:
+\begin{equation}
+    \frac
+    {\sum_i{a_i l_i} \ge A}
+    {\sum_i{ \lceil \frac{a_i}{c} \rceil l_i} \ge \lceil \frac{A}{c} \rceil}
+\end{equation}
+
+O teorema \textit{Division} implementa esse comportamento ao aplicar \textit{map} na lista de coeficientes com a função
+\texttt{mapBoth (ceildiv c)}, que divide ambos elementos do par por $c$, arredondando para cima:
+\begin{minted}[fontsize=\small]{lean4}
+theorem Division
+  {xs : Fin n → Fin 2}
+  {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
+  (c : ℕ)
+  : PBIneq (map (mapBoth (ceildiv c)) as)
+    xs (ceildiv c A)
+\end{minted}
+
+Essa prova foi mais complexa, pois precisamos mostrar o comportamento para a lista toda,
+não sendo suficiente apenas mostrar para algum elemento em particular, como no caso da Saturação.
+Dispondo de ajuda pelo \textit{Zulip do Lean}, chegamos em dois lemas que permitem provar a propriedade.
+
+Mostramos primeiramente que, para dois elementos em isolados, a divisão com teto da soma é menor-ou-igual à soma das divisões com teto:
+\begin{minted}[fontsize=\small]{lean4}
+theorem Nat.add_ceildiv_le_add_ceildiv
+    (a b c : ℕ)
+  : (a + b) ⌈/⌉ c ≤ (a ⌈/⌉ c) + (b ⌈/⌉ c) 
+\end{minted}
+
+Com esse teorema, agora podemos criar uma prova por indução que vai valer para listas de qualquer tamanho:
+\begin{minted}[fontsize=\small]{lean4}
+theorem Finset.ceildiv_le_ceildiv {α : Type*}
+    (as : α → ℕ) (s : Finset α) (c : ℕ)
+  : (∑i in s, as i) ⌈/⌉ c ≤ ∑i in s,(as i ⌈/⌉ c)
+\end{minted}
+
+O último detalhe que tivemos que provar é que podemos distribuir o \textit{ceildiv} sobre a expressão \texttt{if-then-else}:
+\begin{minted}[fontsize=\small]{lean4}
+lemma ceildiv_ite (P : Prop) [Decidable P]
+    (a b c : ℕ)
+  : (if P then b else c) ⌈/⌉ a
+  = if P then (b ⌈/⌉ a) else (c ⌈/⌉ a)
+\end{minted}
+
+Com isso concluímos a prova.
+
+
 
 % * Prova da Regra Addition
 \subsection{Prova da Regra Adição}

From 0a7f6a3f426c9f94027b6389682388715a1cbc41 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Fri, 28 Jun 2024 18:49:44 -0300
Subject: [PATCH 77/93] Proof of Addtion article

---
 article/IEEEtran.cls               | 5623 ++++++++++++++--------------
 article/cutting-planes-article.pdf |  Bin 209408 -> 217469 bytes
 article/cutting-planes-article.tex |   95 +-
 3 files changed, 2898 insertions(+), 2820 deletions(-)

diff --git a/article/IEEEtran.cls b/article/IEEEtran.cls
index b0cf599..6b1e271 100644
--- a/article/IEEEtran.cls
+++ b/article/IEEEtran.cls
@@ -295,35 +295,35 @@
 
 
 \DeclareOption{letterpaper}{\setlength{\paperwidth}{8.5in}%
-                            \setlength{\paperheight}{11in}%
-                            \@IEEEusingAfourpaperfalse
-                            \@IEEEusingcspaperfalse
-                            \def\CLASSOPTIONpaper{letter}%
-                            \def\CLASSINFOpaperwidth{8.5in}%
-                            \def\CLASSINFOpaperheight{11in}}
+    \setlength{\paperheight}{11in}%
+    \@IEEEusingAfourpaperfalse
+    \@IEEEusingcspaperfalse
+    \def\CLASSOPTIONpaper{letter}%
+    \def\CLASSINFOpaperwidth{8.5in}%
+    \def\CLASSINFOpaperheight{11in}}
 
 
 \DeclareOption{a4paper}{\setlength{\paperwidth}{210mm}%
-                        \setlength{\paperheight}{297mm}%
-                        \@IEEEusingAfourpapertrue
-                        \@IEEEusingcspaperfalse
-                        \def\CLASSOPTIONpaper{a4}%
-                        \def\CLASSINFOpaperwidth{210mm}%
-                        \def\CLASSINFOpaperheight{297mm}}
+    \setlength{\paperheight}{297mm}%
+    \@IEEEusingAfourpapertrue
+    \@IEEEusingcspaperfalse
+    \def\CLASSOPTIONpaper{a4}%
+    \def\CLASSINFOpaperwidth{210mm}%
+    \def\CLASSINFOpaperheight{297mm}}
 
 % special paper option for compsoc journals
 \DeclareOption{cspaper}{\setlength{\paperwidth}{7.875in}%
-                        \setlength{\paperheight}{10.75in}%
-                        \@IEEEusingcspapertrue
-                        \@IEEEusingAfourpaperfalse
-                        \def\CLASSOPTIONpaper{ieeecs}%
-                        \def\CLASSINFOpaperwidth{7.875in}%
-                        \def\CLASSINFOpaperheight{10.75in}}
+    \setlength{\paperheight}{10.75in}%
+    \@IEEEusingcspapertrue
+    \@IEEEusingAfourpaperfalse
+    \def\CLASSOPTIONpaper{ieeecs}%
+    \def\CLASSINFOpaperwidth{7.875in}%
+    \def\CLASSINFOpaperheight{10.75in}}
 
 \DeclareOption{oneside}{\@twosidefalse\@mparswitchfalse
-                        \CLASSOPTIONonesidetrue\CLASSOPTIONtwosidefalse}
+    \CLASSOPTIONonesidetrue\CLASSOPTIONtwosidefalse}
 \DeclareOption{twoside}{\@twosidetrue\@mparswitchtrue
-                        \CLASSOPTIONtwosidetrue\CLASSOPTIONonesidefalse}
+    \CLASSOPTIONtwosidetrue\CLASSOPTIONonesidefalse}
 
 \DeclareOption{onecolumn}{\CLASSOPTIONonecolumntrue\CLASSOPTIONtwocolumnfalse}
 \DeclareOption{twocolumn}{\CLASSOPTIONtwocolumntrue\CLASSOPTIONonecolumnfalse}
@@ -331,31 +331,31 @@
 % If the user selects draft, then this class AND any packages
 % will go into draft mode.
 \DeclareOption{draft}{\CLASSOPTIONdrafttrue\CLASSOPTIONdraftclstrue
-                      \CLASSOPTIONdraftclsnofootfalse} 
+    \CLASSOPTIONdraftclsnofootfalse}
 % draftcls is for a draft mode which will not affect any packages
 % used by the document.
 \DeclareOption{draftcls}{\CLASSOPTIONdraftfalse\CLASSOPTIONdraftclstrue
-                         \CLASSOPTIONdraftclsnofootfalse} 
+    \CLASSOPTIONdraftclsnofootfalse}
 % draftclsnofoot is like draftcls, but without the footer.
 \DeclareOption{draftclsnofoot}{\CLASSOPTIONdraftfalse\CLASSOPTIONdraftclstrue
-                               \CLASSOPTIONdraftclsnofoottrue} 
+    \CLASSOPTIONdraftclsnofoottrue}
 \DeclareOption{final}{\CLASSOPTIONdraftfalse\CLASSOPTIONdraftclsfalse
-                      \CLASSOPTIONdraftclsnofootfalse}
+    \CLASSOPTIONdraftclsnofootfalse}
 
 \DeclareOption{journal}{\CLASSOPTIONpeerreviewfalse\CLASSOPTIONpeerreviewcafalse
-                        \CLASSOPTIONjournaltrue\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotefalse}
+    \CLASSOPTIONjournaltrue\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotefalse}
 
 \DeclareOption{conference}{\CLASSOPTIONpeerreviewfalse\CLASSOPTIONpeerreviewcafalse
-                           \CLASSOPTIONjournalfalse\CLASSOPTIONconferencetrue\CLASSOPTIONtechnotefalse}
+    \CLASSOPTIONjournalfalse\CLASSOPTIONconferencetrue\CLASSOPTIONtechnotefalse}
 
 \DeclareOption{technote}{\CLASSOPTIONpeerreviewfalse\CLASSOPTIONpeerreviewcafalse
-                         \CLASSOPTIONjournalfalse\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotetrue}
+    \CLASSOPTIONjournalfalse\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotetrue}
 
 \DeclareOption{peerreview}{\CLASSOPTIONpeerreviewtrue\CLASSOPTIONpeerreviewcafalse
-                           \CLASSOPTIONjournalfalse\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotefalse}
+    \CLASSOPTIONjournalfalse\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotefalse}
 
 \DeclareOption{peerreviewca}{\CLASSOPTIONpeerreviewtrue\CLASSOPTIONpeerreviewcatrue
-                             \CLASSOPTIONjournalfalse\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotefalse}
+    \CLASSOPTIONjournalfalse\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotefalse}
 
 \DeclareOption{nofonttune}{\CLASSOPTIONnofonttunetrue}
 
@@ -382,15 +382,15 @@
 % Sets the category codes for punctuation to their normal values.
 % For local use with argument scanning.
 \def\IEEEnormalcatcodespunct{\catcode`\!=12 \catcode`\,=12 \catcode`\:=12
-\catcode`\;=12 \catcode`\`=12 \catcode`\'=12 \catcode`\"=12 \catcode`\.=12
-\catcode`\/=12 \catcode`\?=12 \catcode`\*=12 \catcode`\+=12 \catcode`\-=12
-\catcode`\<=12 \catcode`\>=12 \catcode`\(=12 \catcode`\)=12 \catcode`\[=12
-\catcode`\]=12 \catcode`\==12 \catcode`\|=12}
+    \catcode`\;=12 \catcode`\`=12 \catcode`\'=12 \catcode`\"=12 \catcode`\.=12
+    \catcode`\/=12 \catcode`\?=12 \catcode`\*=12 \catcode`\+=12 \catcode`\-=12
+    \catcode`\<=12 \catcode`\>=12 \catcode`\(=12 \catcode`\)=12 \catcode`\[=12
+        \catcode`\]=12 \catcode`\==12 \catcode`\|=12}
 % Sets the category codes for numbers to their normal values.
 % For local use with argument scanning.
 \def\IEEEnormalcatcodesnum{\catcode`\0=12 \catcode`\1=12 \catcode`\2=12
-\catcode`\3=12 \catcode`\4=12 \catcode`\5=12 \catcode`\6=12 \catcode`\7=12
-\catcode`\8=12 \catcode`\9=12}
+    \catcode`\3=12 \catcode`\4=12 \catcode`\5=12 \catcode`\6=12 \catcode`\7=12
+    \catcode`\8=12 \catcode`\9=12}
 % combined action of \IEEEnormalcatcodespunct and \IEEEnormalcatcodesnum
 \def\IEEEnormalcatcodes{\IEEEnormalcatcodespunct\IEEEnormalcatcodesnum}
 
@@ -422,31 +422,31 @@
 \def\@IEEEextracttoken{\@ifstar{\let\@IEEEextracttokendef=\def\@@IEEEextracttoken}{\let\@IEEEextracttokendef=\edef\@@IEEEextracttoken}}
 
 \def\@@IEEEextracttoken#1{\@IEEEextracttokendef\@IEEEextracttokenarg{#1}\relax
-\def\@IEEEextractedtokensdiscarded{}\relax % initialize to empty
-% if the macro is unchanged after being acquired as a single undelimited argument
-% with anything after it being stripped off as a delimited argument
-% we know we have one token without any enclosing braces. loop until this is true.
-\let\@IEEEextracttokencurgroup\@IEEEextracttokenarg
-\loop
-  % trap case of an empty argument as this would cause a problem with
-  % \@@@IEEEextracttoken's first (nondelimited) argument acquisition
-  \ifx\@IEEEextracttokencurgroup\@empty
-    \def\@IEEEextractedtokenmacro{}\relax
-  \else
-    \expandafter\@@@IEEEextracttoken\@IEEEextracttokencurgroup\@IEEEgeneralsequenceDELIMITER\relax
-  \fi
-  \ifx\@IEEEextractedtokenmacro\@IEEEextracttokencurgroup
-  \else
+    \def\@IEEEextractedtokensdiscarded{}\relax % initialize to empty
+    % if the macro is unchanged after being acquired as a single undelimited argument
+    % with anything after it being stripped off as a delimited argument
+    % we know we have one token without any enclosing braces. loop until this is true.
+    \let\@IEEEextracttokencurgroup\@IEEEextracttokenarg
+    \loop
+    % trap case of an empty argument as this would cause a problem with
+    % \@@@IEEEextracttoken's first (nondelimited) argument acquisition
+    \ifx\@IEEEextracttokencurgroup\@empty
+        \def\@IEEEextractedtokenmacro{}\relax
+    \else
+        \expandafter\@@@IEEEextracttoken\@IEEEextracttokencurgroup\@IEEEgeneralsequenceDELIMITER\relax
+    \fi
+    \ifx\@IEEEextractedtokenmacro\@IEEEextracttokencurgroup
+    \else
     \let\@IEEEextracttokencurgroup=\@IEEEextractedtokenmacro
-\repeat
-% we can safely do a \let= here because there should be at most one token
-% the relax is needed to handle the case of no token found
-\expandafter\let\expandafter\@IEEEextractedtoken\@IEEEextractedtokenmacro\relax}
+    \repeat
+    % we can safely do a \let= here because there should be at most one token
+    % the relax is needed to handle the case of no token found
+    \expandafter\let\expandafter\@IEEEextractedtoken\@IEEEextractedtokenmacro\relax}
 
 \def\@@@IEEEextracttoken#1#2\@IEEEgeneralsequenceDELIMITER{\def\@IEEEextractedtokenmacro{#1}\relax
-\def\@@IEEEextractedtokensdiscarded{#2}\expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter
-\@IEEEextractedtokensdiscarded\expandafter\expandafter\expandafter
-{\expandafter\@@IEEEextractedtokensdiscarded\@IEEEextractedtokensdiscarded}}
+    \def\@@IEEEextractedtokensdiscarded{#2}\expandafter\expandafter\expandafter\def\expandafter\expandafter\expandafter
+    \@IEEEextractedtokensdiscarded\expandafter\expandafter\expandafter
+    {\expandafter\@@IEEEextractedtokensdiscarded\@IEEEextractedtokensdiscarded}}
 %%
 %% -- End of Command Argument Scanning Support Functions --
 
@@ -464,14 +464,14 @@
 
 % comsoc verify that newtxmath, mtpro2, mt11p or mathtime has been loaded
 \def\@IEEEcomsocverifymathfont{\typeout{-- Verifying Times compatible math font.}\relax
-  \@ifpackageloaded{newtxmath}{\typeout{-- newtxmath loaded, OK.}}{\@@IEEEcomsocverifymathfont}}
+    \@ifpackageloaded{newtxmath}{\typeout{-- newtxmath loaded, OK.}}{\@@IEEEcomsocverifymathfont}}
 \def\@@IEEEcomsocverifymathfont{\@ifpackageloaded{mtpro2}{\typeout{-- mtpro2 loaded, OK.}}{\@@@IEEEcomsocverifymathfont}}
 \def\@@@IEEEcomsocverifymathfont{\@ifpackageloaded{mt11p}{\typeout{-- mt11p2 loaded, OK.}}{\@@@@IEEEcomsocverifymathfont}}
 \def\@@@@IEEEcomsocverifymathfont{\@ifpackageloaded{mathtime}{\typeout{-- mathtime loaded, OK.}}{\@IEEEcomsocenforcemathfont}}
 
 % comsoc, if a Times math font was not loaded by user, enforce it
 \def\@IEEEcomsocenforcemathfont{\typeout{** Times compatible math font not found, forcing.}\relax
-\IfFileExists{newtxmath.sty}{\typeout{-- Found newtxmath, loading.}\RequirePackage{newtxmath}}{\@@IEEEcomsocenforcemathfont}}
+    \IfFileExists{newtxmath.sty}{\typeout{-- Found newtxmath, loading.}\RequirePackage{newtxmath}}{\@@IEEEcomsocenforcemathfont}}
 \def\@@IEEEcomsocenforcemathfont{\IfFileExists{mtpro2.sty}{\typeout{-- Found mtpro2, loading.}\RequirePackage{mtpro2}}{\@@@IEEEcomsocenforcemathfont}}
 \def\@@@IEEEcomsocenforcemathfont{\IfFileExists{mt11p.sty}{\typeout{-- Found mt11p, loading.}\RequirePackage{mt11p}}{\@@@@IEEEcomsocenforcemathfont}}
 \def\@@@@IEEEcomsocenforcemathfont{\IfFileExists{mathtime.sty}{\typeout{-- Found mathtime, loading.}\RequirePackage{mathtime}}{\@@@@@IEEEcomsocenforcemathfont}}
@@ -480,11 +480,11 @@
 
 
 \ifCLASSOPTIONcomsoc
-  % ensure that if newtxmath is used, the cmintegrals option is also invoked
-  \PassOptionsToPackage{cmintegrals}{newtxmath}
-  % comsoc requires a Times like math font
-  % ensure this requirement is satisfied at document start
-  \AtBeginDocument{\@IEEEcomsocverifymathfont}
+    % ensure that if newtxmath is used, the cmintegrals option is also invoked
+    \PassOptionsToPackage{cmintegrals}{newtxmath}
+    % comsoc requires a Times like math font
+    % ensure this requirement is satisfied at document start
+    \AtBeginDocument{\@IEEEcomsocverifymathfont}
 \fi
 
 
@@ -504,26 +504,26 @@
 
 
 \ifCLASSOPTIONcomsoc
-  \typeout{-- Using IEEE Communications Society mode.}
+    \typeout{-- Using IEEE Communications Society mode.}
 \fi
 
 \ifCLASSOPTIONcompsoc
-  \typeout{-- Using IEEE Computer Society mode.}
+    \typeout{-- Using IEEE Computer Society mode.}
 \fi
 
 
 % V1.7 conference notice message hook
 \def\@IEEEconsolenoticeconference{\typeout{}%
-\typeout{** Conference Paper **}%
-\typeout{Before submitting the final camera ready copy, remember to:}%
-\typeout{}%
-\typeout{ 1. Manually equalize the lengths of two columns on the last page}%
-\typeout{ of your paper;}%
-\typeout{}%
-\typeout{ 2. Ensure that any PostScript and/or PDF output post-processing}%
-\typeout{ uses only Type 1 fonts and that every step in the generation}%
-\typeout{ process uses the appropriate paper size.}%
-\typeout{}}
+    \typeout{** Conference Paper **}%
+    \typeout{Before submitting the final camera ready copy, remember to:}%
+    \typeout{}%
+    \typeout{ 1. Manually equalize the lengths of two columns on the last page}%
+    \typeout{ of your paper;}%
+    \typeout{}%
+    \typeout{ 2. Ensure that any PostScript and/or PDF output post-processing}%
+    \typeout{ uses only Type 1 fonts and that every step in the generation}%
+    \typeout{ process uses the appropriate paper size.}%
+    \typeout{}}
 
 
 % we can send console reminder messages to the user here
@@ -532,9 +532,9 @@
 
 % warn about the use of single column other than for draft mode
 \ifCLASSOPTIONtwocolumn\else%
-  \ifCLASSOPTIONdraftcls\else%
-   \typeout{** ATTENTION: Single column mode is not typically used with IEEE publications.}%
-  \fi%
+    \ifCLASSOPTIONdraftcls\else%
+        \typeout{** ATTENTION: Single column mode is not typically used with IEEE publications.}%
+    \fi%
 \fi
 
 
@@ -544,27 +544,28 @@
 % even if only effect is to set them to \relax.
 % if \pdfoutput is undefined or equal to relax, output a dvips special
 {\@ifundefined{pdfoutput}{\AtBeginDvi{\special{papersize=\CLASSINFOpaperwidth,\CLASSINFOpaperheight}}}{%
-% pdfoutput is defined and not equal to \relax
-% check for pdfpageheight existence just in case someone sets pdfoutput
-% under non-pdflatex. If exists, set them regardless of value of \pdfoutput.
-\@ifundefined{pdfpageheight}{\relax}{\global\pdfpagewidth\paperwidth
-\global\pdfpageheight\paperheight}%
-% if using \pdfoutput=0 under pdflatex, send dvips papersize special
-\ifcase\pdfoutput
-\AtBeginDvi{\special{papersize=\CLASSINFOpaperwidth,\CLASSINFOpaperheight}}%
-\else
-% we are using pdf output, set CLASSINFOpdf flag
-\global\CLASSINFOpdftrue
-\fi}}
+    % pdfoutput is defined and not equal to \relax
+    % check for pdfpageheight existence just in case someone sets pdfoutput
+    % under non-pdflatex. If exists, set them regardless of value of \pdfoutput.
+    \@ifundefined{pdfpageheight}{\relax}{\global\pdfpagewidth\paperwidth
+        \global\pdfpageheight\paperheight}%
+    % if using \pdfoutput=0 under pdflatex, send dvips papersize special
+    \ifcase\pdfoutput
+        \AtBeginDvi{\special{papersize=\CLASSINFOpaperwidth,\CLASSINFOpaperheight}}%
+    \else
+        % we are using pdf output, set CLASSINFOpdf flag
+        \global\CLASSINFOpdftrue
+    \fi}}
 
 % let the user know the selected papersize
 \typeout{-- Using \CLASSINFOpaperwidth\space x \CLASSINFOpaperheight\space
-(\CLASSOPTIONpaper)\space paper.}
+    (\CLASSOPTIONpaper)\space paper.}
 
 \ifCLASSINFOpdf
-\typeout{-- Using PDF output.}
+    \typeout{-- Using PDF output.}
 \else
-\typeout{-- Using DVI output.}
+    \typeout{-- Using DVI output.}
+
 \fi
 
 
@@ -623,94 +624,94 @@
 
 % 9pt option defaults
 \ifx\CLASSOPTIONpt\@IEEEptsizenine
-\typeout{-- This is a 9 point document.}
-\def\normalsize{\@setfontsize{\normalsize}{9}{11.0476pt}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{11.0476pt}
-\normalsize
-\abovedisplayskip 1.5ex plus 3pt minus 1pt
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0pt plus 3pt
-\belowdisplayshortskip 1.5ex plus 3pt minus 1pt
-\def\small{\@setfontsize{\small}{8.5}{10pt}}
-\def\footnotesize{\@setfontsize{\footnotesize}{8}{9pt}}
-\def\scriptsize{\@setfontsize{\scriptsize}{7}{8pt}}
-\def\tiny{\@setfontsize{\tiny}{5}{6pt}}
-% sublargesize is the same as large - 10pt
-\def\sublargesize{\@setfontsize{\sublargesize}{10}{12pt}}
-\def\large{\@setfontsize{\large}{10}{12pt}}
-\def\Large{\@setfontsize{\Large}{12}{14pt}}
-\def\LARGE{\@setfontsize{\LARGE}{14}{17pt}}
-\def\huge{\@setfontsize{\huge}{17}{20pt}}
-\def\Huge{\@setfontsize{\Huge}{20}{24pt}}
+    \typeout{-- This is a 9 point document.}
+    \def\normalsize{\@setfontsize{\normalsize}{9}{11.0476pt}}
+    \setlength{\@IEEEnormalsizeunitybaselineskip}{11.0476pt}
+    \normalsize
+    \abovedisplayskip 1.5ex plus 3pt minus 1pt
+    \belowdisplayskip \abovedisplayskip
+    \abovedisplayshortskip 0pt plus 3pt
+    \belowdisplayshortskip 1.5ex plus 3pt minus 1pt
+    \def\small{\@setfontsize{\small}{8.5}{10pt}}
+    \def\footnotesize{\@setfontsize{\footnotesize}{8}{9pt}}
+    \def\scriptsize{\@setfontsize{\scriptsize}{7}{8pt}}
+    \def\tiny{\@setfontsize{\tiny}{5}{6pt}}
+    % sublargesize is the same as large - 10pt
+    \def\sublargesize{\@setfontsize{\sublargesize}{10}{12pt}}
+    \def\large{\@setfontsize{\large}{10}{12pt}}
+    \def\Large{\@setfontsize{\Large}{12}{14pt}}
+    \def\LARGE{\@setfontsize{\LARGE}{14}{17pt}}
+    \def\huge{\@setfontsize{\huge}{17}{20pt}}
+    \def\Huge{\@setfontsize{\Huge}{20}{24pt}}
 \fi
 %
 % 10pt option defaults
 \ifx\CLASSOPTIONpt\@IEEEptsizeten
-\typeout{-- This is a 10 point document.}
-\def\normalsize{\@setfontsize{\normalsize}{10}{12.00pt}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{12pt}
-\normalsize
-\abovedisplayskip 1.5ex plus 4pt minus 2pt
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0pt plus 4pt
-\belowdisplayshortskip 1.5ex plus 4pt minus 2pt
-\def\small{\@setfontsize{\small}{9}{10pt}}
-\def\footnotesize{\@setfontsize{\footnotesize}{8}{9pt}}
-\def\scriptsize{\@setfontsize{\scriptsize}{7}{8pt}}
-\def\tiny{\@setfontsize{\tiny}{5}{6pt}}
-% sublargesize is a tad smaller than large - 11pt
-\def\sublargesize{\@setfontsize{\sublargesize}{11}{13.4pt}}
-\def\large{\@setfontsize{\large}{12}{14pt}}
-\def\Large{\@setfontsize{\Large}{14}{17pt}}
-\def\LARGE{\@setfontsize{\LARGE}{17}{20pt}}
-\def\huge{\@setfontsize{\huge}{20}{24pt}}
-\def\Huge{\@setfontsize{\Huge}{24}{28pt}}
+    \typeout{-- This is a 10 point document.}
+    \def\normalsize{\@setfontsize{\normalsize}{10}{12.00pt}}
+    \setlength{\@IEEEnormalsizeunitybaselineskip}{12pt}
+    \normalsize
+    \abovedisplayskip 1.5ex plus 4pt minus 2pt
+    \belowdisplayskip \abovedisplayskip
+    \abovedisplayshortskip 0pt plus 4pt
+    \belowdisplayshortskip 1.5ex plus 4pt minus 2pt
+    \def\small{\@setfontsize{\small}{9}{10pt}}
+    \def\footnotesize{\@setfontsize{\footnotesize}{8}{9pt}}
+    \def\scriptsize{\@setfontsize{\scriptsize}{7}{8pt}}
+    \def\tiny{\@setfontsize{\tiny}{5}{6pt}}
+    % sublargesize is a tad smaller than large - 11pt
+    \def\sublargesize{\@setfontsize{\sublargesize}{11}{13.4pt}}
+    \def\large{\@setfontsize{\large}{12}{14pt}}
+    \def\Large{\@setfontsize{\Large}{14}{17pt}}
+    \def\LARGE{\@setfontsize{\LARGE}{17}{20pt}}
+    \def\huge{\@setfontsize{\huge}{20}{24pt}}
+    \def\Huge{\@setfontsize{\Huge}{24}{28pt}}
 \fi
 %
 % 11pt option defaults
 \ifx\CLASSOPTIONpt\@IEEEptsizeeleven
-\typeout{-- This is an 11 point document.}
-\def\normalsize{\@setfontsize{\normalsize}{11}{13.3846pt}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{13.3846pt}
-\normalsize
-\abovedisplayskip 1.5ex plus 5pt minus 3pt
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0pt plus 5pt
-\belowdisplayshortskip 1.5ex plus 5pt minus 3pt
-\def\small{\@setfontsize{\small}{10}{12pt}}
-\def\footnotesize{\@setfontsize{\footnotesize}{9}{10.5pt}}
-\def\scriptsize{\@setfontsize{\scriptsize}{8}{9pt}}
-\def\tiny{\@setfontsize{\tiny}{6}{7pt}}
-% sublargesize is the same as large - 12pt
-\def\sublargesize{\@setfontsize{\sublargesize}{12}{14pt}}
-\def\large{\@setfontsize{\large}{12}{14pt}}
-\def\Large{\@setfontsize{\Large}{14}{17pt}}
-\def\LARGE{\@setfontsize{\LARGE}{17}{20pt}}
-\def\huge{\@setfontsize{\huge}{20}{24pt}}
-\def\Huge{\@setfontsize{\Huge}{24}{28pt}}
+    \typeout{-- This is an 11 point document.}
+    \def\normalsize{\@setfontsize{\normalsize}{11}{13.3846pt}}
+    \setlength{\@IEEEnormalsizeunitybaselineskip}{13.3846pt}
+    \normalsize
+    \abovedisplayskip 1.5ex plus 5pt minus 3pt
+    \belowdisplayskip \abovedisplayskip
+    \abovedisplayshortskip 0pt plus 5pt
+    \belowdisplayshortskip 1.5ex plus 5pt minus 3pt
+    \def\small{\@setfontsize{\small}{10}{12pt}}
+    \def\footnotesize{\@setfontsize{\footnotesize}{9}{10.5pt}}
+    \def\scriptsize{\@setfontsize{\scriptsize}{8}{9pt}}
+    \def\tiny{\@setfontsize{\tiny}{6}{7pt}}
+    % sublargesize is the same as large - 12pt
+    \def\sublargesize{\@setfontsize{\sublargesize}{12}{14pt}}
+    \def\large{\@setfontsize{\large}{12}{14pt}}
+    \def\Large{\@setfontsize{\Large}{14}{17pt}}
+    \def\LARGE{\@setfontsize{\LARGE}{17}{20pt}}
+    \def\huge{\@setfontsize{\huge}{20}{24pt}}
+    \def\Huge{\@setfontsize{\Huge}{24}{28pt}}
 \fi
 %
 % 12pt option defaults
 \ifx\CLASSOPTIONpt\@IEEEptsizetwelve
-\typeout{-- This is a 12 point document.}
-\def\normalsize{\@setfontsize{\normalsize}{12}{13.92pt}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{13.92pt}
-\normalsize
-\abovedisplayskip 1.5ex plus 6pt minus 4pt
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0pt plus 6pt
-\belowdisplayshortskip 1.5ex plus 6pt minus 4pt
-\def\small{\@setfontsize{\small}{10}{12pt}}
-\def\footnotesize{\@setfontsize{\footnotesize}{9}{10.5pt}}
-\def\scriptsize{\@setfontsize{\scriptsize}{8}{9pt}}
-\def\tiny{\@setfontsize{\tiny}{6}{7pt}}
-% sublargesize is the same as large - 14pt
-\def\sublargesize{\@setfontsize{\sublargesize}{14}{17pt}}
-\def\large{\@setfontsize{\large}{14}{17pt}}
-\def\Large{\@setfontsize{\Large}{17}{20pt}}
-\def\LARGE{\@setfontsize{\LARGE}{20}{24pt}}
-\def\huge{\@setfontsize{\huge}{22}{26pt}}
-\def\Huge{\@setfontsize{\Huge}{24}{28pt}}
+    \typeout{-- This is a 12 point document.}
+    \def\normalsize{\@setfontsize{\normalsize}{12}{13.92pt}}
+    \setlength{\@IEEEnormalsizeunitybaselineskip}{13.92pt}
+    \normalsize
+    \abovedisplayskip 1.5ex plus 6pt minus 4pt
+    \belowdisplayskip \abovedisplayskip
+    \abovedisplayshortskip 0pt plus 6pt
+    \belowdisplayshortskip 1.5ex plus 6pt minus 4pt
+    \def\small{\@setfontsize{\small}{10}{12pt}}
+    \def\footnotesize{\@setfontsize{\footnotesize}{9}{10.5pt}}
+    \def\scriptsize{\@setfontsize{\scriptsize}{8}{9pt}}
+    \def\tiny{\@setfontsize{\tiny}{6}{7pt}}
+    % sublargesize is the same as large - 14pt
+    \def\sublargesize{\@setfontsize{\sublargesize}{14}{17pt}}
+    \def\large{\@setfontsize{\large}{14}{17pt}}
+    \def\Large{\@setfontsize{\Large}{17}{20pt}}
+    \def\LARGE{\@setfontsize{\LARGE}{20}{24pt}}
+    \def\huge{\@setfontsize{\huge}{22}{26pt}}
+    \def\Huge{\@setfontsize{\Huge}{24}{28pt}}
 \fi
 
 
@@ -719,188 +720,188 @@
 % compsoc font sizes use bp "Postscript" point units (1/72in) 
 % rather than the traditional pt (1/72.27)
 \ifCLASSOPTIONcompsoc
-% -- compsoc defaults --
-% ** will override some of these values later **
-% 9pt
-\ifx\CLASSOPTIONpt\@IEEEptsizenine
-\def\normalsize{\@setfontsize{\normalsize}{9bp}{11bp}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{11bp}
-\normalsize
-\abovedisplayskip 1.5ex plus 3bp minus 1bp
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0bp plus 3bp
-\belowdisplayshortskip 1.5ex plus 3bp minus 1bp
-\def\small{\@setfontsize{\small}{8.5bp}{10bp}}
-\def\footnotesize{\@setfontsize{\footnotesize}{8bp}{9bp}}
-\def\scriptsize{\@setfontsize{\scriptsize}{7bp}{8bp}}
-\def\tiny{\@setfontsize{\tiny}{5bp}{6bp}}
-% sublargesize is the same as large - 10bp
-\def\sublargesize{\@setfontsize{\sublargesize}{10bp}{12bp}}
-\def\large{\@setfontsize{\large}{10bp}{12bp}}
-\def\Large{\@setfontsize{\Large}{12bp}{14bp}}
-\def\LARGE{\@setfontsize{\LARGE}{14bp}{17bp}}
-\def\huge{\@setfontsize{\huge}{17bp}{20bp}}
-\def\Huge{\@setfontsize{\Huge}{20bp}{24bp}}
-\fi
-%
-% 10pt
-\ifx\CLASSOPTIONpt\@IEEEptsizeten
-\def\normalsize{\@setfontsize{\normalsize}{10bp}{12bp}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{12bp}
-\normalsize
-\abovedisplayskip 1.5ex plus 4bp minus 2bp
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0pt plus 4bp
-\belowdisplayshortskip 1.5ex plus 4bp minus 2bp
-\def\small{\@setfontsize{\small}{9bp}{10bp}}
-\def\footnotesize{\@setfontsize{\footnotesize}{8bp}{9bp}}
-\def\scriptsize{\@setfontsize{\scriptsize}{7bp}{8bp}}
-\def\tiny{\@setfontsize{\tiny}{5bp}{6bp}}
-% sublargesize is a tad smaller than large - 11bp
-\def\sublargesize{\@setfontsize{\sublargesize}{11bp}{13.5bp}}
-\def\large{\@setfontsize{\large}{12bp}{14bp}}
-\def\Large{\@setfontsize{\Large}{14bp}{17bp}}
-\def\LARGE{\@setfontsize{\LARGE}{17bp}{20bp}}
-\def\huge{\@setfontsize{\huge}{20bp}{24bp}}
-\def\Huge{\@setfontsize{\Huge}{24bp}{28bp}}
-\fi
-%
-% 11pt
-\ifx\CLASSOPTIONpt\@IEEEptsizeeleven
-\def\normalsize{\@setfontsize{\normalsize}{11bp}{13.5bp}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{13.5bp}
-\normalsize
-\abovedisplayskip 1.5ex plus 5bp minus 3bp
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0pt plus 5bp
-\belowdisplayshortskip 1.5ex plus 5bp minus 3bp
-\def\small{\@setfontsize{\small}{10bp}{12bp}}
-\def\footnotesize{\@setfontsize{\footnotesize}{9bp}{10.5bp}}
-\def\scriptsize{\@setfontsize{\scriptsize}{8bp}{9bp}}
-\def\tiny{\@setfontsize{\tiny}{6bp}{7bp}}
-% sublargesize is the same as large - 12bp
-\def\sublargesize{\@setfontsize{\sublargesize}{12bp}{14bp}}
-\def\large{\@setfontsize{\large}{12bp}{14bp}}
-\def\Large{\@setfontsize{\Large}{14bp}{17bp}}
-\def\LARGE{\@setfontsize{\LARGE}{17bp}{20bp}}
-\def\huge{\@setfontsize{\huge}{20bp}{24bp}}
-\def\Huge{\@setfontsize{\Huge}{24bp}{28bp}}
-\fi
-%
-% 12pt
-\ifx\CLASSOPTIONpt\@IEEEptsizetwelve
-\def\normalsize{\@setfontsize{\normalsize}{12bp}{14bp}}%
-\setlength{\@IEEEnormalsizeunitybaselineskip}{14bp}%
-\normalsize
-\abovedisplayskip 1.5ex plus 6bp minus 4bp
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0pt plus 6bp
-\belowdisplayshortskip 1.5ex plus 6bp minus 4bp
-\def\small{\@setfontsize{\small}{10bp}{12bp}}
-\def\footnotesize{\@setfontsize{\footnotesize}{9bp}{10.5bp}}
-\def\scriptsize{\@setfontsize{\scriptsize}{8bp}{9bp}}
-\def\tiny{\@setfontsize{\tiny}{6bp}{7bp}}
-% sublargesize is the same as large - 14bp
-\def\sublargesize{\@setfontsize{\sublargesize}{14bp}{17bp}}
-\def\large{\@setfontsize{\large}{14bp}{17bp}}
-\def\Large{\@setfontsize{\Large}{17bp}{20bp}}
-\def\LARGE{\@setfontsize{\LARGE}{20bp}{24bp}}
-\def\huge{\@setfontsize{\huge}{22bp}{26bp}}
-\def\Huge{\@setfontsize{\Huge}{24bp}{28bp}}
-\fi
-%
-% -- override defaults: compsoc journals use special normalsizes --
-\ifCLASSOPTIONconference
-% 
-% compsoc conferences
-% 9pt
-\ifx\CLASSOPTIONpt\@IEEEptsizenine
-\def\normalsize{\@setfontsize{\normalsize}{9bp}{10.8bp}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{10.8bp}
-\normalsize
-\abovedisplayskip 1.5ex plus 3bp minus 1bp
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0bp plus 3bp
-\belowdisplayshortskip 1.5ex plus 3bp minus 1bp
-\fi
-% 10pt
-\ifx\CLASSOPTIONpt\@IEEEptsizeten
-\def\normalsize{\@setfontsize{\normalsize}{10bp}{11.2bp}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{11.2bp}
-\normalsize
-\abovedisplayskip 1.5ex plus 4bp minus 2bp
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0pt plus 4bp
-\belowdisplayshortskip 1.5ex plus 4bp minus 2bp
-\fi
-% 11pt
-\ifx\CLASSOPTIONpt\@IEEEptsizeeleven
-\def\normalsize{\@setfontsize{\normalsize}{11bp}{13.2bp}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{13.2bp}
-\normalsize
-\abovedisplayskip 1.5ex plus 5bp minus 3bp
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0pt plus 5bp
-\belowdisplayshortskip 1.5ex plus 5bp minus 3bp
-\fi
-% 12pt
-\ifx\CLASSOPTIONpt\@IEEEptsizetwelve
-\def\normalsize{\@setfontsize{\normalsize}{12bp}{14.4bp}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{14.4bp}
-\normalsize
-\abovedisplayskip 1.5ex plus 6bp minus 4bp
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0pt plus 6bp
-\belowdisplayshortskip 1.5ex plus 6bp minus 4bp
-\fi
-% 
-% compsoc nonconferences
-\else
-% 9pt
-\ifx\CLASSOPTIONpt\@IEEEptsizenine
-\def\normalsize{\@setfontsize{\normalsize}{9bp}{10.8bp}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{10.8bp}
-\normalsize
-\abovedisplayskip 1.5ex plus 3bp minus 1bp
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0bp plus 3bp
-\belowdisplayshortskip 1.5ex plus 3bp minus 1bp
-\fi
-% 10pt
-\ifx\CLASSOPTIONpt\@IEEEptsizeten
-% the official spec is 9.5bp with 11.4bp leading for 10pt,
-% but measurements of proofs suggest upto 11.723bp leading
-% here we'll use 11.54bp which gives 61 lines per column
-% with the standard compsoc margins
-\def\normalsize{\@setfontsize{\normalsize}{9.5bp}{11.54bp}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{11.54bp}
-\normalsize
-\abovedisplayskip 1.5ex plus 4bp minus 2bp
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0pt plus 4bp
-\belowdisplayshortskip 1.5ex plus 4bp minus 2bp
-\fi
-% 11pt
-\ifx\CLASSOPTIONpt\@IEEEptsizeeleven
-\def\normalsize{\@setfontsize{\normalsize}{11bp}{13.2bp}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{13.2bp}
-\normalsize
-\abovedisplayskip 1.5ex plus 5bp minus 3bp
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0pt plus 5bp
-\belowdisplayshortskip 1.5ex plus 5bp minus 3bp
-\fi
-% 12pt
-\ifx\CLASSOPTIONpt\@IEEEptsizetwelve
-\def\normalsize{\@setfontsize{\normalsize}{12bp}{14.4bp}}
-\setlength{\@IEEEnormalsizeunitybaselineskip}{14.4bp}
-\normalsize
-\abovedisplayskip 1.5ex plus 6bp minus 4bp
-\belowdisplayskip \abovedisplayskip
-\abovedisplayshortskip 0pt plus 6bp
-\belowdisplayshortskip 1.5ex plus 6bp minus 4bp
-\fi
-\fi\fi
+    % -- compsoc defaults --
+    % ** will override some of these values later **
+    % 9pt
+    \ifx\CLASSOPTIONpt\@IEEEptsizenine
+        \def\normalsize{\@setfontsize{\normalsize}{9bp}{11bp}}
+        \setlength{\@IEEEnormalsizeunitybaselineskip}{11bp}
+        \normalsize
+        \abovedisplayskip 1.5ex plus 3bp minus 1bp
+        \belowdisplayskip \abovedisplayskip
+        \abovedisplayshortskip 0bp plus 3bp
+        \belowdisplayshortskip 1.5ex plus 3bp minus 1bp
+        \def\small{\@setfontsize{\small}{8.5bp}{10bp}}
+        \def\footnotesize{\@setfontsize{\footnotesize}{8bp}{9bp}}
+        \def\scriptsize{\@setfontsize{\scriptsize}{7bp}{8bp}}
+        \def\tiny{\@setfontsize{\tiny}{5bp}{6bp}}
+        % sublargesize is the same as large - 10bp
+        \def\sublargesize{\@setfontsize{\sublargesize}{10bp}{12bp}}
+        \def\large{\@setfontsize{\large}{10bp}{12bp}}
+        \def\Large{\@setfontsize{\Large}{12bp}{14bp}}
+        \def\LARGE{\@setfontsize{\LARGE}{14bp}{17bp}}
+        \def\huge{\@setfontsize{\huge}{17bp}{20bp}}
+        \def\Huge{\@setfontsize{\Huge}{20bp}{24bp}}
+    \fi
+    %
+    % 10pt
+    \ifx\CLASSOPTIONpt\@IEEEptsizeten
+        \def\normalsize{\@setfontsize{\normalsize}{10bp}{12bp}}
+        \setlength{\@IEEEnormalsizeunitybaselineskip}{12bp}
+        \normalsize
+        \abovedisplayskip 1.5ex plus 4bp minus 2bp
+        \belowdisplayskip \abovedisplayskip
+        \abovedisplayshortskip 0pt plus 4bp
+        \belowdisplayshortskip 1.5ex plus 4bp minus 2bp
+        \def\small{\@setfontsize{\small}{9bp}{10bp}}
+        \def\footnotesize{\@setfontsize{\footnotesize}{8bp}{9bp}}
+        \def\scriptsize{\@setfontsize{\scriptsize}{7bp}{8bp}}
+        \def\tiny{\@setfontsize{\tiny}{5bp}{6bp}}
+        % sublargesize is a tad smaller than large - 11bp
+        \def\sublargesize{\@setfontsize{\sublargesize}{11bp}{13.5bp}}
+        \def\large{\@setfontsize{\large}{12bp}{14bp}}
+        \def\Large{\@setfontsize{\Large}{14bp}{17bp}}
+        \def\LARGE{\@setfontsize{\LARGE}{17bp}{20bp}}
+        \def\huge{\@setfontsize{\huge}{20bp}{24bp}}
+        \def\Huge{\@setfontsize{\Huge}{24bp}{28bp}}
+    \fi
+    %
+    % 11pt
+    \ifx\CLASSOPTIONpt\@IEEEptsizeeleven
+        \def\normalsize{\@setfontsize{\normalsize}{11bp}{13.5bp}}
+        \setlength{\@IEEEnormalsizeunitybaselineskip}{13.5bp}
+        \normalsize
+        \abovedisplayskip 1.5ex plus 5bp minus 3bp
+        \belowdisplayskip \abovedisplayskip
+        \abovedisplayshortskip 0pt plus 5bp
+        \belowdisplayshortskip 1.5ex plus 5bp minus 3bp
+        \def\small{\@setfontsize{\small}{10bp}{12bp}}
+        \def\footnotesize{\@setfontsize{\footnotesize}{9bp}{10.5bp}}
+        \def\scriptsize{\@setfontsize{\scriptsize}{8bp}{9bp}}
+        \def\tiny{\@setfontsize{\tiny}{6bp}{7bp}}
+        % sublargesize is the same as large - 12bp
+        \def\sublargesize{\@setfontsize{\sublargesize}{12bp}{14bp}}
+        \def\large{\@setfontsize{\large}{12bp}{14bp}}
+        \def\Large{\@setfontsize{\Large}{14bp}{17bp}}
+        \def\LARGE{\@setfontsize{\LARGE}{17bp}{20bp}}
+        \def\huge{\@setfontsize{\huge}{20bp}{24bp}}
+        \def\Huge{\@setfontsize{\Huge}{24bp}{28bp}}
+    \fi
+    %
+    % 12pt
+    \ifx\CLASSOPTIONpt\@IEEEptsizetwelve
+        \def\normalsize{\@setfontsize{\normalsize}{12bp}{14bp}}%
+        \setlength{\@IEEEnormalsizeunitybaselineskip}{14bp}%
+        \normalsize
+        \abovedisplayskip 1.5ex plus 6bp minus 4bp
+        \belowdisplayskip \abovedisplayskip
+        \abovedisplayshortskip 0pt plus 6bp
+        \belowdisplayshortskip 1.5ex plus 6bp minus 4bp
+        \def\small{\@setfontsize{\small}{10bp}{12bp}}
+        \def\footnotesize{\@setfontsize{\footnotesize}{9bp}{10.5bp}}
+        \def\scriptsize{\@setfontsize{\scriptsize}{8bp}{9bp}}
+        \def\tiny{\@setfontsize{\tiny}{6bp}{7bp}}
+        % sublargesize is the same as large - 14bp
+        \def\sublargesize{\@setfontsize{\sublargesize}{14bp}{17bp}}
+        \def\large{\@setfontsize{\large}{14bp}{17bp}}
+        \def\Large{\@setfontsize{\Large}{17bp}{20bp}}
+        \def\LARGE{\@setfontsize{\LARGE}{20bp}{24bp}}
+        \def\huge{\@setfontsize{\huge}{22bp}{26bp}}
+        \def\Huge{\@setfontsize{\Huge}{24bp}{28bp}}
+    \fi
+    %
+    % -- override defaults: compsoc journals use special normalsizes --
+    \ifCLASSOPTIONconference
+        % 
+        % compsoc conferences
+        % 9pt
+        \ifx\CLASSOPTIONpt\@IEEEptsizenine
+            \def\normalsize{\@setfontsize{\normalsize}{9bp}{10.8bp}}
+            \setlength{\@IEEEnormalsizeunitybaselineskip}{10.8bp}
+            \normalsize
+            \abovedisplayskip 1.5ex plus 3bp minus 1bp
+            \belowdisplayskip \abovedisplayskip
+            \abovedisplayshortskip 0bp plus 3bp
+            \belowdisplayshortskip 1.5ex plus 3bp minus 1bp
+        \fi
+        % 10pt
+        \ifx\CLASSOPTIONpt\@IEEEptsizeten
+            \def\normalsize{\@setfontsize{\normalsize}{10bp}{11.2bp}}
+            \setlength{\@IEEEnormalsizeunitybaselineskip}{11.2bp}
+            \normalsize
+            \abovedisplayskip 1.5ex plus 4bp minus 2bp
+            \belowdisplayskip \abovedisplayskip
+            \abovedisplayshortskip 0pt plus 4bp
+            \belowdisplayshortskip 1.5ex plus 4bp minus 2bp
+        \fi
+        % 11pt
+        \ifx\CLASSOPTIONpt\@IEEEptsizeeleven
+            \def\normalsize{\@setfontsize{\normalsize}{11bp}{13.2bp}}
+            \setlength{\@IEEEnormalsizeunitybaselineskip}{13.2bp}
+            \normalsize
+            \abovedisplayskip 1.5ex plus 5bp minus 3bp
+            \belowdisplayskip \abovedisplayskip
+            \abovedisplayshortskip 0pt plus 5bp
+            \belowdisplayshortskip 1.5ex plus 5bp minus 3bp
+        \fi
+        % 12pt
+        \ifx\CLASSOPTIONpt\@IEEEptsizetwelve
+            \def\normalsize{\@setfontsize{\normalsize}{12bp}{14.4bp}}
+            \setlength{\@IEEEnormalsizeunitybaselineskip}{14.4bp}
+            \normalsize
+            \abovedisplayskip 1.5ex plus 6bp minus 4bp
+            \belowdisplayskip \abovedisplayskip
+            \abovedisplayshortskip 0pt plus 6bp
+            \belowdisplayshortskip 1.5ex plus 6bp minus 4bp
+        \fi
+        % 
+        % compsoc nonconferences
+    \else
+        % 9pt
+        \ifx\CLASSOPTIONpt\@IEEEptsizenine
+            \def\normalsize{\@setfontsize{\normalsize}{9bp}{10.8bp}}
+            \setlength{\@IEEEnormalsizeunitybaselineskip}{10.8bp}
+            \normalsize
+            \abovedisplayskip 1.5ex plus 3bp minus 1bp
+            \belowdisplayskip \abovedisplayskip
+            \abovedisplayshortskip 0bp plus 3bp
+            \belowdisplayshortskip 1.5ex plus 3bp minus 1bp
+        \fi
+        % 10pt
+        \ifx\CLASSOPTIONpt\@IEEEptsizeten
+            % the official spec is 9.5bp with 11.4bp leading for 10pt,
+            % but measurements of proofs suggest upto 11.723bp leading
+            % here we'll use 11.54bp which gives 61 lines per column
+            % with the standard compsoc margins
+            \def\normalsize{\@setfontsize{\normalsize}{9.5bp}{11.54bp}}
+            \setlength{\@IEEEnormalsizeunitybaselineskip}{11.54bp}
+            \normalsize
+            \abovedisplayskip 1.5ex plus 4bp minus 2bp
+            \belowdisplayskip \abovedisplayskip
+            \abovedisplayshortskip 0pt plus 4bp
+            \belowdisplayshortskip 1.5ex plus 4bp minus 2bp
+        \fi
+        % 11pt
+        \ifx\CLASSOPTIONpt\@IEEEptsizeeleven
+            \def\normalsize{\@setfontsize{\normalsize}{11bp}{13.2bp}}
+            \setlength{\@IEEEnormalsizeunitybaselineskip}{13.2bp}
+            \normalsize
+            \abovedisplayskip 1.5ex plus 5bp minus 3bp
+            \belowdisplayskip \abovedisplayskip
+            \abovedisplayshortskip 0pt plus 5bp
+            \belowdisplayshortskip 1.5ex plus 5bp minus 3bp
+        \fi
+        % 12pt
+        \ifx\CLASSOPTIONpt\@IEEEptsizetwelve
+            \def\normalsize{\@setfontsize{\normalsize}{12bp}{14.4bp}}
+            \setlength{\@IEEEnormalsizeunitybaselineskip}{14.4bp}
+            \normalsize
+            \abovedisplayskip 1.5ex plus 6bp minus 4bp
+            \belowdisplayskip \abovedisplayskip
+            \abovedisplayshortskip 0pt plus 6bp
+            \belowdisplayshortskip 1.5ex plus 6bp minus 4bp
+        \fi
+    \fi\fi
 
 
 
@@ -919,9 +920,9 @@
 % warn the user in case they forget to use the 9pt option with
 % technote
 \ifCLASSOPTIONtechnote%
- \ifx\CLASSOPTIONpt\@IEEEptsizenine\else%
-  \typeout{** ATTENTION: Technotes are normally 9pt documents.}%
- \fi%
+    \ifx\CLASSOPTIONpt\@IEEEptsizenine\else%
+        \typeout{** ATTENTION: Technotes are normally 9pt documents.}%
+    \fi%
 \fi
 
 
@@ -935,12 +936,12 @@
 \def\@IEEEstringpcr{pcr} % Courier typewriter family
 \def\@IEEEstringcmtt{cmtt} % Computer Modern typewriter family
 \DeclareTextCommandDefault{\textunderscore}{\leavevmode
-\ifx\f@family\@IEEEstringpcr\string_\else
-\ifx\f@family\@IEEEstringcmtt\string_\else
-\ifx\f@family\@IEEEstringptm\kern 0em\vbox{\hrule\@width 0.5em\@height 0.5pt\kern -0.3ex}\else
-\ifx\f@family\@IEEEstringppl\kern 0em\vbox{\hrule\@width 0.5em\@height 0.5pt\kern -0.3ex}\else
-\ifx\f@family\@IEEEstringphv\kern -0.03em\vbox{\hrule\@width 0.62em\@height 0.52pt\kern -0.33ex}\kern -0.03em\else
-\kern 0.09em\vbox{\hrule\@width 0.6em\@height 0.44pt\kern -0.63pt\kern -0.42ex}\kern 0.09em\fi\fi\fi\fi\fi\relax}
+    \ifx\f@family\@IEEEstringpcr\string_\else
+        \ifx\f@family\@IEEEstringcmtt\string_\else
+            \ifx\f@family\@IEEEstringptm\kern 0em\vbox{\hrule\@width 0.5em\@height 0.5pt\kern -0.3ex}\else
+                \ifx\f@family\@IEEEstringppl\kern 0em\vbox{\hrule\@width 0.5em\@height 0.5pt\kern -0.3ex}\else
+                    \ifx\f@family\@IEEEstringphv\kern -0.03em\vbox{\hrule\@width 0.62em\@height 0.52pt\kern -0.33ex}\kern -0.03em\else
+                        \kern 0.09em\vbox{\hrule\@width 0.6em\@height 0.44pt\kern -0.63pt\kern -0.42ex}\kern 0.09em\fi\fi\fi\fi\fi\relax}
 
 
 
@@ -948,16 +949,17 @@
 % set the default \baselinestretch
 \def\baselinestretch{1}
 \ifCLASSOPTIONdraftcls
-  \def\baselinestretch{1.5}% default baselinestretch for draft modes
-\fi 
+    \def\baselinestretch{1.5}% default baselinestretch for draft modes
+\fi
 
 
 % process CLASSINPUT baselinestretch
 \ifx\CLASSINPUTbaselinestretch\@IEEEundefined
 \else
-  \edef\baselinestretch{\CLASSINPUTbaselinestretch} % user CLASSINPUT override
-  \typeout{** ATTENTION: Overriding \string\baselinestretch\space to
-           \baselinestretch\space via \string\CLASSINPUT.}
+    \edef\baselinestretch{\CLASSINPUTbaselinestretch} % user CLASSINPUT override
+    \typeout{** ATTENTION: Overriding \string\baselinestretch\space to
+        \baselinestretch\space via \string\CLASSINPUT.}
+
 \fi
 
 \small\normalsize % make \baselinestretch take affect
@@ -1021,16 +1023,17 @@
 % compsoc nonconference papers use Palatino,
 % tweak settings to better match the proofs
 \ifCLASSOPTIONcompsoc
-\ifCLASSOPTIONconference\else
-% for medium (normal weight)
-\def\@IEEEinterspaceratioM{0.28}
-\def\@IEEEinterspaceMINratioM{0.21}
-\def\@IEEEinterspaceMAXratioM{0.47}
-% for bold
-\def\@IEEEinterspaceratioB{0.305}
-\def\@IEEEinterspaceMINratioB{0.21}
-\def\@IEEEinterspaceMAXratioB{0.52}
-\fi\fi
+    \ifCLASSOPTIONconference\else
+        % for medium (normal weight)
+        \def\@IEEEinterspaceratioM{0.28}
+        \def\@IEEEinterspaceMINratioM{0.21}
+        \def\@IEEEinterspaceMAXratioM{0.47}
+        % for bold
+        \def\@IEEEinterspaceratioB{0.305}
+        \def\@IEEEinterspaceMINratioB{0.21}
+        \def\@IEEEinterspaceMAXratioB{0.52}
+
+    \fi\fi
 
 
 % command to revise the interword spacing for the current font under TeX:
@@ -1040,54 +1043,54 @@
 % since all changes to the \fontdimen are global, we can enclose these commands
 % in braces to confine any font attribute or length changes
 \def\@@@IEEEsetfontdimens#1#2#3{{%
-\setlength{\@IEEEtrantmpdimenB}{\f@size pt}% grab the font size in pt, could use 1em instead.
-\setlength{\@IEEEtrantmpdimenA}{#1\@IEEEtrantmpdimenB}%
-\fontdimen2\font=\@IEEEtrantmpdimenA\relax
-\addtolength{\@IEEEtrantmpdimenA}{-#2\@IEEEtrantmpdimenB}%
-\fontdimen3\font=-\@IEEEtrantmpdimenA\relax
-\setlength{\@IEEEtrantmpdimenA}{#1\@IEEEtrantmpdimenB}%
-\addtolength{\@IEEEtrantmpdimenA}{-#3\@IEEEtrantmpdimenB}%
-\fontdimen4\font=\@IEEEtrantmpdimenA\relax}}
+            \setlength{\@IEEEtrantmpdimenB}{\f@size pt}% grab the font size in pt, could use 1em instead.
+            \setlength{\@IEEEtrantmpdimenA}{#1\@IEEEtrantmpdimenB}%
+            \fontdimen2\font=\@IEEEtrantmpdimenA\relax
+            \addtolength{\@IEEEtrantmpdimenA}{-#2\@IEEEtrantmpdimenB}%
+            \fontdimen3\font=-\@IEEEtrantmpdimenA\relax
+            \setlength{\@IEEEtrantmpdimenA}{#1\@IEEEtrantmpdimenB}%
+            \addtolength{\@IEEEtrantmpdimenA}{-#3\@IEEEtrantmpdimenB}%
+            \fontdimen4\font=\@IEEEtrantmpdimenA\relax}}
 
 % revise the interword spacing for each font weight
 \def\@@IEEEsetfontdimens{{%
-\mdseries
-\@@@IEEEsetfontdimens{\@IEEEinterspaceratioM}{\@IEEEinterspaceMAXratioM}{\@IEEEinterspaceMINratioM}%
-\bfseries
-\@@@IEEEsetfontdimens{\@IEEEinterspaceratioB}{\@IEEEinterspaceMAXratioB}{\@IEEEinterspaceMINratioB}%
-}}
+            \mdseries
+            \@@@IEEEsetfontdimens{\@IEEEinterspaceratioM}{\@IEEEinterspaceMAXratioM}{\@IEEEinterspaceMINratioM}%
+            \bfseries
+            \@@@IEEEsetfontdimens{\@IEEEinterspaceratioB}{\@IEEEinterspaceMAXratioB}{\@IEEEinterspaceMINratioB}%
+        }}
 
 % revise the interword spacing for each font shape
 % \slshape is not often used for IEEE work and is not altered here. The \scshape caps are
 % already a tad too large in the free LaTeX fonts (as compared to what the IEEE uses) so we
 % won't alter these either.
 \def\@IEEEsetfontdimens{{%
-\normalfont
-\@@IEEEsetfontdimens
-\normalfont\itshape
-\@@IEEEsetfontdimens
-}}
+            \normalfont
+            \@@IEEEsetfontdimens
+            \normalfont\itshape
+            \@@IEEEsetfontdimens
+        }}
 
 % command to revise the interword spacing for each font size (and shape
 % and weight). Only the \rmfamily is done here as \ttfamily uses a 
 % fixed spacing and \sffamily is not used as the main text of IEEE papers.
 \def\@IEEEtunefonts{{\selectfont\rmfamily
-\tiny\@IEEEsetfontdimens
-\scriptsize\@IEEEsetfontdimens
-\footnotesize\@IEEEsetfontdimens
-\small\@IEEEsetfontdimens
-\normalsize\@IEEEsetfontdimens
-\sublargesize\@IEEEsetfontdimens
-\large\@IEEEsetfontdimens
-\LARGE\@IEEEsetfontdimens
-\huge\@IEEEsetfontdimens
-\Huge\@IEEEsetfontdimens}}
+            \tiny\@IEEEsetfontdimens
+            \scriptsize\@IEEEsetfontdimens
+            \footnotesize\@IEEEsetfontdimens
+            \small\@IEEEsetfontdimens
+            \normalsize\@IEEEsetfontdimens
+            \sublargesize\@IEEEsetfontdimens
+            \large\@IEEEsetfontdimens
+            \LARGE\@IEEEsetfontdimens
+            \huge\@IEEEsetfontdimens
+            \Huge\@IEEEsetfontdimens}}
 
 % if the nofonttune class option is not given, revise the interword spacing
 % now - in case IEEEtran makes any default length measurements, and make
 % sure all the default fonts are loaded
 \ifCLASSOPTIONnofonttune\else
-\@IEEEtunefonts
+    \@IEEEtunefonts
 \fi
 
 % and again at the start of the document in case the user loaded different fonts
@@ -1111,11 +1114,11 @@
 % Sets \textwidth to allow the specified inner and outer margins
 % for the current \paperwidth.
 \def\IEEEsettextwidth#1#2{\@IEEEtrantmpdimenA\paperwidth
-\@IEEEtrantmpdimenB#1\relax
-\advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB
-\@IEEEtrantmpdimenB#2\relax
-\advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB
-\textwidth\@IEEEtrantmpdimenA}
+    \@IEEEtrantmpdimenB#1\relax
+    \advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB
+    \@IEEEtrantmpdimenB#2\relax
+    \advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB
+    \textwidth\@IEEEtrantmpdimenA}
 
 
 
@@ -1131,55 +1134,55 @@
 % \textwidth should be set properly for the given margins before calling this
 % function.
 \def\IEEEsetsidemargin#1#2{\@IEEEtrantmpdimenA #2\relax
-\@IEEEextracttoken{#1}\relax
-% check for mode errors
-\ifx\@IEEEextractedtokenmacro\@empty
-  \@IEEEclspkgerror{Empty mode type in \string\IEEEsetsidemargin\space (line \the\inputlineno).\MessageBreak
-  Defaulting to `i'}{Valid modes for \string\IEEEsetsidemargin\space are: i, o, c and a.}\relax
-  \let\@IEEEextractedtoken=i\relax
-  \def\@IEEEextractedtokenmacro{i}\relax
-\else
-  \ifx\@IEEEextractedtokensdiscarded\@empty\else
-    \typeout{** WARNING: \string\IEEEsetsidemargin\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
-  \fi
-\fi
-% handle each mode
-\if\@IEEEextractedtoken a\relax
-  \advance\oddsidemargin by \@IEEEtrantmpdimenA\relax
-\else
-\if\@IEEEextractedtoken c\relax
-  \oddsidemargin\paperwidth
-  \advance\oddsidemargin by -\textwidth
-  \divide\oddsidemargin by 2\relax
-  \advance\oddsidemargin by -1in\relax
-  \advance\oddsidemargin by \@IEEEtrantmpdimenA\relax
-\else
-\if\@IEEEextractedtoken o\relax
-  \oddsidemargin\paperwidth
-  \advance\oddsidemargin by -\textwidth
-  \advance\oddsidemargin by -\@IEEEtrantmpdimenA
-  \advance\oddsidemargin by -1in\relax
-\else
-  \if\@IEEEextractedtoken i\relax
-  \else
-    \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEsetsidemargin\space (line \the\inputlineno).\MessageBreak
-    Defaulting to `i'}%
-    {Valid modes for \string\IEEEsetsidemargin\space are: i, o, c  and a.}%
-  \fi
-  \oddsidemargin\@IEEEtrantmpdimenA
-  \advance\oddsidemargin by -1in\relax
-\fi\fi\fi
-% odd and even side margins both mean "inner" for single sided pages
-\evensidemargin\oddsidemargin
-% but are mirrors of each other when twosided is in effect
-\if@twoside
-  \evensidemargin\paperwidth
-  \advance\evensidemargin by -\textwidth
-  \advance\evensidemargin by -\oddsidemargin
-  % have to compensate for both the builtin 1in LaTex offset
-  % and the fact we already subtracted this offset from \oddsidemargin
-  \advance\evensidemargin -2in\relax
-\fi}
+    \@IEEEextracttoken{#1}\relax
+    % check for mode errors
+    \ifx\@IEEEextractedtokenmacro\@empty
+        \@IEEEclspkgerror{Empty mode type in \string\IEEEsetsidemargin\space (line \the\inputlineno).\MessageBreak
+            Defaulting to `i'}{Valid modes for \string\IEEEsetsidemargin\space are: i, o, c and a.}\relax
+        \let\@IEEEextractedtoken=i\relax
+        \def\@IEEEextractedtokenmacro{i}\relax
+    \else
+        \ifx\@IEEEextractedtokensdiscarded\@empty\else
+            \typeout{** WARNING: \string\IEEEsetsidemargin\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+        \fi
+    \fi
+    % handle each mode
+    \if\@IEEEextractedtoken a\relax
+        \advance\oddsidemargin by \@IEEEtrantmpdimenA\relax
+    \else
+        \if\@IEEEextractedtoken c\relax
+            \oddsidemargin\paperwidth
+            \advance\oddsidemargin by -\textwidth
+            \divide\oddsidemargin by 2\relax
+            \advance\oddsidemargin by -1in\relax
+            \advance\oddsidemargin by \@IEEEtrantmpdimenA\relax
+        \else
+            \if\@IEEEextractedtoken o\relax
+                \oddsidemargin\paperwidth
+                \advance\oddsidemargin by -\textwidth
+                \advance\oddsidemargin by -\@IEEEtrantmpdimenA
+                \advance\oddsidemargin by -1in\relax
+            \else
+                \if\@IEEEextractedtoken i\relax
+                \else
+                    \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEsetsidemargin\space (line \the\inputlineno).\MessageBreak
+                        Defaulting to `i'}%
+                    {Valid modes for \string\IEEEsetsidemargin\space are: i, o, c  and a.}%
+                \fi
+                \oddsidemargin\@IEEEtrantmpdimenA
+                \advance\oddsidemargin by -1in\relax
+            \fi\fi\fi
+    % odd and even side margins both mean "inner" for single sided pages
+    \evensidemargin\oddsidemargin
+    % but are mirrors of each other when twosided is in effect
+    \if@twoside
+        \evensidemargin\paperwidth
+        \advance\evensidemargin by -\textwidth
+        \advance\evensidemargin by -\oddsidemargin
+        % have to compensate for both the builtin 1in LaTex offset
+        % and the fact we already subtracted this offset from \oddsidemargin
+        \advance\evensidemargin -2in\relax
+    \fi}
 
 
 
@@ -1189,16 +1192,16 @@
 % the actual height and depth of the \IEEEdefaultsampletext text. 
 \def\IEEEsettextheight{\@ifnextchar [{\@IEEEsettextheight}{\@IEEEsettextheight[\IEEEdefaultsampletext]}}
 \def\@IEEEsettextheight[#1]#2#3{\textheight\paperheight\relax
-  \@IEEEtrantmpdimenA #2\relax
-  \advance \textheight by -\@IEEEtrantmpdimenA% subtract top margin
-  \@IEEEtrantmpdimenA #3\relax
-  \advance \textheight by -\@IEEEtrantmpdimenA% subtract bottom margin
-  \advance \textheight by \topskip% add \topskip
-  % subtract off everything above the top, and below the bottom, baselines
-  \settoheight{\@IEEEtrantmpdimenA}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance \textheight by -\@IEEEtrantmpdimenA
-  \settodepth{\@IEEEtrantmpdimenA}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance \textheight by -\@IEEEtrantmpdimenA}
+    \@IEEEtrantmpdimenA #2\relax
+    \advance \textheight by -\@IEEEtrantmpdimenA% subtract top margin
+    \@IEEEtrantmpdimenA #3\relax
+    \advance \textheight by -\@IEEEtrantmpdimenA% subtract bottom margin
+    \advance \textheight by \topskip% add \topskip
+    % subtract off everything above the top, and below the bottom, baselines
+    \settoheight{\@IEEEtrantmpdimenA}{\begingroup #1\relax\relax\relax\endgroup}\relax
+    \advance \textheight by -\@IEEEtrantmpdimenA
+    \settodepth{\@IEEEtrantmpdimenA}{\begingroup #1\relax\relax\relax\endgroup}\relax
+    \advance \textheight by -\@IEEEtrantmpdimenA}
 
 
 
@@ -1224,112 +1227,112 @@
 % In anycase, if the given length is already quantized,
 % \IEEEquantizedlengthdiff will be set to zero.
 \def\IEEEquantizelength#1#2#3{\begingroup
-% work in isolation so as not to externally disturb the \@IEEEtrantmp
-% variables
-% load the argument values indirectly via \IEEEquantizedlengthdiff
-% in case the user refers to our \@IEEEtrantmpdimenX, \IEEEquantizedlength,
-% etc. in the arguments. we also will work with these as counters,
-% i.e., in sp units
-% A has the base unit
-\IEEEquantizedlengthdiff #2\relax\relax\relax\relax
-\@IEEEtrantmpcountA\IEEEquantizedlengthdiff
-% B has the input length
-\IEEEquantizedlengthdiff #3\relax\relax\relax\relax
-\@IEEEtrantmpcountB\IEEEquantizedlengthdiff
-\@IEEEtrantmpdimenA\the\@IEEEtrantmpcountA sp\relax
-\@IEEEtrantmpdimenB\the\@IEEEtrantmpcountB sp\relax
-% \@IEEEtrantmpcountC will have the quantized int
-% \IEEEquantizedlength will have the quantized length
-% \@IEEEtrantmpdimenC will have the quantized diff
-% initialize them to zero as this is what will be
-% exported if an error occurs
-\@IEEEtrantmpcountC 0\relax
-\IEEEquantizedlength 0sp\relax
-\@IEEEtrantmpdimenC 0sp\relax
-% extract mode
-\@IEEEextracttoken{#1}\relax
-% check for mode errors
-\ifx\@IEEEextractedtokenmacro\@empty
-  \@IEEEclspkgerror{Empty mode type in \string\IEEEquantizelength\space (line \the\inputlineno).\MessageBreak
-  Defaulting to `d'}{Valid modes for \string\IEEEquantizelength\space are: d, c and i.}\relax
-  \let\@IEEEextractedtoken=d\relax
-  \def\@IEEEextractedtokenmacro{d}\relax
-\else
-  \ifx\@IEEEextractedtokensdiscarded\@empty\else
-    \typeout{** WARNING: \string\IEEEquantizelength\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
-  \fi
-\fi
-% check for base unit is zero error
-\ifnum\@IEEEtrantmpcountA=0\relax
-\@IEEEclspkgerror{Base unit is zero in \string\IEEEquantizelength\space (line \the\inputlineno).\MessageBreak
-  \string\IEEEquantizedlength\space and \string\IEEEquantizedlengthdiff\space are set to zero}{Division by zero is not allowed.}\relax
-\else% base unit is nonzero
-  % \@IEEEtrantmpcountC carries the number of integer units
-  % in the quantized length (integer length \ base)
-  \@IEEEtrantmpcountC\@IEEEtrantmpcountB\relax
-  \divide\@IEEEtrantmpcountC by \@IEEEtrantmpcountA\relax
-  % \IEEEquantizedlength has the (rounded down) quantized length
-  % = base * int
-  \IEEEquantizedlength\@IEEEtrantmpdimenA\relax
-  \multiply\IEEEquantizedlength by \@IEEEtrantmpcountC\relax
-  % \@IEEEtrantmpdimenC has the difference
-  % = quantized length - length
-  \@IEEEtrantmpdimenC\IEEEquantizedlength\relax
-  \advance\@IEEEtrantmpdimenC by -\@IEEEtrantmpdimenB\relax
-  % trap special case of length being already quantized
-  % to avoid a roundup under i option
-  \ifdim\@IEEEtrantmpdimenC=0sp\relax
-  \else % length not is already quantized
-    % set dimenA to carry the upper quantized (absolute value) difference:
-    % quantizedlength + base - length
-    \advance\@IEEEtrantmpdimenA by \IEEEquantizedlength\relax
-    \advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB\relax
-    % set dimenB to carry the lower quantized (absolute value) difference:
-    % length - quantizedlength
-    \advance\@IEEEtrantmpdimenB by -\IEEEquantizedlength\relax
-    % handle each mode
-    \if\@IEEEextractedtoken c\relax
-      % compare upper and lower amounts, select upper if lower > upper
-      \ifdim\@IEEEtrantmpdimenB>\@IEEEtrantmpdimenA\relax
-        % use upper
-        \advance\IEEEquantizedlength by \the\@IEEEtrantmpcountA sp\relax
-        \advance\@IEEEtrantmpcountC by 1\relax
-        \@IEEEtrantmpdimenC\@IEEEtrantmpdimenA
-      \else% <=. uselower
-        % no need to do anything for lower, use output values already setup
-      \fi
-    \else% not mode c
-    \if\@IEEEextractedtoken i\relax
-      % always round up under i mode
-      \advance\IEEEquantizedlength by \the\@IEEEtrantmpcountA sp\relax
-      \advance\@IEEEtrantmpcountC by 1\relax
-      \@IEEEtrantmpdimenC\@IEEEtrantmpdimenA
+    % work in isolation so as not to externally disturb the \@IEEEtrantmp
+    % variables
+    % load the argument values indirectly via \IEEEquantizedlengthdiff
+    % in case the user refers to our \@IEEEtrantmpdimenX, \IEEEquantizedlength,
+    % etc. in the arguments. we also will work with these as counters,
+    % i.e., in sp units
+    % A has the base unit
+    \IEEEquantizedlengthdiff #2\relax\relax\relax\relax
+    \@IEEEtrantmpcountA\IEEEquantizedlengthdiff
+    % B has the input length
+    \IEEEquantizedlengthdiff #3\relax\relax\relax\relax
+    \@IEEEtrantmpcountB\IEEEquantizedlengthdiff
+    \@IEEEtrantmpdimenA\the\@IEEEtrantmpcountA sp\relax
+    \@IEEEtrantmpdimenB\the\@IEEEtrantmpcountB sp\relax
+    % \@IEEEtrantmpcountC will have the quantized int
+    % \IEEEquantizedlength will have the quantized length
+    % \@IEEEtrantmpdimenC will have the quantized diff
+    % initialize them to zero as this is what will be
+    % exported if an error occurs
+    \@IEEEtrantmpcountC 0\relax
+    \IEEEquantizedlength 0sp\relax
+    \@IEEEtrantmpdimenC 0sp\relax
+    % extract mode
+    \@IEEEextracttoken{#1}\relax
+    % check for mode errors
+    \ifx\@IEEEextractedtokenmacro\@empty
+        \@IEEEclspkgerror{Empty mode type in \string\IEEEquantizelength\space (line \the\inputlineno).\MessageBreak
+            Defaulting to `d'}{Valid modes for \string\IEEEquantizelength\space are: d, c and i.}\relax
+        \let\@IEEEextractedtoken=d\relax
+        \def\@IEEEextractedtokenmacro{d}\relax
     \else
-      \if\@IEEEextractedtoken d\relax
-      \else
-        \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEquantizelength\space (line \the\inputlineno).\MessageBreak
-        Defaulting to `d'}%
-        {Valid modes for \string\IEEEquantizelength\space are: d, c, and i.}\relax
-      \fi % if d
-      % no need to do anything for d, use output values already setup
-    \fi\fi % if i, c
-  \fi % if length is already quantized
-\fi% if base unit is zero
-% globally assign the results to macros we use here to escape the enclosing
-% group without needing to call \global on any of the \@IEEEtrantmp variables.
-% \@IEEEtrantmpcountC has the quantized int
-% \IEEEquantizedlength has the quantized length
-% \@IEEEtrantmpdimenC has the quantized diff
-\xdef\@IEEEquantizedlengthintmacro{\the\@IEEEtrantmpcountC}\relax
-\@IEEEtrantmpcountC\IEEEquantizedlength\relax
-\xdef\@IEEEquantizedlengthmacro{\the\@IEEEtrantmpcountC}\relax
-\@IEEEtrantmpcountC\@IEEEtrantmpdimenC\relax
-\xdef\@IEEEquantizedlengthdiffmacro{\the\@IEEEtrantmpcountC}\relax
-\endgroup
-% locally assign the outputs here from the macros
-\expandafter\IEEEquantizedlength\@IEEEquantizedlengthmacro sp\relax
-\expandafter\IEEEquantizedlengthdiff\@IEEEquantizedlengthdiffmacro sp\relax
-\edef\IEEEquantizedlengthint{\@IEEEquantizedlengthintmacro}\relax}
+        \ifx\@IEEEextractedtokensdiscarded\@empty\else
+            \typeout{** WARNING: \string\IEEEquantizelength\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+        \fi
+    \fi
+    % check for base unit is zero error
+    \ifnum\@IEEEtrantmpcountA=0\relax
+        \@IEEEclspkgerror{Base unit is zero in \string\IEEEquantizelength\space (line \the\inputlineno).\MessageBreak
+            \string\IEEEquantizedlength\space and \string\IEEEquantizedlengthdiff\space are set to zero}{Division by zero is not allowed.}\relax
+    \else% base unit is nonzero
+        % \@IEEEtrantmpcountC carries the number of integer units
+        % in the quantized length (integer length \ base)
+        \@IEEEtrantmpcountC\@IEEEtrantmpcountB\relax
+        \divide\@IEEEtrantmpcountC by \@IEEEtrantmpcountA\relax
+        % \IEEEquantizedlength has the (rounded down) quantized length
+        % = base * int
+        \IEEEquantizedlength\@IEEEtrantmpdimenA\relax
+        \multiply\IEEEquantizedlength by \@IEEEtrantmpcountC\relax
+        % \@IEEEtrantmpdimenC has the difference
+        % = quantized length - length
+        \@IEEEtrantmpdimenC\IEEEquantizedlength\relax
+        \advance\@IEEEtrantmpdimenC by -\@IEEEtrantmpdimenB\relax
+        % trap special case of length being already quantized
+        % to avoid a roundup under i option
+        \ifdim\@IEEEtrantmpdimenC=0sp\relax
+        \else % length not is already quantized
+            % set dimenA to carry the upper quantized (absolute value) difference:
+            % quantizedlength + base - length
+            \advance\@IEEEtrantmpdimenA by \IEEEquantizedlength\relax
+            \advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB\relax
+            % set dimenB to carry the lower quantized (absolute value) difference:
+            % length - quantizedlength
+            \advance\@IEEEtrantmpdimenB by -\IEEEquantizedlength\relax
+            % handle each mode
+            \if\@IEEEextractedtoken c\relax
+                % compare upper and lower amounts, select upper if lower > upper
+                \ifdim\@IEEEtrantmpdimenB>\@IEEEtrantmpdimenA\relax
+                    % use upper
+                    \advance\IEEEquantizedlength by \the\@IEEEtrantmpcountA sp\relax
+                    \advance\@IEEEtrantmpcountC by 1\relax
+                    \@IEEEtrantmpdimenC\@IEEEtrantmpdimenA
+                \else% <=. uselower
+                    % no need to do anything for lower, use output values already setup
+                \fi
+            \else% not mode c
+                \if\@IEEEextractedtoken i\relax
+                    % always round up under i mode
+                    \advance\IEEEquantizedlength by \the\@IEEEtrantmpcountA sp\relax
+                    \advance\@IEEEtrantmpcountC by 1\relax
+                    \@IEEEtrantmpdimenC\@IEEEtrantmpdimenA
+                \else
+                    \if\@IEEEextractedtoken d\relax
+                    \else
+                        \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEquantizelength\space (line \the\inputlineno).\MessageBreak
+                            Defaulting to `d'}%
+                        {Valid modes for \string\IEEEquantizelength\space are: d, c, and i.}\relax
+                    \fi % if d
+                    % no need to do anything for d, use output values already setup
+                \fi\fi % if i, c
+        \fi % if length is already quantized
+    \fi% if base unit is zero
+    % globally assign the results to macros we use here to escape the enclosing
+    % group without needing to call \global on any of the \@IEEEtrantmp variables.
+    % \@IEEEtrantmpcountC has the quantized int
+    % \IEEEquantizedlength has the quantized length
+    % \@IEEEtrantmpdimenC has the quantized diff
+    \xdef\@IEEEquantizedlengthintmacro{\the\@IEEEtrantmpcountC}\relax
+    \@IEEEtrantmpcountC\IEEEquantizedlength\relax
+    \xdef\@IEEEquantizedlengthmacro{\the\@IEEEtrantmpcountC}\relax
+    \@IEEEtrantmpcountC\@IEEEtrantmpdimenC\relax
+    \xdef\@IEEEquantizedlengthdiffmacro{\the\@IEEEtrantmpcountC}\relax
+    \endgroup
+    % locally assign the outputs here from the macros
+    \expandafter\IEEEquantizedlength\@IEEEquantizedlengthmacro sp\relax
+    \expandafter\IEEEquantizedlengthdiff\@IEEEquantizedlengthdiffmacro sp\relax
+    \edef\IEEEquantizedlengthint{\@IEEEquantizedlengthintmacro}\relax}
 
 
 
@@ -1353,27 +1356,27 @@
 % Depends on: \IEEEquantizelength
 \def\IEEEquantizetextheight{\@ifnextchar [{\@IEEEquantizetextheight}{\@IEEEquantizetextheight[\baselineskip]}}
 \def\@IEEEquantizetextheight[#1]#2{\begingroup
-% use our \IEEEquantizedtextheightdiff as a scratch pad
-% we need to subtract off \topskip before quantization
-\IEEEquantizedtextheightdiff\textheight
-\advance\IEEEquantizedtextheightdiff by -\topskip\relax
-\IEEEquantizelength{#2}{#1}{\IEEEquantizedtextheightdiff}
-% add back \topskip line
-\advance\IEEEquantizedlength by \topskip
-\@IEEEtrantmpcountC\IEEEquantizedlengthint\relax
-\advance\@IEEEtrantmpcountC by 1\relax
-% globally assign the results to macros we use here to escape the enclosing
-% group without needing to call \global on any of the \@IEEEtrantmp variables.
-\xdef\@IEEEquantizedtextheightlpcmacro{\the\@IEEEtrantmpcountC}\relax
-\@IEEEtrantmpcountC\IEEEquantizedlength\relax
-\xdef\@IEEEquantizedtextheightmacro{\the\@IEEEtrantmpcountC}\relax
-\@IEEEtrantmpcountC\IEEEquantizedlengthdiff\relax
-\xdef\@IEEEquantizedtextheightdiffmacro{\the\@IEEEtrantmpcountC}\relax
-\endgroup
-% locally assign the outputs here from the macros
-\textheight\@IEEEquantizedtextheightmacro sp\relax
-\IEEEquantizedtextheightdiff\@IEEEquantizedtextheightdiffmacro sp\relax
-\edef\IEEEquantizedtextheightlpc{\@IEEEquantizedtextheightlpcmacro}}
+    % use our \IEEEquantizedtextheightdiff as a scratch pad
+    % we need to subtract off \topskip before quantization
+    \IEEEquantizedtextheightdiff\textheight
+    \advance\IEEEquantizedtextheightdiff by -\topskip\relax
+    \IEEEquantizelength{#2}{#1}{\IEEEquantizedtextheightdiff}
+    % add back \topskip line
+    \advance\IEEEquantizedlength by \topskip
+    \@IEEEtrantmpcountC\IEEEquantizedlengthint\relax
+    \advance\@IEEEtrantmpcountC by 1\relax
+    % globally assign the results to macros we use here to escape the enclosing
+    % group without needing to call \global on any of the \@IEEEtrantmp variables.
+    \xdef\@IEEEquantizedtextheightlpcmacro{\the\@IEEEtrantmpcountC}\relax
+    \@IEEEtrantmpcountC\IEEEquantizedlength\relax
+    \xdef\@IEEEquantizedtextheightmacro{\the\@IEEEtrantmpcountC}\relax
+    \@IEEEtrantmpcountC\IEEEquantizedlengthdiff\relax
+    \xdef\@IEEEquantizedtextheightdiffmacro{\the\@IEEEtrantmpcountC}\relax
+    \endgroup
+    % locally assign the outputs here from the macros
+    \textheight\@IEEEquantizedtextheightmacro sp\relax
+    \IEEEquantizedtextheightdiff\@IEEEquantizedtextheightdiffmacro sp\relax
+    \edef\IEEEquantizedtextheightlpc{\@IEEEquantizedtextheightlpcmacro}}
 
 
 
@@ -1393,69 +1396,69 @@
 % given margins before calling this function.
 \def\IEEEsettopmargin{\@ifnextchar [{\@IEEEsettopmargin}{\@IEEEsettopmargin[\IEEEdefaultsampletext]}}
 \def\@IEEEsettopmargin[#1]#2#3{\@IEEEtrantmpdimenA #3\relax
-\@IEEEextracttoken{#2}\relax
-% check for mode errors
-\ifx\@IEEEextractedtokenmacro\@empty
-  \@IEEEclspkgerror{Empty mode type in \string\IEEEsettopmargin\space (line \the\inputlineno).\MessageBreak
-  Defaulting to `t'}{Valid modes for \string\IEEEsettopmargin\space are: t, b, c, a and q.}\relax
-  \let\@IEEEextractedtoken=t\relax
-  \def\@IEEEextractedtokenmacro{t}\relax
-\else
-  \ifx\@IEEEextractedtokensdiscarded\@empty\else
-    \typeout{** WARNING: \string\IEEEsettopmargin\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
-  \fi
-\fi
-% handle each mode
-\if\@IEEEextractedtoken a\relax
-  \advance\topmargin by \@IEEEtrantmpdimenA\relax
-\else
-\if\@IEEEextractedtoken q\relax
-  % we need to adjust by half the \IEEEquantizedtextheightdiff value
-  \@IEEEtrantmpdimenB\IEEEquantizedtextheightdiff\relax
-  \divide\@IEEEtrantmpdimenB by 2\relax
-  % a positive \IEEEquantizedtextheightdiff means we need to reduce \topmargin
-  % because \textheight has been lenghtened
-  \advance\topmargin by -\@IEEEtrantmpdimenB\relax
-  \advance\topmargin by \@IEEEtrantmpdimenA\relax
-\else
-\if\@IEEEextractedtoken c\relax
-  \topmargin\paperheight
-  \advance\topmargin by -\textheight
-  % \textheight includes \topskip, but we should not count topskip whitespace here, backout
-  \advance \topmargin by \topskip
-  \settoheight{\@IEEEtrantmpdimenB}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance\topmargin by -\@IEEEtrantmpdimenB\relax
-  \settodepth{\@IEEEtrantmpdimenB}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance\topmargin by -\@IEEEtrantmpdimenB\relax
-  \divide\topmargin by 2\relax
-  \advance\topmargin by \@IEEEtrantmpdimenA\relax
-\else
-\if\@IEEEextractedtoken b\relax
-  \topmargin\paperheight
-  \advance\topmargin by -\textheight
-  % \textheight includes \topskip, but we should not count topskip whitespace here, backout
-  \advance \topmargin by \topskip
-  \settodepth{\@IEEEtrantmpdimenB}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance\topmargin by -\@IEEEtrantmpdimenB\relax
-  \advance\topmargin by -\@IEEEtrantmpdimenA\relax
-\else
-  \if\@IEEEextractedtoken t\relax
-  \else
-    \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEsettopmargin\space (line \the\inputlineno).\MessageBreak
-    Defaulting to `t'}%
-    {Valid modes for \string\IEEEsettopmargin\space are: t, b, c, a and q.}\relax
-  \fi
-  \topmargin\@IEEEtrantmpdimenA\relax
-  \settoheight{\@IEEEtrantmpdimenB}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance\topmargin by \@IEEEtrantmpdimenB\relax
-\fi\fi % if t, b, c
-% convert desired top margin into actual \topmargin
-% this is not done for the q or a modes because they are only adjustments
-\advance \topmargin by -\topskip
-\advance \topmargin by -1in
-\advance \topmargin by -\headheight
-\advance \topmargin by -\headsep
-\fi\fi % if q, a
+    \@IEEEextracttoken{#2}\relax
+    % check for mode errors
+    \ifx\@IEEEextractedtokenmacro\@empty
+        \@IEEEclspkgerror{Empty mode type in \string\IEEEsettopmargin\space (line \the\inputlineno).\MessageBreak
+            Defaulting to `t'}{Valid modes for \string\IEEEsettopmargin\space are: t, b, c, a and q.}\relax
+        \let\@IEEEextractedtoken=t\relax
+        \def\@IEEEextractedtokenmacro{t}\relax
+    \else
+        \ifx\@IEEEextractedtokensdiscarded\@empty\else
+            \typeout{** WARNING: \string\IEEEsettopmargin\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+        \fi
+    \fi
+    % handle each mode
+    \if\@IEEEextractedtoken a\relax
+        \advance\topmargin by \@IEEEtrantmpdimenA\relax
+    \else
+        \if\@IEEEextractedtoken q\relax
+            % we need to adjust by half the \IEEEquantizedtextheightdiff value
+            \@IEEEtrantmpdimenB\IEEEquantizedtextheightdiff\relax
+            \divide\@IEEEtrantmpdimenB by 2\relax
+            % a positive \IEEEquantizedtextheightdiff means we need to reduce \topmargin
+            % because \textheight has been lenghtened
+            \advance\topmargin by -\@IEEEtrantmpdimenB\relax
+            \advance\topmargin by \@IEEEtrantmpdimenA\relax
+        \else
+            \if\@IEEEextractedtoken c\relax
+                \topmargin\paperheight
+                \advance\topmargin by -\textheight
+                % \textheight includes \topskip, but we should not count topskip whitespace here, backout
+                \advance \topmargin by \topskip
+                \settoheight{\@IEEEtrantmpdimenB}{\begingroup #1\relax\relax\relax\endgroup}\relax
+                \advance\topmargin by -\@IEEEtrantmpdimenB\relax
+                \settodepth{\@IEEEtrantmpdimenB}{\begingroup #1\relax\relax\relax\endgroup}\relax
+                \advance\topmargin by -\@IEEEtrantmpdimenB\relax
+                \divide\topmargin by 2\relax
+                \advance\topmargin by \@IEEEtrantmpdimenA\relax
+            \else
+                \if\@IEEEextractedtoken b\relax
+                    \topmargin\paperheight
+                    \advance\topmargin by -\textheight
+                    % \textheight includes \topskip, but we should not count topskip whitespace here, backout
+                    \advance \topmargin by \topskip
+                    \settodepth{\@IEEEtrantmpdimenB}{\begingroup #1\relax\relax\relax\endgroup}\relax
+                    \advance\topmargin by -\@IEEEtrantmpdimenB\relax
+                    \advance\topmargin by -\@IEEEtrantmpdimenA\relax
+                \else
+                    \if\@IEEEextractedtoken t\relax
+                    \else
+                        \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEsettopmargin\space (line \the\inputlineno).\MessageBreak
+                            Defaulting to `t'}%
+                        {Valid modes for \string\IEEEsettopmargin\space are: t, b, c, a and q.}\relax
+                    \fi
+                    \topmargin\@IEEEtrantmpdimenA\relax
+                    \settoheight{\@IEEEtrantmpdimenB}{\begingroup #1\relax\relax\relax\endgroup}\relax
+                    \advance\topmargin by \@IEEEtrantmpdimenB\relax
+                \fi\fi % if t, b, c
+            % convert desired top margin into actual \topmargin
+            % this is not done for the q or a modes because they are only adjustments
+            \advance \topmargin by -\topskip
+            \advance \topmargin by -1in
+            \advance \topmargin by -\headheight
+            \advance \topmargin by -\headsep
+        \fi\fi % if q, a
 }
 
 
@@ -1478,77 +1481,77 @@
 \def\IEEEsetheadermargin{\@ifnextchar [{\@IEEEsetheadermargin}{\@IEEEsetheadermargin[\IEEEdefaultheadersampletext]}}
 \def\@IEEEsetheadermargin[#1]{\@ifnextchar [{\@@IEEEsetheadermargin[#1]}{\@@IEEEsetheadermargin[#1][\IEEEdefaultsampletext]}}
 \def\@@IEEEsetheadermargin[#1][#2]#3#4{\@IEEEtrantmpdimenA #4\relax
-\@IEEEextracttoken{#3}\relax
-% check for mode errors
-\ifx\@IEEEextractedtokenmacro\@empty
-  \@IEEEclspkgerror{Empty mode type in \string\IEEEsetheadermargin\space (line \the\inputlineno).\MessageBreak
-  Defaulting to `t'}{Valid modes for \string\IEEEsetheadermargin\space are: t, b, c, and a.}\relax
-  \let\@IEEEextractedtoken=t\relax
-  \def\@IEEEextractedtokenmacro{t}\relax
-\else
-  \ifx\@IEEEextractedtokensdiscarded\@empty\else
-    \typeout{** WARNING: \string\IEEEsetheadermargin\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
-  \fi
-\fi
-% handle each mode
-\if\@IEEEextractedtoken a\relax
-  % No need to do anything here and can pass through the adjustment
-  % value as is. The end adjustment of \topmargin and \headsep will
-  % do all that is needed
-\else
-\if\@IEEEextractedtoken c\relax
-  % get the bottom margin
-  \@IEEEtrantmpdimenB\headsep\relax
-  \settodepth{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
-  \advance\@IEEEtrantmpdimenB by \topskip
-  \settoheight{\@IEEEtrantmpdimenC}{\begingroup #2\relax\relax\relax\endgroup}\relax
-  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
-  % at this point \@IEEEtrantmpdimenB has the actual header bottom margin
-  % subtract from it the top header margin
-  \advance\@IEEEtrantmpdimenB -1in\relax % take into consideration the system 1in offset of the top margin
-  \advance\@IEEEtrantmpdimenB by -\topmargin
-  \advance\@IEEEtrantmpdimenB by -\headheight
-  \settoheight{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance\@IEEEtrantmpdimenB by \@IEEEtrantmpdimenC
-  % at this point \@IEEEtrantmpdimenB has the difference between the bottom and top margins
-  % we need to adjust by half this amount to center the header
-  \divide\@IEEEtrantmpdimenB by 2\relax
-  % and add to offset
-  \advance\@IEEEtrantmpdimenA by \@IEEEtrantmpdimenB
-\else
-\if\@IEEEextractedtoken b\relax
-  \@IEEEtrantmpdimenB\headsep\relax
-  \settodepth{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
-  \advance\@IEEEtrantmpdimenB by \topskip
-  \settoheight{\@IEEEtrantmpdimenC}{\begingroup #2\relax\relax\relax\endgroup}\relax
-  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
-  % at this point \@IEEEtrantmpdimenB has the actual header bottom margin
-  % get the difference between the actual and the desired
-  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenA
-  \@IEEEtrantmpdimenA\@IEEEtrantmpdimenB
-\else
-  \if\@IEEEextractedtoken t\relax
-  \else
-    \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEsetheadermargin\space (line \the\inputlineno).\MessageBreak
-    Defaulting to `t'}%
-    {Valid modes for \string\IEEEsetheadermargin\space are: t, b, c and a.}\relax
-  \fi
-  \@IEEEtrantmpdimenB 1in\relax % take into consideration the system 1in offset of the top margin
-  \advance\@IEEEtrantmpdimenB by \topmargin
-  \advance\@IEEEtrantmpdimenB by \headheight
-  \settoheight{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
-  % at this point \@IEEEtrantmpdimenB has the actual header top margin
-  % get the difference between the desired and the actual
-  \advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB
-\fi\fi % if t, b, c
-\fi % if a
-% advance \topmargin by the needed amount and reduce \headsep by the same
-% so as not to disturb the location of the main text
-\advance\topmargin by \@IEEEtrantmpdimenA\relax
-\advance\headsep by -\@IEEEtrantmpdimenA\relax
+    \@IEEEextracttoken{#3}\relax
+    % check for mode errors
+    \ifx\@IEEEextractedtokenmacro\@empty
+        \@IEEEclspkgerror{Empty mode type in \string\IEEEsetheadermargin\space (line \the\inputlineno).\MessageBreak
+            Defaulting to `t'}{Valid modes for \string\IEEEsetheadermargin\space are: t, b, c, and a.}\relax
+        \let\@IEEEextractedtoken=t\relax
+        \def\@IEEEextractedtokenmacro{t}\relax
+    \else
+        \ifx\@IEEEextractedtokensdiscarded\@empty\else
+            \typeout{** WARNING: \string\IEEEsetheadermargin\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+        \fi
+    \fi
+    % handle each mode
+    \if\@IEEEextractedtoken a\relax
+        % No need to do anything here and can pass through the adjustment
+        % value as is. The end adjustment of \topmargin and \headsep will
+        % do all that is needed
+    \else
+        \if\@IEEEextractedtoken c\relax
+            % get the bottom margin
+            \@IEEEtrantmpdimenB\headsep\relax
+            \settodepth{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+            \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+            \advance\@IEEEtrantmpdimenB by \topskip
+            \settoheight{\@IEEEtrantmpdimenC}{\begingroup #2\relax\relax\relax\endgroup}\relax
+            \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+            % at this point \@IEEEtrantmpdimenB has the actual header bottom margin
+            % subtract from it the top header margin
+            \advance\@IEEEtrantmpdimenB -1in\relax % take into consideration the system 1in offset of the top margin
+            \advance\@IEEEtrantmpdimenB by -\topmargin
+            \advance\@IEEEtrantmpdimenB by -\headheight
+            \settoheight{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+            \advance\@IEEEtrantmpdimenB by \@IEEEtrantmpdimenC
+            % at this point \@IEEEtrantmpdimenB has the difference between the bottom and top margins
+            % we need to adjust by half this amount to center the header
+            \divide\@IEEEtrantmpdimenB by 2\relax
+            % and add to offset
+            \advance\@IEEEtrantmpdimenA by \@IEEEtrantmpdimenB
+        \else
+            \if\@IEEEextractedtoken b\relax
+                \@IEEEtrantmpdimenB\headsep\relax
+                \settodepth{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+                \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+                \advance\@IEEEtrantmpdimenB by \topskip
+                \settoheight{\@IEEEtrantmpdimenC}{\begingroup #2\relax\relax\relax\endgroup}\relax
+                \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+                % at this point \@IEEEtrantmpdimenB has the actual header bottom margin
+                % get the difference between the actual and the desired
+                \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenA
+                \@IEEEtrantmpdimenA\@IEEEtrantmpdimenB
+            \else
+                \if\@IEEEextractedtoken t\relax
+                \else
+                    \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEsetheadermargin\space (line \the\inputlineno).\MessageBreak
+                        Defaulting to `t'}%
+                    {Valid modes for \string\IEEEsetheadermargin\space are: t, b, c and a.}\relax
+                \fi
+                \@IEEEtrantmpdimenB 1in\relax % take into consideration the system 1in offset of the top margin
+                \advance\@IEEEtrantmpdimenB by \topmargin
+                \advance\@IEEEtrantmpdimenB by \headheight
+                \settoheight{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+                \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+                % at this point \@IEEEtrantmpdimenB has the actual header top margin
+                % get the difference between the desired and the actual
+                \advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB
+            \fi\fi % if t, b, c
+    \fi % if a
+    % advance \topmargin by the needed amount and reduce \headsep by the same
+    % so as not to disturb the location of the main text
+    \advance\topmargin by \@IEEEtrantmpdimenA\relax
+    \advance\headsep by -\@IEEEtrantmpdimenA\relax
 }
 
 
@@ -1570,85 +1573,85 @@
 \def\IEEEsetfootermargin{\@ifnextchar [{\@IEEEsetfootermargin}{\@IEEEsetfootermargin[\IEEEdefaultfootersampletext]}}
 \def\@IEEEsetfootermargin[#1]{\@ifnextchar [{\@@IEEEsetfootermargin[#1]}{\@@IEEEsetfootermargin[#1][\IEEEdefaultsampletext]}}
 \def\@@IEEEsetfootermargin[#1][#2]#3#4{\@IEEEtrantmpdimenA #4\relax
-\@IEEEextracttoken{#3}\relax
-% check for mode errors
-\ifx\@IEEEextractedtokenmacro\@empty
-  \@IEEEclspkgerror{Empty mode type in \string\IEEEsetfootermargin\space (line \the\inputlineno).\MessageBreak
-  Defaulting to `t'}{Valid modes for \string\IEEEsetfootermargin\space are: t, b, c, and a.}\relax
-  \let\@IEEEextractedtoken=t\relax
-  \def\@IEEEextractedtokenmacro{t}\relax
-\else
-  \ifx\@IEEEextractedtokensdiscarded\@empty\else
-    \typeout{** WARNING: \string\IEEEsetfootermargin\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
-  \fi
-\fi
-% handle each mode
-\if\@IEEEextractedtoken a\relax
-  % No need to do anything here and can pass through the adjustment
-  % value as is. The end adjustment of \footskip will do all that
-  % is needed
-\else
-\if\@IEEEextractedtoken c\relax
-  % calculate the bottom margin
-  \@IEEEtrantmpdimenB 1in\relax % system 1in offset
-  \advance\@IEEEtrantmpdimenB\topmargin\relax
-  \advance\@IEEEtrantmpdimenB\headheight\relax
-  \advance\@IEEEtrantmpdimenB\headsep\relax
-  \advance\@IEEEtrantmpdimenB\textheight\relax
-  \advance\@IEEEtrantmpdimenB\footskip\relax
-  \settodepth{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance\@IEEEtrantmpdimenC by \@IEEEtrantmpdimenB
-  \@IEEEtrantmpdimenB\paperheight
-  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
-  % at this point \@IEEEtrantmpdimenB has the actual footer bottom margin
-  % now subtract off the footer top margin
-  \advance\@IEEEtrantmpdimenB -\footskip\relax
-  \settodepth{\@IEEEtrantmpdimenC}{\begingroup #2\relax\relax\relax\endgroup}\relax
-  \advance\@IEEEtrantmpdimenB by \@IEEEtrantmpdimenC
-  \settoheight{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance\@IEEEtrantmpdimenB by \@IEEEtrantmpdimenC
-  % at this point \@IEEEtrantmpdimenB has the difference between the bottom
-  % and top footer margins
-  % our adjustment must be half this value to center the footer
-  \divide\@IEEEtrantmpdimenB by 2\relax
-  % add to the offset
-  \advance\@IEEEtrantmpdimenA by \@IEEEtrantmpdimenB
-\else
-\if\@IEEEextractedtoken b\relax
-  % calculate the bottom margin
-  \@IEEEtrantmpdimenB 1in\relax % system 1in offset
-  \advance\@IEEEtrantmpdimenB\topmargin\relax
-  \advance\@IEEEtrantmpdimenB\headheight\relax
-  \advance\@IEEEtrantmpdimenB\headsep\relax
-  \advance\@IEEEtrantmpdimenB\textheight\relax
-  \advance\@IEEEtrantmpdimenB\footskip\relax
-  \settodepth{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance\@IEEEtrantmpdimenC by \@IEEEtrantmpdimenB
-  \@IEEEtrantmpdimenB\paperheight
-  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
-  % at this point \@IEEEtrantmpdimenB has the actual footer bottom margin
-  % get the difference between the actual and the desired
-  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenA
-  \@IEEEtrantmpdimenA\@IEEEtrantmpdimenB
-\else
-  \if\@IEEEextractedtoken t\relax
-  \else
-    \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEsetfootermargin\space (line \the\inputlineno).\MessageBreak
-    Defaulting to `t'}%
-    {Valid modes for \string\IEEEsetfootermargin\space are: t, b, c and a.}\relax
-  \fi
-  \@IEEEtrantmpdimenB\footskip\relax
-  \settodepth{\@IEEEtrantmpdimenC}{\begingroup #2\relax\relax\relax\endgroup}\relax
-  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
-  \settoheight{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
-  \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
-  % at this point \@IEEEtrantmpdimenB has the actual footer top margin
-  % get the difference between the desired and the actual
-  \advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB
-\fi\fi % if t, b, c
-\fi % if a
-% advance \footskip by the needed amount
-\advance\footskip by \@IEEEtrantmpdimenA\relax
+    \@IEEEextracttoken{#3}\relax
+    % check for mode errors
+    \ifx\@IEEEextractedtokenmacro\@empty
+        \@IEEEclspkgerror{Empty mode type in \string\IEEEsetfootermargin\space (line \the\inputlineno).\MessageBreak
+            Defaulting to `t'}{Valid modes for \string\IEEEsetfootermargin\space are: t, b, c, and a.}\relax
+        \let\@IEEEextractedtoken=t\relax
+        \def\@IEEEextractedtokenmacro{t}\relax
+    \else
+        \ifx\@IEEEextractedtokensdiscarded\@empty\else
+            \typeout{** WARNING: \string\IEEEsetfootermargin\space mode specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+        \fi
+    \fi
+    % handle each mode
+    \if\@IEEEextractedtoken a\relax
+        % No need to do anything here and can pass through the adjustment
+        % value as is. The end adjustment of \footskip will do all that
+        % is needed
+    \else
+        \if\@IEEEextractedtoken c\relax
+            % calculate the bottom margin
+            \@IEEEtrantmpdimenB 1in\relax % system 1in offset
+            \advance\@IEEEtrantmpdimenB\topmargin\relax
+            \advance\@IEEEtrantmpdimenB\headheight\relax
+            \advance\@IEEEtrantmpdimenB\headsep\relax
+            \advance\@IEEEtrantmpdimenB\textheight\relax
+            \advance\@IEEEtrantmpdimenB\footskip\relax
+            \settodepth{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+            \advance\@IEEEtrantmpdimenC by \@IEEEtrantmpdimenB
+            \@IEEEtrantmpdimenB\paperheight
+            \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+            % at this point \@IEEEtrantmpdimenB has the actual footer bottom margin
+            % now subtract off the footer top margin
+            \advance\@IEEEtrantmpdimenB -\footskip\relax
+            \settodepth{\@IEEEtrantmpdimenC}{\begingroup #2\relax\relax\relax\endgroup}\relax
+            \advance\@IEEEtrantmpdimenB by \@IEEEtrantmpdimenC
+            \settoheight{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+            \advance\@IEEEtrantmpdimenB by \@IEEEtrantmpdimenC
+            % at this point \@IEEEtrantmpdimenB has the difference between the bottom
+            % and top footer margins
+            % our adjustment must be half this value to center the footer
+            \divide\@IEEEtrantmpdimenB by 2\relax
+            % add to the offset
+            \advance\@IEEEtrantmpdimenA by \@IEEEtrantmpdimenB
+        \else
+            \if\@IEEEextractedtoken b\relax
+                % calculate the bottom margin
+                \@IEEEtrantmpdimenB 1in\relax % system 1in offset
+                \advance\@IEEEtrantmpdimenB\topmargin\relax
+                \advance\@IEEEtrantmpdimenB\headheight\relax
+                \advance\@IEEEtrantmpdimenB\headsep\relax
+                \advance\@IEEEtrantmpdimenB\textheight\relax
+                \advance\@IEEEtrantmpdimenB\footskip\relax
+                \settodepth{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+                \advance\@IEEEtrantmpdimenC by \@IEEEtrantmpdimenB
+                \@IEEEtrantmpdimenB\paperheight
+                \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+                % at this point \@IEEEtrantmpdimenB has the actual footer bottom margin
+                % get the difference between the actual and the desired
+                \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenA
+                \@IEEEtrantmpdimenA\@IEEEtrantmpdimenB
+            \else
+                \if\@IEEEextractedtoken t\relax
+                \else
+                    \@IEEEclspkgerror{Unknown mode type `\@IEEEextractedtokenmacro' in \string\IEEEsetfootermargin\space (line \the\inputlineno).\MessageBreak
+                        Defaulting to `t'}%
+                    {Valid modes for \string\IEEEsetfootermargin\space are: t, b, c and a.}\relax
+                \fi
+                \@IEEEtrantmpdimenB\footskip\relax
+                \settodepth{\@IEEEtrantmpdimenC}{\begingroup #2\relax\relax\relax\endgroup}\relax
+                \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+                \settoheight{\@IEEEtrantmpdimenC}{\begingroup #1\relax\relax\relax\endgroup}\relax
+                \advance\@IEEEtrantmpdimenB by -\@IEEEtrantmpdimenC
+                % at this point \@IEEEtrantmpdimenB has the actual footer top margin
+                % get the difference between the desired and the actual
+                \advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpdimenB
+            \fi\fi % if t, b, c
+    \fi % if a
+    % advance \footskip by the needed amount
+    \advance\footskip by \@IEEEtrantmpdimenA\relax
 }
 
 % -- End V1.8a page setup commands --
@@ -1667,7 +1670,7 @@
 % TeX default is 50
 \hyphenpenalty=750
 \ifCLASSOPTIONcompsoc
-\hyphenpenalty 500
+    \hyphenpenalty 500
 \fi
 % If we didn't adjust the interword spacing, 2200 might be better.
 % The TeX default is 1000
@@ -1707,7 +1710,7 @@
 
 \parindent    1.0em
 \ifCLASSOPTIONcompsoc
-  \parindent    1.5em
+    \parindent    1.5em
 \fi
 
 \headheight   12pt
@@ -1740,62 +1743,62 @@
 
 % adjust margins for default conference mode
 \ifCLASSOPTIONconference
- \textheight        9.25in % The standard for conferences (668.4975pt)
- \IEEEsettopmargin{t}{0.75in}
- % tweak textheight to a perfect integer number of lines/page.
- % standard is: 9pt/61 lpc; 10pt/56 lpc; 11pt/50 lpc; 12pt/48 lpc
- \IEEEquantizetextheight{c}
- % tweak top margin so that the error is shared equally at the top and bottom
- \IEEEsettopmargin{q}{0sp}
+    \textheight        9.25in % The standard for conferences (668.4975pt)
+    \IEEEsettopmargin{t}{0.75in}
+    % tweak textheight to a perfect integer number of lines/page.
+    % standard is: 9pt/61 lpc; 10pt/56 lpc; 11pt/50 lpc; 12pt/48 lpc
+    \IEEEquantizetextheight{c}
+    % tweak top margin so that the error is shared equally at the top and bottom
+    \IEEEsettopmargin{q}{0sp}
 \fi
 
 
 % compsoc text sizes, margins and spacings
 \ifCLASSOPTIONcompsoc
- \columnsep         12bp
- % CS specs for \textwdith are 6.875in
- % \textwidth         6.875in
- % however, measurements from proofs show they are using 3.5in columns
- \textwidth         7in
- \advance\textwidth by \columnsep
- % set the side margins to center the text
- \IEEEsetsidemargin{c}{0pt}
- % top/bottom margins to center
- % could just set \textheight to 9.75in for all the different paper sizes
- % and then quantize, but we'll do it the long way here to allow for easy
- % future per-paper size adjustments
- \IEEEsettextheight{0.625in}{0.625in}% 11in - 2 * 0.625in = 9.75in is the standard text height for compsoc journals
- \IEEEsettopmargin{t}{0.625in}
- \if@IEEEusingcspaper
-   \IEEEsettextheight{0.5in}{0.5in}% 10.75in - 2 * 0.5in = 9.75in
-   \IEEEsettopmargin{t}{0.5in}
- \fi
- \if@IEEEusingAfourpaper
-   \IEEEsettextheight{24.675mm}{24.675mm}% 297mm - 2 * 24.675mm = 247.650mm (9.75in)
-   \IEEEsettopmargin{t}{24.675mm}
- \fi
- % tweak textheight to a perfect integer number of lines/page.
- % standard is: 9pt/65 lpc; 10pt/61 lpc; 11pt/53 lpc; 12pt/49 lpc
- \IEEEquantizetextheight{c}
- % tweak top margin so that the error is shared equally at the top and bottom
- \IEEEsettopmargin{q}{0sp}
-
-% compsoc conference
- \ifCLASSOPTIONconference
-  % compsoc conference use a larger value for columnsep
-  \columnsep 0.25in
-  \IEEEsettextwidth{0.75in}{0.75in}
-  % set the side margins to center the text (0.75in for letterpaper)
-  \IEEEsetsidemargin{c}{0pt}
-  % compsoc conferences want 1in top and bottom margin
-  \IEEEsettextheight{1in}{1in}
-  \IEEEsettopmargin{t}{1in}
-  % tweak textheight to a perfect integer number of lines/page.
-  % standard is: 9pt/58 lpc; 10pt/53 lpc; 11pt/48 lpc; 12pt/46 lpc
-  \IEEEquantizetextheight{c}
-  % tweak top margin so that the error is shared equally at the top and bottom
-  \IEEEsettopmargin{q}{0sp}
- \fi
+    \columnsep         12bp
+    % CS specs for \textwdith are 6.875in
+    % \textwidth         6.875in
+    % however, measurements from proofs show they are using 3.5in columns
+    \textwidth         7in
+    \advance\textwidth by \columnsep
+    % set the side margins to center the text
+    \IEEEsetsidemargin{c}{0pt}
+    % top/bottom margins to center
+    % could just set \textheight to 9.75in for all the different paper sizes
+    % and then quantize, but we'll do it the long way here to allow for easy
+    % future per-paper size adjustments
+    \IEEEsettextheight{0.625in}{0.625in}% 11in - 2 * 0.625in = 9.75in is the standard text height for compsoc journals
+    \IEEEsettopmargin{t}{0.625in}
+    \if@IEEEusingcspaper
+        \IEEEsettextheight{0.5in}{0.5in}% 10.75in - 2 * 0.5in = 9.75in
+        \IEEEsettopmargin{t}{0.5in}
+    \fi
+    \if@IEEEusingAfourpaper
+        \IEEEsettextheight{24.675mm}{24.675mm}% 297mm - 2 * 24.675mm = 247.650mm (9.75in)
+        \IEEEsettopmargin{t}{24.675mm}
+    \fi
+    % tweak textheight to a perfect integer number of lines/page.
+    % standard is: 9pt/65 lpc; 10pt/61 lpc; 11pt/53 lpc; 12pt/49 lpc
+    \IEEEquantizetextheight{c}
+    % tweak top margin so that the error is shared equally at the top and bottom
+    \IEEEsettopmargin{q}{0sp}
+    
+    % compsoc conference
+    \ifCLASSOPTIONconference
+        % compsoc conference use a larger value for columnsep
+        \columnsep 0.25in
+        \IEEEsettextwidth{0.75in}{0.75in}
+        % set the side margins to center the text (0.75in for letterpaper)
+        \IEEEsetsidemargin{c}{0pt}
+        % compsoc conferences want 1in top and bottom margin
+        \IEEEsettextheight{1in}{1in}
+        \IEEEsettopmargin{t}{1in}
+        % tweak textheight to a perfect integer number of lines/page.
+        % standard is: 9pt/58 lpc; 10pt/53 lpc; 11pt/48 lpc; 12pt/46 lpc
+        \IEEEquantizetextheight{c}
+        % tweak top margin so that the error is shared equally at the top and bottom
+        \IEEEsettopmargin{q}{0sp}
+    \fi
 \fi
 
 
@@ -1804,17 +1807,17 @@
 % provides a nice 1in margin all around the paper and extra
 % space between the lines for editor's comments
 \ifCLASSOPTIONdraftcls
-  % we want 1in side margins regardless of paper type
-  \IEEEsettextwidth{1in}{1in}
-  \IEEEsetsidemargin{c}{0pt}
-  % want 1in top and bottom margins
-  \IEEEsettextheight{1in}{1in}
-  \IEEEsettopmargin{t}{1in}
-  % digitize textheight to be an integer number of lines.
-  % this may cause the top and bottom margins to be off a tad
-  \IEEEquantizetextheight{c}
-  % tweak top margin so that the error is shared equally at the top and bottom
-  \IEEEsettopmargin{q}{0sp}
+    % we want 1in side margins regardless of paper type
+    \IEEEsettextwidth{1in}{1in}
+    \IEEEsetsidemargin{c}{0pt}
+    % want 1in top and bottom margins
+    \IEEEsettextheight{1in}{1in}
+    \IEEEsettopmargin{t}{1in}
+    % digitize textheight to be an integer number of lines.
+    % this may cause the top and bottom margins to be off a tad
+    \IEEEquantizetextheight{c}
+    % tweak top margin so that the error is shared equally at the top and bottom
+    \IEEEsettopmargin{q}{0sp}
 \fi
 
 
@@ -1823,21 +1826,22 @@
 % if inner margin defined, but outer margin not, set outer to inner.
 \ifx\CLASSINPUTinnersidemargin\@IEEEundefined
 \else
-  \ifx\CLASSINPUToutersidemargin\@IEEEundefined
-    \edef\CLASSINPUToutersidemargin{\CLASSINPUTinnersidemargin}
-  \fi
+    \ifx\CLASSINPUToutersidemargin\@IEEEundefined
+        \edef\CLASSINPUToutersidemargin{\CLASSINPUTinnersidemargin}
+    \fi
 \fi
 
 \ifx\CLASSINPUToutersidemargin\@IEEEundefined
 \else
-  % if outer margin defined, but inner margin not, set inner to outer.
-  \ifx\CLASSINPUTinnersidemargin\@IEEEundefined
-    \edef\CLASSINPUTinnersidemargin{\CLASSINPUToutersidemargin}
-  \fi
-  \IEEEsettextwidth{\CLASSINPUTinnersidemargin}{\CLASSINPUToutersidemargin}
-  \IEEEsetsidemargin{i}{\CLASSINPUTinnersidemargin}
-  \typeout{** ATTENTION: Overriding inner side margin to \CLASSINPUTinnersidemargin\space and 
-           outer side margin to \CLASSINPUToutersidemargin\space via \string\CLASSINPUT.}
+    % if outer margin defined, but inner margin not, set inner to outer.
+    \ifx\CLASSINPUTinnersidemargin\@IEEEundefined
+        \edef\CLASSINPUTinnersidemargin{\CLASSINPUToutersidemargin}
+    \fi
+    \IEEEsettextwidth{\CLASSINPUTinnersidemargin}{\CLASSINPUToutersidemargin}
+    \IEEEsetsidemargin{i}{\CLASSINPUTinnersidemargin}
+    \typeout{** ATTENTION: Overriding inner side margin to \CLASSINPUTinnersidemargin\space and
+        outer side margin to \CLASSINPUToutersidemargin\space via \string\CLASSINPUT.}
+
 \fi
 
 
@@ -1846,21 +1850,22 @@
 % if toptext margin defined, but bottomtext margin not, set bottomtext to toptext margin
 \ifx\CLASSINPUTtoptextmargin\@IEEEundefined
 \else
-  \ifx\CLASSINPUTbottomtextmargin\@IEEEundefined
-    \edef\CLASSINPUTbottomtextmargin{\CLASSINPUTtoptextmargin}
-  \fi
+    \ifx\CLASSINPUTbottomtextmargin\@IEEEundefined
+        \edef\CLASSINPUTbottomtextmargin{\CLASSINPUTtoptextmargin}
+    \fi
 \fi
 
 \ifx\CLASSINPUTbottomtextmargin\@IEEEundefined
 \else
-  % if bottomtext margin defined, but toptext margin not, set toptext to bottomtext margin
-  \ifx\CLASSINPUTtoptextmargin\@IEEEundefined
-    \edef\CLASSINPUTtoptextmargin{\CLASSINPUTbottomtextmargin}
-  \fi
-  \IEEEsettextheight{\CLASSINPUTtoptextmargin}{\CLASSINPUTbottomtextmargin}
-  \IEEEsettopmargin{t}{\CLASSINPUTtoptextmargin}
-  \typeout{** ATTENTION: Overriding top text margin to \CLASSINPUTtoptextmargin\space and 
-           bottom text margin to \CLASSINPUTbottomtextmargin\space via \string\CLASSINPUT.}
+    % if bottomtext margin defined, but toptext margin not, set toptext to bottomtext margin
+    \ifx\CLASSINPUTtoptextmargin\@IEEEundefined
+        \edef\CLASSINPUTtoptextmargin{\CLASSINPUTbottomtextmargin}
+    \fi
+    \IEEEsettextheight{\CLASSINPUTtoptextmargin}{\CLASSINPUTbottomtextmargin}
+    \IEEEsettopmargin{t}{\CLASSINPUTtoptextmargin}
+    \typeout{** ATTENTION: Overriding top text margin to \CLASSINPUTtoptextmargin\space and
+        bottom text margin to \CLASSINPUTbottomtextmargin\space via \string\CLASSINPUT.}
+
 \fi
 
 
@@ -1871,37 +1876,38 @@
 
 % adjust header and footer positions for compsoc journals
 \ifCLASSOPTIONcompsoc
-  \ifCLASSOPTIONjournal
-    \IEEEsetheadermargin{b}{\@IEEEnormalsizeunitybaselineskip}
-    \IEEEsetfootermargin{t}{\@IEEEnormalsizeunitybaselineskip}
-  \fi
+    \ifCLASSOPTIONjournal
+        \IEEEsetheadermargin{b}{\@IEEEnormalsizeunitybaselineskip}
+        \IEEEsetfootermargin{t}{\@IEEEnormalsizeunitybaselineskip}
+    \fi
 \fi
 
 
 % V1.8a display lines per column info message on user's console
 \def\IEEEdisplayinfolinespercolumn{\@IEEEtrantmpdimenA=\textheight
-% topskip represents only one line even if > baselineskip
-\advance\@IEEEtrantmpdimenA by -1\topskip
-\@IEEEtrantmpcountA=\@IEEEtrantmpdimenA
-\@IEEEtrantmpcountB=\@IEEEtrantmpdimenA
-\divide\@IEEEtrantmpcountB by \baselineskip
-% need to add one line to include topskip (first) line
-\advance\@IEEEtrantmpcountB by 1
-% save lines per column value as text
-\edef\@IEEEnumlinespercolumninfotxt{\the\@IEEEtrantmpcountB}
-% backout topskip advance to allow direct \@IEEEtrantmpcountA comparison
-\advance\@IEEEtrantmpcountB by -1
-% restore value as text height (without topskip) rather than just as number of lines
-\multiply\@IEEEtrantmpcountB by \baselineskip
-% is the column height an integer number of lines per column?
-\ifnum\@IEEEtrantmpcountA=\@IEEEtrantmpcountB
-\edef\@IEEEnumlinespercolumnexactinfotxt{exact}
-\else
-\@IEEEtrantmpdimenA\@IEEEtrantmpcountA sp\relax
-\advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpcountB sp\relax
-\edef\@IEEEnumlinespercolumnexactinfotxt{approximate, difference = \the\@IEEEtrantmpdimenA}
-\fi
-\typeout{-- Lines per column: \@IEEEnumlinespercolumninfotxt\space (\@IEEEnumlinespercolumnexactinfotxt).}}
+    % topskip represents only one line even if > baselineskip
+    \advance\@IEEEtrantmpdimenA by -1\topskip
+    \@IEEEtrantmpcountA=\@IEEEtrantmpdimenA
+    \@IEEEtrantmpcountB=\@IEEEtrantmpdimenA
+    \divide\@IEEEtrantmpcountB by \baselineskip
+    % need to add one line to include topskip (first) line
+    \advance\@IEEEtrantmpcountB by 1
+    % save lines per column value as text
+    \edef\@IEEEnumlinespercolumninfotxt{\the\@IEEEtrantmpcountB}
+    % backout topskip advance to allow direct \@IEEEtrantmpcountA comparison
+    \advance\@IEEEtrantmpcountB by -1
+    % restore value as text height (without topskip) rather than just as number of lines
+    \multiply\@IEEEtrantmpcountB by \baselineskip
+    % is the column height an integer number of lines per column?
+    \ifnum\@IEEEtrantmpcountA=\@IEEEtrantmpcountB
+        \edef\@IEEEnumlinespercolumnexactinfotxt{exact}
+    \else
+        \@IEEEtrantmpdimenA\@IEEEtrantmpcountA sp\relax
+        \advance\@IEEEtrantmpdimenA by -\@IEEEtrantmpcountB sp\relax
+        \edef\@IEEEnumlinespercolumnexactinfotxt{approximate, difference = \the\@IEEEtrantmpdimenA}
+
+    \fi
+    \typeout{-- Lines per column: \@IEEEnumlinespercolumninfotxt\space (\@IEEEnumlinespercolumnexactinfotxt).}}
 % delay execution till start of document to allow for user changes
 \AtBeginDocument{\IEEEdisplayinfolinespercolumn}
 
@@ -2125,8 +2131,8 @@
 % \leftmargin = argument + \labelwidth + \labelsep
 % Note controlled spacing here, shield end of lines with %
 \def\IEEEcalcleftmargin#1{\setlength{\leftmargin}{#1}%
-\addtolength{\leftmargin}{\labelwidth}%
-\addtolength{\leftmargin}{\labelsep}}
+    \addtolength{\leftmargin}{\labelwidth}%
+    \addtolength{\leftmargin}{\labelsep}}
 
 % This command provides an easy way to set \labelwidth to the
 % width of the given text. It is the same as
@@ -2190,27 +2196,27 @@
 % V1.6 use \edefs instead of dimen's to conserve dimen registers
 % Note controlled spacing here, shield end of lines with %
 \def\@IEEEsavelistparams{\edef\@IEEEiedtopsep{\the\topsep}%
-\edef\@IEEEiedlabelwidth{\the\labelwidth}%
-\edef\@IEEEiedlabelsep{\the\labelsep}%
-\edef\@IEEEiedleftmargin{\the\leftmargin}%
-\edef\@IEEEiedpartopsep{\the\partopsep}%
-\edef\@IEEEiedparsep{\the\parsep}%
-\edef\@IEEEieditemsep{\the\itemsep}%
-\edef\@IEEEiedrightmargin{\the\rightmargin}%
-\edef\@IEEEiedlistparindent{\the\listparindent}%
-\edef\@IEEEieditemindent{\the\itemindent}}
+    \edef\@IEEEiedlabelwidth{\the\labelwidth}%
+    \edef\@IEEEiedlabelsep{\the\labelsep}%
+    \edef\@IEEEiedleftmargin{\the\leftmargin}%
+    \edef\@IEEEiedpartopsep{\the\partopsep}%
+    \edef\@IEEEiedparsep{\the\parsep}%
+    \edef\@IEEEieditemsep{\the\itemsep}%
+    \edef\@IEEEiedrightmargin{\the\rightmargin}%
+    \edef\@IEEEiedlistparindent{\the\listparindent}%
+    \edef\@IEEEieditemindent{\the\itemindent}}
 
 % Note controlled spacing here
 \def\@IEEErestorelistparams{\topsep\@IEEEiedtopsep\relax%
-\labelwidth\@IEEEiedlabelwidth\relax%
-\labelsep\@IEEEiedlabelsep\relax%
-\leftmargin\@IEEEiedleftmargin\relax%
-\partopsep\@IEEEiedpartopsep\relax%
-\parsep\@IEEEiedparsep\relax%
-\itemsep\@IEEEieditemsep\relax%
-\rightmargin\@IEEEiedrightmargin\relax%
-\listparindent\@IEEEiedlistparindent\relax%
-\itemindent\@IEEEieditemindent\relax}
+    \labelwidth\@IEEEiedlabelwidth\relax%
+    \labelsep\@IEEEiedlabelsep\relax%
+    \leftmargin\@IEEEiedleftmargin\relax%
+    \partopsep\@IEEEiedpartopsep\relax%
+    \parsep\@IEEEiedparsep\relax%
+    \itemsep\@IEEEieditemsep\relax%
+    \rightmargin\@IEEEiedrightmargin\relax%
+    \listparindent\@IEEEiedlistparindent\relax%
+    \itemindent\@IEEEieditemindent\relax}
 
 
 % v1.6b provide original LaTeX IED list environments
@@ -2224,11 +2230,11 @@
 
 % provide original LaTeX description environment from article.cls
 \newenvironment{LaTeXdescription}
-               {\list{}{\labelwidth\z@ \itemindent-\leftmargin
-                        \let\makelabel\descriptionlabel}}
-               {\endlist}
+{\list{}{\labelwidth\z@ \itemindent-\leftmargin
+        \let\makelabel\descriptionlabel}}
+{\endlist}
 \newcommand*\descriptionlabel[1]{\hspace\labelsep
-                                 \normalfont\bfseries #1}
+    \normalfont\bfseries #1}
 
 
 % override LaTeX's default IED lists
@@ -2264,35 +2270,35 @@
 % IEEEtran itemized list MDS 1/2001
 % Note controlled spacing here, shield end of lines with %
 \def\@@IEEEitemize[#1]{%
-                \ifnum\@itemdepth>3\relax\@toodeep\else%
-                \ifnum\@listdepth>5\relax\@toodeep\else%
-                \advance\@itemdepth\@ne%
-                \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
-                % get the IEEElabelindentfactor for this level
-                \advance\@listdepth\@ne% we need to know what the level WILL be
-                \edef\IEEElabelindentfactor{\csname IEEElabelindentfactor\romannumeral\the\@listdepth\endcsname}%
-                \advance\@listdepth-\@ne% undo our increment
-                \def\@IEEEiedjustify{2}% right justified labels are default
-                % set other defaults
-                \IEEEnocalcleftmarginfalse%
-                \IEEEnolabelindentfactorfalse%
-                \topsep\IEEEiedtopsep%
-                \IEEElabelindent\IEEEilabelindent%
-                \labelsep\IEEEiednormlabelsep%
-                \partopsep 0ex%
-                \parsep 0ex%
-                \itemsep 0ex%
-                \rightmargin 0em%
-                \listparindent 0em%
-                \itemindent 0em%
-                % calculate the label width
-                % the user can override this later if
-                % they specified a \labelwidth
-                \settowidth{\labelwidth}{\csname labelitem\romannumeral\the\@itemdepth\endcsname}%
-                \@IEEEsavelistparams% save our list parameters
-                \list{\csname\@itemitem\endcsname}{%
+    \ifnum\@itemdepth>3\relax\@toodeep\else%
+        \ifnum\@listdepth>5\relax\@toodeep\else%
+            \advance\@itemdepth\@ne%
+            \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
+            % get the IEEElabelindentfactor for this level
+            \advance\@listdepth\@ne% we need to know what the level WILL be
+            \edef\IEEElabelindentfactor{\csname IEEElabelindentfactor\romannumeral\the\@listdepth\endcsname}%
+            \advance\@listdepth-\@ne% undo our increment
+            \def\@IEEEiedjustify{2}% right justified labels are default
+            % set other defaults
+            \IEEEnocalcleftmarginfalse%
+            \IEEEnolabelindentfactorfalse%
+            \topsep\IEEEiedtopsep%
+            \IEEElabelindent\IEEEilabelindent%
+            \labelsep\IEEEiednormlabelsep%
+            \partopsep 0ex%
+            \parsep 0ex%
+            \itemsep 0ex%
+            \rightmargin 0em%
+            \listparindent 0em%
+            \itemindent 0em%
+            % calculate the label width
+            % the user can override this later if
+            % they specified a \labelwidth
+            \settowidth{\labelwidth}{\csname labelitem\romannumeral\the\@itemdepth\endcsname}%
+            \@IEEEsavelistparams% save our list parameters
+            \list{\csname\@itemitem\endcsname}{%
                 \@IEEErestorelistparams% override any list{} changes
-                                       % to our globals
+                % to our globals
                 \let\makelabel\@IEEEiedmakelabel% v1.6b setup \makelabel
                 \IEEEiedlistdecl% let user alter parameters
                 #1\relax%
@@ -2315,36 +2321,36 @@
 % IEEEtran enumerate list MDS 1/2001
 % Note controlled spacing here, shield end of lines with %
 \def\@@IEEEenumerate[#1]{%
-                \ifnum\@enumdepth>3\relax\@toodeep\else%
-                \ifnum\@listdepth>5\relax\@toodeep\else%
-                \advance\@enumdepth\@ne%
-                \edef\@enumctr{enum\romannumeral\the\@enumdepth}%
-                % get the IEEElabelindentfactor for this level
-                \advance\@listdepth\@ne% we need to know what the level WILL be
-                \edef\IEEElabelindentfactor{\csname IEEElabelindentfactor\romannumeral\the\@listdepth\endcsname}%
-                \advance\@listdepth-\@ne% undo our increment
-                \def\@IEEEiedjustify{2}% right justified labels are default
-                % set other defaults
-                \IEEEnocalcleftmarginfalse%
-                \IEEEnolabelindentfactorfalse%
-                \topsep\IEEEiedtopsep%
-                \IEEElabelindent\IEEEelabelindent%
-                \labelsep\IEEEiednormlabelsep%
-                \partopsep 0ex%
-                \parsep 0ex%
-                \itemsep 0ex%
-                \rightmargin 0em%
-                \listparindent 0em%
-                \itemindent 0em%
-                % calculate the label width
-                % We'll set it to the width suitable for all labels using
-                % normalfont 1) to 9)
-                % The user can override this later
-                \settowidth{\labelwidth}{9)}%
-                \@IEEEsavelistparams% save our list parameters
-                \list{\csname label\@enumctr\endcsname}{\usecounter{\@enumctr}%
+    \ifnum\@enumdepth>3\relax\@toodeep\else%
+        \ifnum\@listdepth>5\relax\@toodeep\else%
+            \advance\@enumdepth\@ne%
+            \edef\@enumctr{enum\romannumeral\the\@enumdepth}%
+            % get the IEEElabelindentfactor for this level
+            \advance\@listdepth\@ne% we need to know what the level WILL be
+            \edef\IEEElabelindentfactor{\csname IEEElabelindentfactor\romannumeral\the\@listdepth\endcsname}%
+            \advance\@listdepth-\@ne% undo our increment
+            \def\@IEEEiedjustify{2}% right justified labels are default
+            % set other defaults
+            \IEEEnocalcleftmarginfalse%
+            \IEEEnolabelindentfactorfalse%
+            \topsep\IEEEiedtopsep%
+            \IEEElabelindent\IEEEelabelindent%
+            \labelsep\IEEEiednormlabelsep%
+            \partopsep 0ex%
+            \parsep 0ex%
+            \itemsep 0ex%
+            \rightmargin 0em%
+            \listparindent 0em%
+            \itemindent 0em%
+            % calculate the label width
+            % We'll set it to the width suitable for all labels using
+            % normalfont 1) to 9)
+            % The user can override this later
+            \settowidth{\labelwidth}{9)}%
+            \@IEEEsavelistparams% save our list parameters
+            \list{\csname label\@enumctr\endcsname}{\usecounter{\@enumctr}%
                 \@IEEErestorelistparams% override any list{} changes
-                                       % to our globals
+                % to our globals
                 \let\makelabel\@IEEEiedmakelabel% v1.6b setup \makelabel
                 \IEEEiedlistdecl% let user alter parameters 
                 #1\relax%
@@ -2367,65 +2373,65 @@
 % IEEEtran description list MDS 1/2001
 % Note controlled spacing here, shield end of lines with %
 \def\@@IEEEdescription[#1]{%
-                \ifnum\@listdepth>5\relax\@toodeep\else%
-                % get the IEEElabelindentfactor for this level
-                \advance\@listdepth\@ne% we need to know what the level WILL be
-                \edef\IEEElabelindentfactor{\csname IEEElabelindentfactor\romannumeral\the\@listdepth\endcsname}%
-                \advance\@listdepth-\@ne% undo our increment
-                \def\@IEEEiedjustify{0}% left justified labels are default
-                % set other defaults
-                \IEEEnocalcleftmarginfalse%
-                \IEEEnolabelindentfactorfalse%
-                \topsep\IEEEiedtopsep% 
-                \IEEElabelindent\IEEEdlabelindent%
-                % assume normal labelsep
-                \labelsep\IEEEiednormlabelsep%
-                \partopsep 0ex%
-                \parsep 0ex%
-                \itemsep 0ex%
-                \rightmargin 0em%
-                \listparindent 0em%
-                \itemindent 0em%
-                % Bogus label width in case the user forgets
-                % to set it.
-                % TIP: If you want to see what a variable's width is you
-                % can use the TeX command \showthe\width-variable to 
-                % display it on the screen during compilation 
-                % (This might be helpful to know when you need to find out
-                % which label is the widest)
-                \settowidth{\labelwidth}{Hello}%
-                \@IEEEsavelistparams% save our list parameters
-                \list{}{\@IEEErestorelistparams% override any list{} changes
-                                               % to our globals
-                \let\makelabel\@IEEEiedmakelabel% v1.6b setup \makelabel
-                \IEEEiedlistdecl% let user alter parameters 
-                #1\relax%
-                % If the user has requested not to use the
-                % labelindent factor, don't revise \IEEElabelindent
-                \ifIEEEnolabelindentfactor\relax%
-                \else\IEEElabelindent=\IEEElabelindentfactor\IEEElabelindent%
-                \fi%
-                % Unless the user has requested otherwise,
-                % calculate our left margin based
-                % on \IEEElabelindent, \labelwidth and
-                % \labelsep
-                \ifIEEEnocalcleftmargin\relax%
-                \else\IEEEcalcleftmargin{\IEEElabelindent}\relax%
-                \fi}\fi}
+    \ifnum\@listdepth>5\relax\@toodeep\else%
+        % get the IEEElabelindentfactor for this level
+        \advance\@listdepth\@ne% we need to know what the level WILL be
+        \edef\IEEElabelindentfactor{\csname IEEElabelindentfactor\romannumeral\the\@listdepth\endcsname}%
+        \advance\@listdepth-\@ne% undo our increment
+        \def\@IEEEiedjustify{0}% left justified labels are default
+        % set other defaults
+        \IEEEnocalcleftmarginfalse%
+        \IEEEnolabelindentfactorfalse%
+        \topsep\IEEEiedtopsep% 
+        \IEEElabelindent\IEEEdlabelindent%
+        % assume normal labelsep
+        \labelsep\IEEEiednormlabelsep%
+        \partopsep 0ex%
+        \parsep 0ex%
+        \itemsep 0ex%
+        \rightmargin 0em%
+        \listparindent 0em%
+        \itemindent 0em%
+        % Bogus label width in case the user forgets
+        % to set it.
+        % TIP: If you want to see what a variable's width is you
+        % can use the TeX command \showthe\width-variable to 
+        % display it on the screen during compilation 
+        % (This might be helpful to know when you need to find out
+        % which label is the widest)
+        \settowidth{\labelwidth}{Hello}%
+        \@IEEEsavelistparams% save our list parameters
+        \list{}{\@IEEErestorelistparams% override any list{} changes
+            % to our globals
+            \let\makelabel\@IEEEiedmakelabel% v1.6b setup \makelabel
+            \IEEEiedlistdecl% let user alter parameters 
+            #1\relax%
+            % If the user has requested not to use the
+            % labelindent factor, don't revise \IEEElabelindent
+            \ifIEEEnolabelindentfactor\relax%
+            \else\IEEElabelindent=\IEEElabelindentfactor\IEEElabelindent%
+            \fi%
+            % Unless the user has requested otherwise,
+            % calculate our left margin based
+            % on \IEEElabelindent, \labelwidth and
+            % \labelsep
+            \ifIEEEnocalcleftmargin\relax%
+            \else\IEEEcalcleftmargin{\IEEElabelindent}\relax%
+            \fi}\fi}
 
 % v1.6b we use one makelabel that does justification as needed.
 \def\@IEEEiedmakelabel#1{\relax\if\@IEEEiedjustify 0\relax
-\makebox[\labelwidth][l]{\normalfont #1}\else
-\if\@IEEEiedjustify 1\relax
-\makebox[\labelwidth][c]{\normalfont #1}\else
-\makebox[\labelwidth][r]{\normalfont #1}\fi\fi}
+        \makebox[\labelwidth][l]{\normalfont #1}\else
+        \if\@IEEEiedjustify 1\relax
+            \makebox[\labelwidth][c]{\normalfont #1}\else
+            \makebox[\labelwidth][r]{\normalfont #1}\fi\fi}
 
 
 % compsoc uses a larger value for the normal labelsep
 % and also extra spacing above and below each list
 \ifCLASSOPTIONcompsoc
-  \IEEEiednormlabelsep 1.2em
-  \IEEEiedtopsep 6pt plus 3pt minus 3pt
+    \IEEEiednormlabelsep 1.2em
+    \IEEEiedtopsep 6pt plus 3pt minus 3pt
 \fi
 
 
@@ -2433,13 +2439,13 @@
 % V1.7 define environments with newenvironment
 \newenvironment{verse}{\let\\=\@centercr
     \list{}{\itemsep\z@ \itemindent -1.5em \listparindent \itemindent
-    \rightmargin\leftmargin\advance\leftmargin 1.5em}\item\relax}
-    {\endlist}
+        \rightmargin\leftmargin\advance\leftmargin 1.5em}\item\relax}
+{\endlist}
 \newenvironment{quotation}{\list{}{\listparindent 1.5em \itemindent\listparindent
-    \rightmargin\leftmargin \parsep 0pt plus 1pt}\item\relax}
-    {\endlist}
+        \rightmargin\leftmargin \parsep 0pt plus 1pt}\item\relax}
+{\endlist}
 \newenvironment{quote}{\list{}{\rightmargin\leftmargin}\item\relax}
-    {\endlist}
+{\endlist}
 
 
 % \titlepage
@@ -2478,7 +2484,7 @@
 % the text to clear the footnote marks. Note that \thanks in IEEEtran
 % uses its own value of \footnotesep which is set in \maketitle.
 {\footnotesize
-\global\footnotesep 0.8\baselineskip}
+    \global\footnotesep 0.8\baselineskip}
 
 
 \skip\@mpfootins = \skip\footins
@@ -2490,8 +2496,8 @@
 \long\def\@makefntext#1{\parindent 1em\indent\hbox{\@makefnmark}#1}% V1.6 use 1em
 % V1.7 compsoc does not use superscipts for footnote marks
 \ifCLASSOPTIONcompsoc
-\def\@IEEEcompsocmakefnmark{\hbox{\normalfont\@thefnmark.\ }}
-\long\def\@makefntext#1{\parindent 1em\indent\hbox{\@IEEEcompsocmakefnmark}#1}
+    \def\@IEEEcompsocmakefnmark{\hbox{\normalfont\@thefnmark.\ }}
+    \long\def\@makefntext#1{\parindent 1em\indent\hbox{\@IEEEcompsocmakefnmark}#1}
 \fi
 
 % The IEEE does not use footnote rules
@@ -2502,14 +2508,14 @@
 \newif\if@IEEEenableoneshotfootnoterule
 \@IEEEenableoneshotfootnoterulefalse
 \ifCLASSOPTIONcompsoc
-\def\footnoterule{\relax\if@IEEEenableoneshotfootnoterule
-\kern-5pt
-\hbox to \columnwidth{\hfill\vrule width 0.5\columnwidth height 0.4pt\hfill}
-\kern4.6pt
-\global\@IEEEenableoneshotfootnoterulefalse
-\else
-\relax
-\fi}
+    \def\footnoterule{\relax\if@IEEEenableoneshotfootnoterule
+            \kern-5pt
+            \hbox to \columnwidth{\hfill\vrule width 0.5\columnwidth height 0.4pt\hfill}
+            \kern4.6pt
+            \global\@IEEEenableoneshotfootnoterulefalse
+        \else
+            \relax
+        \fi}
 \fi
 
 % V1.6 do not allow LaTeX to break a footnote across multiple pages
@@ -2525,7 +2531,7 @@
 
 % technotes do not allow /paragraph
 \ifCLASSOPTIONtechnote
-   \setcounter{secnumdepth}{3}
+    \setcounter{secnumdepth}{3}
 \fi
 % neither do compsoc conferences
 \@IEEEcompsocconfonly{\setcounter{secnumdepth}{3}}
@@ -2544,18 +2550,18 @@
 \def\theequation{\arabic{equation}}                          % 1
 \def\theIEEEsubequation{\theequation\alph{IEEEsubequation}}  % 1a (used only by IEEEtran's IEEEeqnarray)
 \ifCLASSOPTIONcompsoc
-% compsoc is all arabic
-\def\thesection{\arabic{section}}                
-\def\thesubsection{\thesection.\arabic{subsection}}
-\def\thesubsubsection{\thesubsection.\arabic{subsubsection}}
-\def\theparagraph{\thesubsubsection.\arabic{paragraph}}
+    % compsoc is all arabic
+    \def\thesection{\arabic{section}}
+    \def\thesubsection{\thesection.\arabic{subsection}}
+    \def\thesubsubsection{\thesubsection.\arabic{subsubsection}}
+    \def\theparagraph{\thesubsubsection.\arabic{paragraph}}
 \else
-\def\thesection{\Roman{section}}                             % I
-% V1.7, \mbox prevents breaks around - 
-\def\thesubsection{\mbox{\thesection-\Alph{subsection}}}     % I-A
-% V1.7 use I-A1 format used by the IEEE rather than I-A.1
-\def\thesubsubsection{\thesubsection\arabic{subsubsection}}  % I-A1
-\def\theparagraph{\thesubsubsection\alph{paragraph}}         % I-A1a
+    \def\thesection{\Roman{section}}                             % I
+    % V1.7, \mbox prevents breaks around - 
+    \def\thesubsection{\mbox{\thesection-\Alph{subsection}}}     % I-A
+    % V1.7 use I-A1 format used by the IEEE rather than I-A.1
+    \def\thesubsubsection{\thesubsection\arabic{subsubsection}}  % I-A1
+    \def\theparagraph{\thesubsubsection\alph{paragraph}}         % I-A1a
 \fi
 
 % From Heiko Oberdiek. Because of the \mbox in \thesubsection, we need to
@@ -2570,22 +2576,23 @@
 % V1.6, using \thesection in \thesectiondis allows changes
 % in the former to automatically appear in the latter
 \ifCLASSOPTIONcompsoc
-  \ifCLASSOPTIONconference% compsoc conference
-    \def\thesectiondis{\thesection.}
-    \def\thesubsectiondis{\thesectiondis\arabic{subsection}.}
-    \def\thesubsubsectiondis{\thesubsectiondis\arabic{subsubsection}.}
-    \def\theparagraphdis{\thesubsubsectiondis\arabic{paragraph}.}
-  \else% compsoc not conferencs
-    \def\thesectiondis{\thesection}
-    \def\thesubsectiondis{\thesectiondis.\arabic{subsection}}
-    \def\thesubsubsectiondis{\thesubsectiondis.\arabic{subsubsection}}
-    \def\theparagraphdis{\thesubsubsectiondis.\arabic{paragraph}}
-  \fi
+    \ifCLASSOPTIONconference% compsoc conference
+        \def\thesectiondis{\thesection.}
+        \def\thesubsectiondis{\thesectiondis\arabic{subsection}.}
+        \def\thesubsubsectiondis{\thesubsectiondis\arabic{subsubsection}.}
+        \def\theparagraphdis{\thesubsubsectiondis\arabic{paragraph}.}
+    \else% compsoc not conferencs
+        \def\thesectiondis{\thesection}
+        \def\thesubsectiondis{\thesectiondis.\arabic{subsection}}
+        \def\thesubsubsectiondis{\thesubsectiondis.\arabic{subsubsection}}
+        \def\theparagraphdis{\thesubsubsectiondis.\arabic{paragraph}}
+
+    \fi
 \else% not compsoc
-  \def\thesectiondis{\thesection.}                   % I.
-  \def\thesubsectiondis{\Alph{subsection}.}          % B.
-  \def\thesubsubsectiondis{\arabic{subsubsection})}  % 3)
-  \def\theparagraphdis{\alph{paragraph})}            % d)
+    \def\thesectiondis{\thesection.}                   % I.
+    \def\thesubsectiondis{\Alph{subsection}.}          % B.
+    \def\thesubsubsectiondis{\arabic{subsubsection})}  % 3)
+    \def\theparagraphdis{\alph{paragraph})}            % d)
 \fi
 
 % just like LaTeX2e's \@eqnnum
@@ -2608,7 +2615,7 @@
 \def\tablename{TABLE}
 \@IEEEcompsocconfonly{\def\figurename{Figure}}
 \def\partname{Part}
-\def\appendixname{Appendix}
+\def\appendixname{Apêndice}
 \def\abstractname{Abstract}
 % IEEE specific names
 \def\IEEEkeywordsname{Index Terms}
@@ -2697,8 +2704,8 @@
 \setlength\abovecaptionskip{0.5\baselineskip}
 % compsoc journals are a little more generous
 \ifCLASSOPTIONcompsoc\ifCLASSOPTIONjournal
-  \setlength\abovecaptionskip{0.75\baselineskip}
-\fi\fi
+        \setlength\abovecaptionskip{0.75\baselineskip}
+    \fi\fi
 \setlength\belowcaptionskip{0pt}
 % V1.6 create hooks in case the caption spacing ever needs to be
 % overridden by a user
@@ -2715,9 +2722,9 @@
 % here we calculate a space equal to the amount \topskip exceeds the main text height
 % we hook in at \@floatboxreset
 \def\@IEEEfiguretopskipspace{\ifdim\prevdepth=-1000pt\relax
-\setlength{\@IEEEtrantmpdimenA}{1\topskip}\relax
-\addtolength{\@IEEEtrantmpdimenA}{-0.7\@IEEEnormalsizeunitybaselineskip}\relax
-\vspace*{\@IEEEtrantmpdimenA}\fi}
+        \setlength{\@IEEEtrantmpdimenA}{1\topskip}\relax
+        \addtolength{\@IEEEtrantmpdimenA}{-0.7\@IEEEnormalsizeunitybaselineskip}\relax
+        \vspace*{\@IEEEtrantmpdimenA}\fi}
 % V1.8 compensate for \topskip at the top of top tables so caption text is on main text baseline
 % use a strut set on the caption baseline within \@makecaption
 \def\@IEEEtabletopskipstrut{\ifdim\prevdepth=-1000pt\rule{0pt}{\topskip}\fi}
@@ -2728,69 +2735,71 @@
 
 
 \ifCLASSOPTIONcompsoc
-% V1.7 compsoc \@makecaption
-\ifCLASSOPTIONconference% compsoc conference
-\long\def\@makecaption#1#2{%
-% test if is a for a figure or table
-\ifx\@captype\@IEEEtablestring%
-% if a table, do table caption
-\footnotesize\bgroup\par\centering\@IEEEtabletopskipstrut{\normalfont\footnotesize {#1.}\nobreakspace\scshape #2}\par\addvspace{0.5\baselineskip}\egroup%
-\@IEEEtablecaptionsepspace
-% if not a table, format it as a figure
-\else
-\@IEEEfigurecaptionsepspace
-\setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}\nobreakspace #2}%
-\ifdim \wd\@tempboxa >\hsize%
-% if caption is longer than a line, let it wrap around
-\setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}\nobreakspace}%
-\parbox[t]{\hsize}{\normalfont\footnotesize \noindent\unhbox\@tempboxa#2}%
-% if caption is shorter than a line, center
-\else%
-\hbox to\hsize{\normalfont\footnotesize\hfil\box\@tempboxa\hfil}%
-\fi\fi}
-%
-\else% nonconference compsoc
-\long\def\@makecaption#1#2{%
-% test if is a for a figure or table
-\ifx\@captype\@IEEEtablestring%
-% if a table, do table caption
-\footnotesize\bgroup\par\centering\@IEEEtabletopskipstrut{\normalfont\sffamily\footnotesize #1}\\{\normalfont\sffamily\footnotesize #2}\par\addvspace{0.5\baselineskip}\egroup%
-\@IEEEtablecaptionsepspace
-% if not a table, format it as a figure
-\else
-\@IEEEfigurecaptionsepspace
-\setbox\@tempboxa\hbox{\normalfont\sffamily\footnotesize {#1.}\nobreakspace #2}%
-\ifdim \wd\@tempboxa >\hsize%
-% if caption is longer than a line, let it wrap around
-\setbox\@tempboxa\hbox{\normalfont\sffamily\footnotesize {#1.}\nobreakspace}%
-\parbox[t]{\hsize}{\normalfont\sffamily\footnotesize \noindent\unhbox\@tempboxa#2}%
-% if caption is shorter than a line, left justify
-\else%
-\hbox to\hsize{\normalfont\sffamily\footnotesize\box\@tempboxa\hfil}%
-\fi\fi}
-\fi
-%
+    % V1.7 compsoc \@makecaption
+    \ifCLASSOPTIONconference% compsoc conference
+        \long\def\@makecaption#1#2{%
+            % test if is a for a figure or table
+            \ifx\@captype\@IEEEtablestring%
+                % if a table, do table caption
+                \footnotesize\bgroup\par\centering\@IEEEtabletopskipstrut{\normalfont\footnotesize {#1.}\nobreakspace\scshape #2}\par\addvspace{0.5\baselineskip}\egroup%
+                \@IEEEtablecaptionsepspace
+                % if not a table, format it as a figure
+            \else
+                \@IEEEfigurecaptionsepspace
+                \setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}\nobreakspace #2}%
+                \ifdim \wd\@tempboxa >\hsize%
+                    % if caption is longer than a line, let it wrap around
+                    \setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}\nobreakspace}%
+                    \parbox[t]{\hsize}{\normalfont\footnotesize \noindent\unhbox\@tempboxa#2}%
+                    % if caption is shorter than a line, center
+                \else%
+                    \hbox to\hsize{\normalfont\footnotesize\hfil\box\@tempboxa\hfil}%
+                \fi\fi}
+        %
+    \else% nonconference compsoc
+        \long\def\@makecaption#1#2{%
+            % test if is a for a figure or table
+            \ifx\@captype\@IEEEtablestring%
+                % if a table, do table caption
+                \footnotesize\bgroup\par\centering\@IEEEtabletopskipstrut{\normalfont\sffamily\footnotesize #1}\\{\normalfont\sffamily\footnotesize #2}\par\addvspace{0.5\baselineskip}\egroup%
+                \@IEEEtablecaptionsepspace
+                % if not a table, format it as a figure
+            \else
+                \@IEEEfigurecaptionsepspace
+                \setbox\@tempboxa\hbox{\normalfont\sffamily\footnotesize {#1.}\nobreakspace #2}%
+                \ifdim \wd\@tempboxa >\hsize%
+                    % if caption is longer than a line, let it wrap around
+                    \setbox\@tempboxa\hbox{\normalfont\sffamily\footnotesize {#1.}\nobreakspace}%
+                    \parbox[t]{\hsize}{\normalfont\sffamily\footnotesize \noindent\unhbox\@tempboxa#2}%
+                    % if caption is shorter than a line, left justify
+                \else%
+                    \hbox to\hsize{\normalfont\sffamily\footnotesize\box\@tempboxa\hfil}%
+                \fi\fi}
+
+    \fi
+    %
 \else% traditional noncompsoc \@makecaption
-\long\def\@makecaption#1#2{%
-% test if is a for a figure or table
-\ifx\@captype\@IEEEtablestring%
-% if a table, do table caption
-\footnotesize\bgroup\par\centering\@IEEEtabletopskipstrut{\normalfont\footnotesize #1}\\{\normalfont\footnotesize\scshape #2}\par\addvspace{0.5\baselineskip}\egroup%
-\@IEEEtablecaptionsepspace
-% if not a table, format it as a figure
-\else
-\@IEEEfigurecaptionsepspace
-% 3/2001 use footnotesize, not small; use two nonbreaking spaces, not one
-\setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}\nobreakspace\nobreakspace #2}%
-\ifdim \wd\@tempboxa >\hsize%
-% if caption is longer than a line, let it wrap around
-\setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}\nobreakspace\nobreakspace}%
-\parbox[t]{\hsize}{\normalfont\footnotesize\noindent\unhbox\@tempboxa#2}%
-% if caption is shorter than a line, center if conference, left justify otherwise
-\else%
-\ifCLASSOPTIONconference \hbox to\hsize{\normalfont\footnotesize\hfil\box\@tempboxa\hfil}%
-\else \hbox to\hsize{\normalfont\footnotesize\box\@tempboxa\hfil}%
-\fi\fi\fi}
+    \long\def\@makecaption#1#2{%
+        % test if is a for a figure or table
+        \ifx\@captype\@IEEEtablestring%
+            % if a table, do table caption
+            \footnotesize\bgroup\par\centering\@IEEEtabletopskipstrut{\normalfont\footnotesize #1}\\{\normalfont\footnotesize\scshape #2}\par\addvspace{0.5\baselineskip}\egroup%
+            \@IEEEtablecaptionsepspace
+            % if not a table, format it as a figure
+        \else
+            \@IEEEfigurecaptionsepspace
+            % 3/2001 use footnotesize, not small; use two nonbreaking spaces, not one
+            \setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}\nobreakspace\nobreakspace #2}%
+            \ifdim \wd\@tempboxa >\hsize%
+                % if caption is longer than a line, let it wrap around
+                \setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}\nobreakspace\nobreakspace}%
+                \parbox[t]{\hsize}{\normalfont\footnotesize\noindent\unhbox\@tempboxa#2}%
+                % if caption is shorter than a line, center if conference, left justify otherwise
+            \else%
+                \ifCLASSOPTIONconference \hbox to\hsize{\normalfont\footnotesize\hfil\box\@tempboxa\hfil}%
+                \else \hbox to\hsize{\normalfont\footnotesize\box\@tempboxa\hfil}%
+                \fi\fi\fi}
+
 \fi
 
 
@@ -2798,13 +2807,13 @@
 % V1.7 disable captions class option, do so in a way that retains operation of \label
 % within \caption
 \ifCLASSOPTIONcaptionsoff
-\long\def\@makecaption#1#2{\vspace*{2em}\footnotesize\bgroup\par\addvspace{0.5\baselineskip}\centering{\footnotesize #1}\par\addvspace{0.5\baselineskip}\egroup%
-\let\@IEEEtemporiglabeldefsave\label
-\let\@IEEEtemplabelargsave\relax
-\def\label##1{\gdef\@IEEEtemplabelargsave{##1}}%
-\setbox\@tempboxa\hbox{#2}%
-\let\label\@IEEEtemporiglabeldefsave
-\ifx\@IEEEtemplabelargsave\relax\else\label{\@IEEEtemplabelargsave}\fi}
+    \long\def\@makecaption#1#2{\vspace*{2em}\footnotesize\bgroup\par\addvspace{0.5\baselineskip}\centering{\footnotesize #1}\par\addvspace{0.5\baselineskip}\egroup%
+        \let\@IEEEtemporiglabeldefsave\label
+        \let\@IEEEtemplabelargsave\relax
+        \def\label##1{\gdef\@IEEEtemplabelargsave{##1}}%
+        \setbox\@tempboxa\hbox{#2}%
+        \let\label\@IEEEtemporiglabeldefsave
+        \ifx\@IEEEtemplabelargsave\relax\else\label{\@IEEEtemplabelargsave}\fi}
 \fi
 
 
@@ -2825,9 +2834,10 @@
 
 \newcounter{table}
 \ifCLASSOPTIONcompsoc
-\def\thetable{\arabic{table}}
+    \def\thetable{\arabic{table}}
 \else
-\def\thetable{\@Roman\c@table}
+    \def\thetable{\@Roman\c@table}
+
 \fi
 \def\fps@table{tbp}
 \def\ftype@table{2}
@@ -2865,15 +2875,15 @@
 \def\@IEEEstripouterbraces{\@ifstar{\let\@IEEEstripouterbracesdef=\def\@@IEEEstripouterbraces}{\let\@IEEEstripouterbracesdef=\edef\@@IEEEstripouterbraces}}
 
 \def\@@IEEEstripouterbraces#1{\@IEEEstripouterbracesdef\@IEEEstripouterbracesarg{#1}\relax
-% If the macro is unchanged after being acquired as a single delimited
-% argument, we know we have one sequence of tokens without any enclosing
-% braces. Loop until this is true.
-\loop
-  \expandafter\@@@IEEEstripouterbraces\@IEEEstripouterbracesarg\@IEEEgeneralsequenceDELIMITER
-\ifx\@IEEEstrippedouterbraces\@IEEEstripouterbracesarg
-\else
-  \let\@IEEEstripouterbracesarg\@IEEEstrippedouterbraces
-\repeat}
+    % If the macro is unchanged after being acquired as a single delimited
+    % argument, we know we have one sequence of tokens without any enclosing
+    % braces. Loop until this is true.
+    \loop
+    \expandafter\@@@IEEEstripouterbraces\@IEEEstripouterbracesarg\@IEEEgeneralsequenceDELIMITER
+    \ifx\@IEEEstrippedouterbraces\@IEEEstripouterbracesarg
+    \else
+    \let\@IEEEstripouterbracesarg\@IEEEstrippedouterbraces
+    \repeat}
 
 \def\@@@IEEEstripouterbraces#1\@IEEEgeneralsequenceDELIMITER{\def\@IEEEstrippedouterbraces{#1}}
 
@@ -2903,22 +2913,22 @@
 \def\@IEEEextractgroup{\@ifstar{\let\@IEEEextractgroupdef=\def\@@IEEEextractgroup}{\let\@IEEEextractgroupdef=\edef\@@IEEEextractgroup}}
 
 \def\@@IEEEextractgroup#1{\@IEEEextractgroupdef\@IEEEextractgrouparg{#1}\relax
-% trap the case of an empty extracted group as this would cause problems with
-% \@IEEEextractgroupremain's argument acquisition
-\ifx\@IEEEextractgrouparg\@empty
-  \def\@IEEEextractedgroup{}\relax
-  \def\@IEEEextractedgroupremain{}\relax
-\else
-  % We have to use some dirty tricks here. We want to insert {} around
-  % whatever remains after the first group so that TeX's argument scanner
-  % will preserve any originally enclosing braces as well as provide an
-  % empty argument to acquire even if there isn't a second group.
-  % In this first of two dirty tricks, we put a } at the end of the structure
-  % we are going to extract from. The \ifnum0=`{\fi keeps TeX happy to allow
-  % what would otherwise be an unbalanced macro definition for
-  % \@@IEEEextractgroup to be acceptable to it.
-  \ifnum0=`{\fi\expandafter\@IEEEextractgroupremain\@IEEEextractgrouparg}\relax
-\fi}
+    % trap the case of an empty extracted group as this would cause problems with
+    % \@IEEEextractgroupremain's argument acquisition
+    \ifx\@IEEEextractgrouparg\@empty
+        \def\@IEEEextractedgroup{}\relax
+        \def\@IEEEextractedgroupremain{}\relax
+    \else
+        % We have to use some dirty tricks here. We want to insert {} around
+        % whatever remains after the first group so that TeX's argument scanner
+        % will preserve any originally enclosing braces as well as provide an
+        % empty argument to acquire even if there isn't a second group.
+        % In this first of two dirty tricks, we put a } at the end of the structure
+        % we are going to extract from. The \ifnum0=`{\fi keeps TeX happy to allow
+        % what would otherwise be an unbalanced macro definition for
+        % \@@IEEEextractgroup to be acceptable to it.
+        \ifnum0=`{\fi\expandafter\@IEEEextractgroupremain\@IEEEextractgrouparg}\relax
+    \fi}
 
 % In the second part of the dirty tricks, we insert a leading { right after
 % the first group is acquired, but before the remainder is. Again, the
@@ -2983,28 +2993,28 @@
 % Depends on: \@IEEEextractgroup, \@IEEEextracttoken
 \def\@IEEEextracttokengroups{\@ifstar{\let\@IEEEextracttokengroupsdef=\def\@@IEEEextracttokengroups}{\let\@IEEEextracttokengroupsdef=\edef\@@IEEEextracttokengroups}}
 \def\@@IEEEextracttokengroups#1{\@IEEEextracttokengroupsdef\@IEEEextracttokengroupsarg{#1}\relax
-% begin extraction, these functions are safe with empty arguments
-% first group
-\expandafter\@IEEEextractgroup\expandafter*\expandafter{\@IEEEextracttokengroupsarg}\relax
-\let\@IEEEextractedfirstgroup\@IEEEextractedgroup
-\let\@IEEEextractedafterfirstgroupremain\@IEEEextractedgroupremain
-\expandafter\@IEEEextracttoken\expandafter*\expandafter{\@IEEEextractedfirstgroup}\relax
-\let\@IEEEextractedfirsttoken\@IEEEextractedtoken
-\let\@IEEEextractedfirsttokenmacro\@IEEEextractedtokenmacro
-\let\@IEEEextractedfirsttokensdiscarded\@IEEEextractedtokensdiscarded
-% first first group
-\expandafter\@IEEEextractgroup\expandafter*\expandafter{\@IEEEextractedfirstgroup}\relax
-\let\@IEEEextractedfirstfirstgroup\@IEEEextractedgroup
-% next group
-\expandafter\@IEEEextractgroup\expandafter*\expandafter{\@IEEEextractedafterfirstgroupremain}\relax
-\let\@IEEEextractednextgroup\@IEEEextractedgroup
-\expandafter\@IEEEextracttoken\expandafter*\expandafter{\@IEEEextractednextgroup}\relax
-\let\@IEEEextractednextfirsttoken\@IEEEextractedtoken
-\let\@IEEEextractednextfirsttokenmacro\@IEEEextractedtokenmacro
-\let\@IEEEextractednextfirsttokensdiscarded\@IEEEextractedtokensdiscarded
-% next first group
-\expandafter\@IEEEextractgroup\expandafter*\expandafter{\@IEEEextractednextgroup}\relax
-\let\@IEEEextractednextfirstgroup\@IEEEextractedgroup}
+    % begin extraction, these functions are safe with empty arguments
+    % first group
+    \expandafter\@IEEEextractgroup\expandafter*\expandafter{\@IEEEextracttokengroupsarg}\relax
+    \let\@IEEEextractedfirstgroup\@IEEEextractedgroup
+    \let\@IEEEextractedafterfirstgroupremain\@IEEEextractedgroupremain
+    \expandafter\@IEEEextracttoken\expandafter*\expandafter{\@IEEEextractedfirstgroup}\relax
+    \let\@IEEEextractedfirsttoken\@IEEEextractedtoken
+    \let\@IEEEextractedfirsttokenmacro\@IEEEextractedtokenmacro
+    \let\@IEEEextractedfirsttokensdiscarded\@IEEEextractedtokensdiscarded
+    % first first group
+    \expandafter\@IEEEextractgroup\expandafter*\expandafter{\@IEEEextractedfirstgroup}\relax
+    \let\@IEEEextractedfirstfirstgroup\@IEEEextractedgroup
+    % next group
+    \expandafter\@IEEEextractgroup\expandafter*\expandafter{\@IEEEextractedafterfirstgroupremain}\relax
+    \let\@IEEEextractednextgroup\@IEEEextractedgroup
+    \expandafter\@IEEEextracttoken\expandafter*\expandafter{\@IEEEextractednextgroup}\relax
+    \let\@IEEEextractednextfirsttoken\@IEEEextractedtoken
+    \let\@IEEEextractednextfirsttokenmacro\@IEEEextractedtokenmacro
+    \let\@IEEEextractednextfirsttokensdiscarded\@IEEEextractedtokensdiscarded
+    % next first group
+    \expandafter\@IEEEextractgroup\expandafter*\expandafter{\@IEEEextractednextgroup}\relax
+    \let\@IEEEextractednextfirstgroup\@IEEEextractedgroup}
 
 
 %% -- End of Command Argument Scanning Support Functions --
@@ -3074,14 +3084,14 @@
 \def\IEEEyesnumber{\@ifstar{\global\@IEEEeqnumpersisttrue\global\@IEEEsubeqnumpersistfalse\@IEEEyesnumber}{\@IEEEyesnumber}}
 
 \def\@IEEEyesnumber{\global\@eqnswtrue
-\if@IEEEeqnarrayISinner% alter counters and label only inside an IEEEeqnarray
-\ifnum\c@IEEEsubequation>0\relax
-   \stepcounter{equation}\setcounter{IEEEsubequation}{0}\gdef\@currentlabel{\p@equation\theequation}\relax
-   \gdef\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
-\fi
-% even if we reached this eqn num via a preadv, it is legit now
-\global\@IEEEeqnumpreadvfalse\global\@IEEEsubeqnumpreadvfalse
-\fi}
+    \if@IEEEeqnarrayISinner% alter counters and label only inside an IEEEeqnarray
+        \ifnum\c@IEEEsubequation>0\relax
+            \stepcounter{equation}\setcounter{IEEEsubequation}{0}\gdef\@currentlabel{\p@equation\theequation}\relax
+            \gdef\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
+        \fi
+        % even if we reached this eqn num via a preadv, it is legit now
+        \global\@IEEEeqnumpreadvfalse\global\@IEEEsubeqnumpreadvfalse
+    \fi}
 
 \def\IEEEnonumber{\@ifstar{\global\@IEEEeqnumpersistfalse\global\@IEEEsubeqnumpersistfalse\global\@eqnswfalse}{\global\@eqnswfalse}}
 
@@ -3089,33 +3099,33 @@
 \def\IEEEyessubnumber{\@ifstar{\global\@IEEEsubeqnumpersisttrue\@IEEEyessubnumber}{\@IEEEyessubnumber}}
 %
 \def\@IEEEyessubnumber{\if@IEEEeqnarrayISinner% alter counters and label only inside an IEEEeqnarray
-  \ifnum\c@IEEEsubequation>0\relax% if it already is a subequation, we are good to go as-is
-  \else% if we are a regular equation we have to watch out for two cases
-    \if@IEEEeqnumpreadv% if this equation is the result of a preadvance, backout and bump the sub eqnnum
-       \global\advance\c@equation\m@ne\global\c@IEEEsubequation=\@IEEEsubeqnnumrollback\addtocounter{IEEEsubequation}{1}\relax
-    \else% non-preadvanced equations just need initialization of their sub eqnnum
-       \setcounter{IEEEsubequation}{1}\relax
-    \fi
-  \fi% fi already is subequation
-  \gdef\@currentlabel{\p@IEEEsubequation\theIEEEsubequation}\relax
-  \gdef\@currentHref{\@IEEEtheHrefsubequation}% setup hyperref label
-  \global\@IEEEeqnumpreadvfalse\global\@IEEEsubeqnumpreadvfalse% no longer a preadv anymore
-  \global\@eqnswtrue
-\fi}
+        \ifnum\c@IEEEsubequation>0\relax% if it already is a subequation, we are good to go as-is
+        \else% if we are a regular equation we have to watch out for two cases
+            \if@IEEEeqnumpreadv% if this equation is the result of a preadvance, backout and bump the sub eqnnum
+                \global\advance\c@equation\m@ne\global\c@IEEEsubequation=\@IEEEsubeqnnumrollback\addtocounter{IEEEsubequation}{1}\relax
+            \else% non-preadvanced equations just need initialization of their sub eqnnum
+                \setcounter{IEEEsubequation}{1}\relax
+            \fi
+        \fi% fi already is subequation
+        \gdef\@currentlabel{\p@IEEEsubequation\theIEEEsubequation}\relax
+        \gdef\@currentHref{\@IEEEtheHrefsubequation}% setup hyperref label
+        \global\@IEEEeqnumpreadvfalse\global\@IEEEsubeqnumpreadvfalse% no longer a preadv anymore
+        \global\@eqnswtrue
+    \fi}
 
 
 \def\IEEEnosubnumber{\@ifstar{\global\@IEEEsubeqnumpersistfalse\@IEEEnosubnumber}{\@IEEEnosubnumber}}
 %
 \def\@IEEEnosubnumber{\if@IEEEeqnarrayISinner% alter counters and label only inside an IEEEeqnarray
-  \if@eqnsw % we do nothing unless we know we will display because we play with the counters here
-    % if it currently is a subequation, bump up to the next equation number and turn off the subequation
-    \ifnum\c@IEEEsubequation>0\relax\addtocounter{equation}{1}\setcounter{IEEEsubequation}{0}\relax
-    \fi
-    \global\@IEEEeqnumpreadvfalse\global\@IEEEsubeqnumpreadvfalse% no longer a preadv anymore
-    \gdef\@currentlabel{\p@equation\theequation}\relax
-    \gdef\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
-  \fi
-\fi}
+        \if@eqnsw % we do nothing unless we know we will display because we play with the counters here
+            % if it currently is a subequation, bump up to the next equation number and turn off the subequation
+            \ifnum\c@IEEEsubequation>0\relax\addtocounter{equation}{1}\setcounter{IEEEsubequation}{0}\relax
+            \fi
+            \global\@IEEEeqnumpreadvfalse\global\@IEEEsubeqnumpreadvfalse% no longer a preadv anymore
+            \gdef\@currentlabel{\p@equation\theequation}\relax
+            \gdef\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
+        \fi
+    \fi}
 
 
 
@@ -3127,35 +3137,35 @@
 % column counter - so as to work within \IEEEeqnarraybox
 % usage: \IEEEeqnarraymulticol{number cols. to span}{col type}{cell text}
 \long\def\IEEEeqnarraymulticol#1#2#3{\multispan{#1}\relax
-% check if column is defined for the precolumn definition
-% We have to be careful here because TeX scans for & even within an \iffalse
-% where it does not expand macros. So, if we used only one \ifx and a #3
-% appeared in the false branch and the user inserted another alignment
-% structure that uses & in the \IEEEeqnarraymulticol{}, TeX will not see that
-% there is an inner alignment in the false branch yet still will see any &
-% there and will think that they apply to the outer alignment resulting in an
-% incomplete \ifx error.
-% So, here we use separate checks for the pre and post parts in order to keep
-% the #3 outside of all conditionals.
-\relax\expandafter\ifx\csname @IEEEeqnarraycolDEF#2\endcsname\@IEEEeqnarraycolisdefined\relax
-\csname @IEEEeqnarraycolPRE#2\endcsname
-\else% if not, error and use default type
-\@IEEEclspkgerror{Invalid column type "#2" in \string\IEEEeqnarraymulticol.\MessageBreak
-Using a default centering column instead}%
-{You must define IEEEeqnarray column types before use.}%
-\csname @IEEEeqnarraycolPRE@IEEEdefault\endcsname
-\fi
-% The ten \relax are to help prevent misleading error messages in case a user
-% accidently inserted a macro that tries to acquire additional arguments.
-#3\relax\relax\relax\relax\relax\relax\relax\relax\relax\relax
-% check if column is defined for the postcolumn definition
-\expandafter\ifx\csname @IEEEeqnarraycolDEF#2\endcsname\@IEEEeqnarraycolisdefined\relax
-\csname @IEEEeqnarraycolPOST#2\endcsname
-\else% if not, use the default type
-\csname @IEEEeqnarraycolPOST@IEEEdefault\endcsname
-\fi
-% advance column counter only if the IEEEeqnarray environment wants it
-\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by #1\relax\fi}
+    % check if column is defined for the precolumn definition
+    % We have to be careful here because TeX scans for & even within an \iffalse
+    % where it does not expand macros. So, if we used only one \ifx and a #3
+    % appeared in the false branch and the user inserted another alignment
+    % structure that uses & in the \IEEEeqnarraymulticol{}, TeX will not see that
+    % there is an inner alignment in the false branch yet still will see any &
+    % there and will think that they apply to the outer alignment resulting in an
+    % incomplete \ifx error.
+    % So, here we use separate checks for the pre and post parts in order to keep
+    % the #3 outside of all conditionals.
+    \relax\expandafter\ifx\csname @IEEEeqnarraycolDEF#2\endcsname\@IEEEeqnarraycolisdefined\relax
+        \csname @IEEEeqnarraycolPRE#2\endcsname
+    \else% if not, error and use default type
+        \@IEEEclspkgerror{Invalid column type "#2" in \string\IEEEeqnarraymulticol.\MessageBreak
+            Using a default centering column instead}%
+        {You must define IEEEeqnarray column types before use.}%
+        \csname @IEEEeqnarraycolPRE@IEEEdefault\endcsname
+    \fi
+    % The ten \relax are to help prevent misleading error messages in case a user
+    % accidently inserted a macro that tries to acquire additional arguments.
+    #3\relax\relax\relax\relax\relax\relax\relax\relax\relax\relax
+    % check if column is defined for the postcolumn definition
+    \expandafter\ifx\csname @IEEEeqnarraycolDEF#2\endcsname\@IEEEeqnarraycolisdefined\relax
+        \csname @IEEEeqnarraycolPOST#2\endcsname
+    \else% if not, use the default type
+        \csname @IEEEeqnarraycolPOST@IEEEdefault\endcsname
+    \fi
+    % advance column counter only if the IEEEeqnarray environment wants it
+    \if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by #1\relax\fi}
 
 % like \omit, but maintains track of the column counter for \IEEEeqnarray
 \def\IEEEeqnarrayomit{\omit\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by 1\relax\fi}
@@ -3164,14 +3174,14 @@ Using a default centering column instead}%
 % provides a way to define a letter referenced column type
 % usage: \IEEEeqnarraydefcol{col. type letter/name}{pre insertion text}{post insertion text}
 \def\IEEEeqnarraydefcol#1#2#3{\expandafter\def\csname @IEEEeqnarraycolPRE#1\endcsname{#2}%
-\expandafter\def\csname @IEEEeqnarraycolPOST#1\endcsname{#3}%
-\expandafter\def\csname @IEEEeqnarraycolDEF#1\endcsname{1}}
+    \expandafter\def\csname @IEEEeqnarraycolPOST#1\endcsname{#3}%
+    \expandafter\def\csname @IEEEeqnarraycolDEF#1\endcsname{1}}
 
 
 % provides a way to define a numerically referenced inter-column glue types
 % usage: \IEEEeqnarraydefcolsep{col. glue number}{glue definition}
 \def\IEEEeqnarraydefcolsep#1#2{\expandafter\def\csname @IEEEeqnarraycolSEP\romannumeral #1\endcsname{#2}%
-\expandafter\def\csname @IEEEeqnarraycolSEPDEF\romannumeral #1\endcsname{1}}
+    \expandafter\def\csname @IEEEeqnarraycolSEPDEF\romannumeral #1\endcsname{1}}
 
 
 \def\@IEEEeqnarraycolisdefined{1}% just a macro for 1, used for checking undefined column types
@@ -3180,13 +3190,13 @@ Using a default centering column instead}%
 % expands and appends the given argument to the \@IEEEtrantmptoksA token list
 % used to build up the \halign preamble
 \def\@IEEEappendtoksA#1{\edef\@@IEEEappendtoksA{\@IEEEtrantmptoksA={\the\@IEEEtrantmptoksA #1}}%
-\@@IEEEappendtoksA}
+    \@@IEEEappendtoksA}
 
 % also appends to \@IEEEtrantmptoksA, but does not expand the argument
 % uses \toks8 as a scratchpad register
 \def\@IEEEappendNOEXPANDtoksA#1{\toks8={#1}%
-\edef\@@IEEEappendNOEXPANDtoksA{\@IEEEtrantmptoksA={\the\@IEEEtrantmptoksA\the\toks8}}%
-\@@IEEEappendNOEXPANDtoksA}
+    \edef\@@IEEEappendNOEXPANDtoksA{\@IEEEtrantmptoksA={\the\@IEEEtrantmptoksA\the\toks8}}%
+    \@@IEEEappendNOEXPANDtoksA}
 
 % define some common column types for the user
 % math
@@ -3247,15 +3257,15 @@ Using a default centering column instead}%
 \def\IEEEeqnarrayseprow{\relax\@ifnextchar[{\@IEEEeqnarrayseprow}{\@IEEEeqnarrayseprow[0.25\normalbaselineskip]}}
 \def\@IEEEeqnarrayseprow[#1]{\relax\@ifnextchar[{\@@IEEEeqnarrayseprow[#1]}{\@@IEEEeqnarrayseprow[#1][\relax]}}
 \def\@@IEEEeqnarrayseprow[#1][#2]{\def\@IEEEeqnarrayseprowARGONE{#1}%
-\ifx\@IEEEeqnarrayseprowARGONE\@empty%
-% get the skip value, based on the font commands
-% use skip5 because \IEEEeqnarraystrutsize uses \skip0, \skip2, \skip3
-% assign within a bogus box to confine the font changes
-{\setbox0=\hbox{#2\relax\global\skip5=0.25\normalbaselineskip}}%
-\else%
-{\setbox0=\hbox{#2\relax\global\skip5=#1}}%
-\fi%
-\@IEEEeqnarrayhoptolastcolumn\IEEEeqnarraystrutsize{\skip5}{0pt}[\relax]\relax}
+    \ifx\@IEEEeqnarrayseprowARGONE\@empty%
+        % get the skip value, based on the font commands
+        % use skip5 because \IEEEeqnarraystrutsize uses \skip0, \skip2, \skip3
+        % assign within a bogus box to confine the font changes
+        {\setbox0=\hbox{#2\relax\global\skip5=0.25\normalbaselineskip}}%
+    \else%
+        {\setbox0=\hbox{#2\relax\global\skip5=#1}}%
+    \fi%
+    \@IEEEeqnarrayhoptolastcolumn\IEEEeqnarraystrutsize{\skip5}{0pt}[\relax]\relax}
 
 % creates a blank separator row, but omits all the column templates
 % usage: \IEEEeqnarrayseprowcut[separation length][font size commands]
@@ -3263,20 +3273,20 @@ Using a default centering column instead}%
 % blank arguments inherit the default values
 % uses \skip5 as a scratch register - calls \@IEEEeqnarraystrutsize which uses more scratch registers
 \def\IEEEeqnarrayseprowcut{\multispan{\@IEEEeqnnumcols}\relax% span all the cols
-% advance column counter only if the IEEEeqnarray environment wants it
-\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
-\@ifnextchar[{\@IEEEeqnarrayseprowcut}{\@IEEEeqnarrayseprowcut[0.25\normalbaselineskip]}}
+    % advance column counter only if the IEEEeqnarray environment wants it
+    \if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+    \@ifnextchar[{\@IEEEeqnarrayseprowcut}{\@IEEEeqnarrayseprowcut[0.25\normalbaselineskip]}}
 \def\@IEEEeqnarrayseprowcut[#1]{\relax\@ifnextchar[{\@@IEEEeqnarrayseprowcut[#1]}{\@@IEEEeqnarrayseprowcut[#1][\relax]}}
 \def\@@IEEEeqnarrayseprowcut[#1][#2]{\def\@IEEEeqnarrayseprowARGONE{#1}%
-\ifx\@IEEEeqnarrayseprowARGONE\@empty%
-% get the skip value, based on the font commands
-% use skip5 because \IEEEeqnarraystrutsize uses \skip0, \skip2, \skip3
-% assign within a bogus box to confine the font changes
-{\setbox0=\hbox{#2\relax\global\skip5=0.25\normalbaselineskip}}%
-\else%
-{\setbox0=\hbox{#2\relax\global\skip5=#1}}%
-\fi%
-\IEEEeqnarraystrutsize{\skip5}{0pt}[\relax]\relax}
+    \ifx\@IEEEeqnarrayseprowARGONE\@empty%
+        % get the skip value, based on the font commands
+        % use skip5 because \IEEEeqnarraystrutsize uses \skip0, \skip2, \skip3
+        % assign within a bogus box to confine the font changes
+        {\setbox0=\hbox{#2\relax\global\skip5=0.25\normalbaselineskip}}%
+    \else%
+        {\setbox0=\hbox{#2\relax\global\skip5=#1}}%
+    \fi%
+    \IEEEeqnarraystrutsize{\skip5}{0pt}[\relax]\relax}
 
 
 
@@ -3285,12 +3295,12 @@ Using a default centering column instead}%
 % updates column counter as needed and turns off struts
 % usage: \IEEEeqnarrayrulerow[rule line thickness]
 \def\IEEEeqnarrayrulerow{\multispan{\@IEEEeqnnumcols}\relax% span all the cols
-% advance column counter only if the IEEEeqnarray environment wants it
-\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
-\@ifnextchar[{\@IEEEeqnarrayrulerow}{\@IEEEeqnarrayrulerow[\arrayrulewidth]}}
+    % advance column counter only if the IEEEeqnarray environment wants it
+    \if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+    \@ifnextchar[{\@IEEEeqnarrayrulerow}{\@IEEEeqnarrayrulerow[\arrayrulewidth]}}
 \def\@IEEEeqnarrayrulerow[#1]{\leaders\hrule height#1\hfil\relax% put in our rule 
-% turn off any struts
-\IEEEeqnarraystrutsize{0pt}{0pt}[\relax]\relax}
+    % turn off any struts
+    \IEEEeqnarraystrutsize{0pt}{0pt}[\relax]\relax}
 
 
 % draws a double rule by using a single rule row, a separator row, and then
@@ -3299,33 +3309,33 @@ Using a default centering column instead}%
 % second optional argument determines the rule spacing, \doublerulesep is the default
 % usage: \IEEEeqnarraydblrulerow[rule line thickness][rule spacing]
 \def\IEEEeqnarraydblrulerow{\multispan{\@IEEEeqnnumcols}\relax% span all the cols
-% advance column counter only if the IEEEeqnarray environment wants it
-\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
-\@ifnextchar[{\@IEEEeqnarraydblrulerow}{\@IEEEeqnarraydblrulerow[\arrayrulewidth]}}
+    % advance column counter only if the IEEEeqnarray environment wants it
+    \if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+    \@ifnextchar[{\@IEEEeqnarraydblrulerow}{\@IEEEeqnarraydblrulerow[\arrayrulewidth]}}
 \def\@IEEEeqnarraydblrulerow[#1]{\relax\@ifnextchar[{\@@IEEEeqnarraydblrulerow[#1]}%
-{\@@IEEEeqnarraydblrulerow[#1][\doublerulesep]}}
+    {\@@IEEEeqnarraydblrulerow[#1][\doublerulesep]}}
 \def\@@IEEEeqnarraydblrulerow[#1][#2]{\def\@IEEEeqnarraydblrulerowARG{#1}%
-% we allow the user to say \IEEEeqnarraydblrulerow[][]
-\ifx\@IEEEeqnarraydblrulerowARG\@empty%
-\@IEEEeqnarrayrulerow[\arrayrulewidth]%
-\else%
-\@IEEEeqnarrayrulerow[#1]\relax%
-\fi%
-\def\@IEEEeqnarraydblrulerowARG{#2}%
-\ifx\@IEEEeqnarraydblrulerowARG\@empty%
-\\\IEEEeqnarrayseprow[\doublerulesep][\relax]%
-\else%
-\\\IEEEeqnarrayseprow[#2][\relax]%
-\fi%
-\\\multispan{\@IEEEeqnnumcols}%
-% advance column counter only if the IEEEeqnarray environment wants it
-\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
-\def\@IEEEeqnarraydblrulerowARG{#1}%
-\ifx\@IEEEeqnarraydblrulerowARG\@empty%
-\@IEEEeqnarrayrulerow[\arrayrulewidth]%
-\else%
-\@IEEEeqnarrayrulerow[#1]%
-\fi%
+    % we allow the user to say \IEEEeqnarraydblrulerow[][]
+    \ifx\@IEEEeqnarraydblrulerowARG\@empty%
+        \@IEEEeqnarrayrulerow[\arrayrulewidth]%
+    \else%
+        \@IEEEeqnarrayrulerow[#1]\relax%
+    \fi%
+    \def\@IEEEeqnarraydblrulerowARG{#2}%
+    \ifx\@IEEEeqnarraydblrulerowARG\@empty%
+        \\\IEEEeqnarrayseprow[\doublerulesep][\relax]%
+    \else%
+        \\\IEEEeqnarrayseprow[#2][\relax]%
+    \fi%
+    \\\multispan{\@IEEEeqnnumcols}%
+    % advance column counter only if the IEEEeqnarray environment wants it
+    \if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+    \def\@IEEEeqnarraydblrulerowARG{#1}%
+    \ifx\@IEEEeqnarraydblrulerowARG\@empty%
+        \@IEEEeqnarrayrulerow[\arrayrulewidth]%
+    \else%
+        \@IEEEeqnarrayrulerow[#1]%
+    \fi%
 }
 
 % draws a double rule by using a single rule row, a separator (cutting) row, and then
@@ -3334,33 +3344,33 @@ Using a default centering column instead}%
 % second optional argument determines the rule spacing, \doublerulesep is the default
 % usage: \IEEEeqnarraydblrulerow[rule line thickness][rule spacing]
 \def\IEEEeqnarraydblrulerowcut{\multispan{\@IEEEeqnnumcols}\relax% span all the cols
-% advance column counter only if the IEEEeqnarray environment wants it
-\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
-\@ifnextchar[{\@IEEEeqnarraydblrulerowcut}{\@IEEEeqnarraydblrulerowcut[\arrayrulewidth]}}
+    % advance column counter only if the IEEEeqnarray environment wants it
+    \if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+    \@ifnextchar[{\@IEEEeqnarraydblrulerowcut}{\@IEEEeqnarraydblrulerowcut[\arrayrulewidth]}}
 \def\@IEEEeqnarraydblrulerowcut[#1]{\relax\@ifnextchar[{\@@IEEEeqnarraydblrulerowcut[#1]}%
-{\@@IEEEeqnarraydblrulerowcut[#1][\doublerulesep]}}
+    {\@@IEEEeqnarraydblrulerowcut[#1][\doublerulesep]}}
 \def\@@IEEEeqnarraydblrulerowcut[#1][#2]{\def\@IEEEeqnarraydblrulerowARG{#1}%
-% we allow the user to say \IEEEeqnarraydblrulerow[][]
-\ifx\@IEEEeqnarraydblrulerowARG\@empty%
-\@IEEEeqnarrayrulerow[\arrayrulewidth]%
-\else%
-\@IEEEeqnarrayrulerow[#1]%
-\fi%
-\def\@IEEEeqnarraydblrulerowARG{#2}%
-\ifx\@IEEEeqnarraydblrulerowARG\@empty%
-\\\IEEEeqnarrayseprowcut[\doublerulesep][\relax]%
-\else%
-\\\IEEEeqnarrayseprowcut[#2][\relax]%
-\fi%
-\\\multispan{\@IEEEeqnnumcols}%
-% advance column counter only if the IEEEeqnarray environment wants it
-\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
-\def\@IEEEeqnarraydblrulerowARG{#1}%
-\ifx\@IEEEeqnarraydblrulerowARG\@empty%
-\@IEEEeqnarrayrulerow[\arrayrulewidth]%
-\else%
-\@IEEEeqnarrayrulerow[#1]%
-\fi%
+    % we allow the user to say \IEEEeqnarraydblrulerow[][]
+    \ifx\@IEEEeqnarraydblrulerowARG\@empty%
+        \@IEEEeqnarrayrulerow[\arrayrulewidth]%
+    \else%
+        \@IEEEeqnarrayrulerow[#1]%
+    \fi%
+    \def\@IEEEeqnarraydblrulerowARG{#2}%
+    \ifx\@IEEEeqnarraydblrulerowARG\@empty%
+        \\\IEEEeqnarrayseprowcut[\doublerulesep][\relax]%
+    \else%
+        \\\IEEEeqnarrayseprowcut[#2][\relax]%
+    \fi%
+    \\\multispan{\@IEEEeqnnumcols}%
+    % advance column counter only if the IEEEeqnarray environment wants it
+    \if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+    \def\@IEEEeqnarraydblrulerowARG{#1}%
+    \ifx\@IEEEeqnarraydblrulerowARG\@empty%
+        \@IEEEeqnarrayrulerow[\arrayrulewidth]%
+    \else%
+        \@IEEEeqnarrayrulerow[#1]%
+    \fi%
 }
 
 
@@ -3369,12 +3379,12 @@ Using a default centering column instead}%
 % relies on \@IEEEeqnnumcols to provide the correct number of columns
 % uses \@IEEEtrantmptoksA, \count0 as scratch registers
 \def\@IEEEeqnarrayhoptolastcolumn{\@IEEEtrantmptoksA={}\count0=1\relax%
-\loop% add cols if the user did not use them all
-\ifnum\count0<\@IEEEeqnnumcols\relax%
-\@IEEEappendtoksA{&}%
-\advance\count0 by 1\relax% update the col count
-\repeat%
-\the\@IEEEtrantmptoksA%execute the &'s
+    \loop% add cols if the user did not use them all
+    \ifnum\count0<\@IEEEeqnnumcols\relax%
+    \@IEEEappendtoksA{&}%
+    \advance\count0 by 1\relax% update the col count
+    \repeat%
+    \the\@IEEEtrantmptoksA%execute the &'s
 }
 
 
@@ -3398,39 +3408,39 @@ Using a default centering column instead}%
 
 % saves the strut height and depth of the master strut
 \def\@IEEEeqnarraymasterstrutsave{\relax%
-\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
-\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
-% remove stretchability
-\dimen0\skip0\relax%
-\dimen2\skip2\relax%
-% save values
-\edef\@IEEEeqnarrayTHEmasterstrutHSAVE{\the\dimen0}%
-\edef\@IEEEeqnarrayTHEmasterstrutDSAVE{\the\dimen2}}
+    \expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
+    \expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
+    % remove stretchability
+    \dimen0\skip0\relax%
+    \dimen2\skip2\relax%
+    % save values
+    \edef\@IEEEeqnarrayTHEmasterstrutHSAVE{\the\dimen0}%
+    \edef\@IEEEeqnarrayTHEmasterstrutDSAVE{\the\dimen2}}
 
 % restores the strut height and depth of the master strut
 \def\@IEEEeqnarraymasterstrutrestore{\relax%
-\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutHSAVE\relax%
-\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutDSAVE\relax%
-% remove stretchability
-\dimen0\skip0\relax%
-\dimen2\skip2\relax%
-% restore values
-\edef\@IEEEeqnarrayTHEmasterstrutheight{\the\dimen0}%
-\edef\@IEEEeqnarrayTHEmasterstrutdepth{\the\dimen2}}
+    \expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutHSAVE\relax%
+    \expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutDSAVE\relax%
+    % remove stretchability
+    \dimen0\skip0\relax%
+    \dimen2\skip2\relax%
+    % restore values
+    \edef\@IEEEeqnarrayTHEmasterstrutheight{\the\dimen0}%
+    \edef\@IEEEeqnarrayTHEmasterstrutdepth{\the\dimen2}}
 
 
 % globally restores the strut height and depth to the 
 % master values and sets the master strut flag to true
 \def\@IEEEeqnarraystrutreset{\relax%
-\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
-\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
-% remove stretchability
-\dimen0\skip0\relax%
-\dimen2\skip2\relax%
-% restore values
-\xdef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
-\xdef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
-\global\@IEEEeqnarrayusemasterstruttrue}
+    \expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
+    \expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
+    % remove stretchability
+    \dimen0\skip0\relax%
+    \dimen2\skip2\relax%
+    % restore values
+    \xdef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+    \xdef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+    \global\@IEEEeqnarrayusemasterstruttrue}
 
 
 % if the master strut is not to be used, make the current
@@ -3439,11 +3449,11 @@ Using a default centering column instead}%
 % this allows user strut commands issued in the last column to be carried
 % into the isolation/strut column
 \def\@IEEEeqnarrayglobalizestrutstatus{\relax%
-\if@IEEEeqnarrayusemasterstrut\else%
-\xdef\@IEEEeqnarrayTHEstrutheight{\@IEEEeqnarrayTHEstrutheight}%
-\xdef\@IEEEeqnarrayTHEstrutdepth{\@IEEEeqnarrayTHEstrutdepth}%
-\global\@IEEEeqnarrayusemasterstrutfalse%
-\fi}
+    \if@IEEEeqnarrayusemasterstrut\else%
+        \xdef\@IEEEeqnarrayTHEstrutheight{\@IEEEeqnarrayTHEstrutheight}%
+        \xdef\@IEEEeqnarrayTHEstrutdepth{\@IEEEeqnarrayTHEstrutdepth}%
+        \global\@IEEEeqnarrayusemasterstrutfalse%
+    \fi}
 
 
 
@@ -3460,36 +3470,36 @@ Using a default centering column instead}%
 % uses scratch registers \skip0, \skip2, \skip3, \dimen0, \dimen2
 \def\IEEEeqnarraystrutsize#1#2{\relax\@ifnextchar[{\@IEEEeqnarraystrutsize{#1}{#2}}{\@IEEEeqnarraystrutsize{#1}{#2}[\relax]}}
 \def\@IEEEeqnarraystrutsize#1#2[#3]{\def\@IEEEeqnarraystrutsizeARG{#1}%
-\ifx\@IEEEeqnarraystrutsizeARG\@empty%
-{\setbox0=\hbox{#3\relax\global\skip3=0.7\normalbaselineskip}}%
-\skip0=\skip3\relax%
-\else% arg one present
-{\setbox0=\hbox{#3\relax\global\skip3=#1\relax}}%
-\skip0=\skip3\relax%
-\fi% if null arg
-\def\@IEEEeqnarraystrutsizeARG{#2}%
-\ifx\@IEEEeqnarraystrutsizeARG\@empty%
-{\setbox0=\hbox{#3\relax\global\skip3=0.3\normalbaselineskip}}%
-\skip2=\skip3\relax%
-\else% arg two present
-{\setbox0=\hbox{#3\relax\global\skip3=#2\relax}}%
-\skip2=\skip3\relax%
-\fi% if null arg
-% remove stretchability, just to be safe
-\dimen0\skip0\relax%
-\dimen2\skip2\relax%
-% dimen0 = height, dimen2 = depth
-\if@IEEEeqnarrayISinner% inner does not touch master strut size
-\edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
-\edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
-\@IEEEeqnarrayusemasterstrutfalse% do not use master
-\else% outer, have to set master strut too
-\edef\@IEEEeqnarrayTHEmasterstrutheight{\the\dimen0}%
-\edef\@IEEEeqnarrayTHEmasterstrutdepth{\the\dimen2}%
-\edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
-\edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
-\@IEEEeqnarrayusemasterstruttrue% use master strut
-\fi}
+    \ifx\@IEEEeqnarraystrutsizeARG\@empty%
+        {\setbox0=\hbox{#3\relax\global\skip3=0.7\normalbaselineskip}}%
+        \skip0=\skip3\relax%
+    \else% arg one present
+        {\setbox0=\hbox{#3\relax\global\skip3=#1\relax}}%
+        \skip0=\skip3\relax%
+    \fi% if null arg
+    \def\@IEEEeqnarraystrutsizeARG{#2}%
+    \ifx\@IEEEeqnarraystrutsizeARG\@empty%
+        {\setbox0=\hbox{#3\relax\global\skip3=0.3\normalbaselineskip}}%
+        \skip2=\skip3\relax%
+    \else% arg two present
+        {\setbox0=\hbox{#3\relax\global\skip3=#2\relax}}%
+        \skip2=\skip3\relax%
+    \fi% if null arg
+    % remove stretchability, just to be safe
+    \dimen0\skip0\relax%
+    \dimen2\skip2\relax%
+    % dimen0 = height, dimen2 = depth
+    \if@IEEEeqnarrayISinner% inner does not touch master strut size
+        \edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+        \edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+        \@IEEEeqnarrayusemasterstrutfalse% do not use master
+    \else% outer, have to set master strut too
+        \edef\@IEEEeqnarrayTHEmasterstrutheight{\the\dimen0}%
+        \edef\@IEEEeqnarrayTHEmasterstrutdepth{\the\dimen2}%
+        \edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+        \edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+        \@IEEEeqnarrayusemasterstruttrue% use master strut
+    \fi}
 
 
 % usage: \IEEEeqnarraystrutsizeadd{added height}{added depth}[font size commands]
@@ -3505,48 +3515,48 @@ Using a default centering column instead}%
 % uses scratch registers \skip0, \skip2, \skip3, \dimen0, \dimen2
 \def\IEEEeqnarraystrutsizeadd#1#2{\relax\@ifnextchar[{\@IEEEeqnarraystrutsizeadd{#1}{#2}}{\@IEEEeqnarraystrutsizeadd{#1}{#2}[\relax]}}
 \def\@IEEEeqnarraystrutsizeadd#1#2[#3]{\def\@IEEEeqnarraystrutsizearg{#1}%
-\ifx\@IEEEeqnarraystrutsizearg\@empty%
-\skip0=0pt\relax%
-\else% arg one present
-{\setbox0=\hbox{#3\relax\global\skip3=#1}}%
-\skip0=\skip3\relax%
-\fi% if null arg
-\def\@IEEEeqnarraystrutsizearg{#2}%
-\ifx\@IEEEeqnarraystrutsizearg\@empty%
-\skip2=0pt\relax%
-\else% arg two present
-{\setbox0=\hbox{#3\relax\global\skip3=#2}}%
-\skip2=\skip3\relax%
-\fi% if null arg
-% remove stretchability, just to be safe
-\dimen0\skip0\relax%
-\dimen2\skip2\relax%
-% dimen0 = height, dimen2 = depth
-\if@IEEEeqnarrayISinner% inner does not touch master strut size
-% get local strut size
-\expandafter\skip0=\@IEEEeqnarrayTHEstrutheight\relax%
-\expandafter\skip2=\@IEEEeqnarrayTHEstrutdepth\relax%
-% add it to the user supplied values
-\advance\dimen0 by \skip0\relax%
-\advance\dimen2 by \skip2\relax%
-% update the local strut size
-\edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
-\edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
-\@IEEEeqnarrayusemasterstrutfalse% do not use master
-\else% outer, have to set master strut too
-% get master strut size
-\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
-\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
-% add it to the user supplied values
-\advance\dimen0 by \skip0\relax%
-\advance\dimen2 by \skip2\relax%
-% update the local and master strut sizes
-\edef\@IEEEeqnarrayTHEmasterstrutheight{\the\dimen0}%
-\edef\@IEEEeqnarrayTHEmasterstrutdepth{\the\dimen2}%
-\edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
-\edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
-\@IEEEeqnarrayusemasterstruttrue% use master strut
-\fi}
+    \ifx\@IEEEeqnarraystrutsizearg\@empty%
+        \skip0=0pt\relax%
+    \else% arg one present
+        {\setbox0=\hbox{#3\relax\global\skip3=#1}}%
+        \skip0=\skip3\relax%
+    \fi% if null arg
+    \def\@IEEEeqnarraystrutsizearg{#2}%
+    \ifx\@IEEEeqnarraystrutsizearg\@empty%
+        \skip2=0pt\relax%
+    \else% arg two present
+        {\setbox0=\hbox{#3\relax\global\skip3=#2}}%
+        \skip2=\skip3\relax%
+    \fi% if null arg
+    % remove stretchability, just to be safe
+    \dimen0\skip0\relax%
+    \dimen2\skip2\relax%
+    % dimen0 = height, dimen2 = depth
+    \if@IEEEeqnarrayISinner% inner does not touch master strut size
+        % get local strut size
+        \expandafter\skip0=\@IEEEeqnarrayTHEstrutheight\relax%
+        \expandafter\skip2=\@IEEEeqnarrayTHEstrutdepth\relax%
+        % add it to the user supplied values
+        \advance\dimen0 by \skip0\relax%
+        \advance\dimen2 by \skip2\relax%
+        % update the local strut size
+        \edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+        \edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+        \@IEEEeqnarrayusemasterstrutfalse% do not use master
+    \else% outer, have to set master strut too
+        % get master strut size
+        \expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
+        \expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
+        % add it to the user supplied values
+        \advance\dimen0 by \skip0\relax%
+        \advance\dimen2 by \skip2\relax%
+        % update the local and master strut sizes
+        \edef\@IEEEeqnarrayTHEmasterstrutheight{\the\dimen0}%
+        \edef\@IEEEeqnarrayTHEmasterstrutdepth{\the\dimen2}%
+        \edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+        \edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+        \@IEEEeqnarrayusemasterstruttrue% use master strut
+    \fi}
 
 
 % allow user a way to see the struts
@@ -3556,24 +3566,24 @@ Using a default centering column instead}%
 % inserts an invisible strut using the master or local strut values
 % uses scratch registers \skip0, \skip2, \dimen0, \dimen2
 \def\@IEEEeqnarrayinsertstrut{\relax%
-\if@IEEEeqnarrayusemasterstrut
-% get master strut size
-\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
-\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
-\else%
-% get local strut size
-\expandafter\skip0=\@IEEEeqnarrayTHEstrutheight\relax%
-\expandafter\skip2=\@IEEEeqnarrayTHEstrutdepth\relax%
-\fi%
-% remove stretchability, probably not needed
-\dimen0\skip0\relax%
-\dimen2\skip2\relax%
-% dimen0 = height, dimen2 = depth
-% allow user to see struts if desired
-\ifIEEEvisiblestruts%
-\vrule width0.2pt height\dimen0 depth\dimen2\relax%
-\else%
-\vrule width0pt height\dimen0 depth\dimen2\relax\fi}
+    \if@IEEEeqnarrayusemasterstrut
+        % get master strut size
+        \expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
+        \expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
+    \else%
+        % get local strut size
+        \expandafter\skip0=\@IEEEeqnarrayTHEstrutheight\relax%
+        \expandafter\skip2=\@IEEEeqnarrayTHEstrutdepth\relax%
+    \fi%
+    % remove stretchability, probably not needed
+    \dimen0\skip0\relax%
+    \dimen2\skip2\relax%
+    % dimen0 = height, dimen2 = depth
+    % allow user to see struts if desired
+    \ifIEEEvisiblestruts%
+        \vrule width0.2pt height\dimen0 depth\dimen2\relax%
+    \else%
+        \vrule width0pt height\dimen0 depth\dimen2\relax\fi}
 
 
 % creates an invisible strut, useable even outside \IEEEeqnarray
@@ -3586,31 +3596,31 @@ Using a default centering column instead}%
 \def\@IEEEstrut[#1]{\relax\@ifnextchar[{\@@IEEEstrut[#1]}{\@@IEEEstrut[#1][0.3\normalbaselineskip]}}
 \def\@@IEEEstrut[#1][#2]{\relax\@ifnextchar[{\@@@IEEEstrut[#1][#2]}{\@@@IEEEstrut[#1][#2][\relax]}}
 \def\@@@IEEEstrut[#1][#2][#3]{\mbox{#3\relax%
-\def\@IEEEstrutARG{#1}%
-\ifx\@IEEEstrutARG\@empty%
-\skip0=0.7\normalbaselineskip\relax%
-\else%
-\skip0=#1\relax%
-\fi%
-\def\@IEEEstrutARG{#2}%
-\ifx\@IEEEstrutARG\@empty%
-\skip2=0.3\normalbaselineskip\relax%
-\else%
-\skip2=#2\relax%
-\fi%
-% remove stretchability, probably not needed
-\dimen0\skip0\relax%
-\dimen2\skip2\relax%
-\ifIEEEvisiblestruts%
-\vrule width0.2pt height\dimen0 depth\dimen2\relax%
-\else%
-\vrule width0.0pt height\dimen0 depth\dimen2\relax\fi}}
+        \def\@IEEEstrutARG{#1}%
+        \ifx\@IEEEstrutARG\@empty%
+            \skip0=0.7\normalbaselineskip\relax%
+        \else%
+            \skip0=#1\relax%
+        \fi%
+        \def\@IEEEstrutARG{#2}%
+        \ifx\@IEEEstrutARG\@empty%
+            \skip2=0.3\normalbaselineskip\relax%
+        \else%
+            \skip2=#2\relax%
+        \fi%
+        % remove stretchability, probably not needed
+        \dimen0\skip0\relax%
+        \dimen2\skip2\relax%
+        \ifIEEEvisiblestruts%
+            \vrule width0.2pt height\dimen0 depth\dimen2\relax%
+        \else%
+            \vrule width0.0pt height\dimen0 depth\dimen2\relax\fi}}
 
 
 % enables strut mode by setting a default strut size and then zeroing the
 % \baselineskip, \lineskip, \lineskiplimit and \jot
 \def\IEEEeqnarraystrutmode{\IEEEeqnarraystrutsize{0.7\normalbaselineskip}{0.3\normalbaselineskip}[\relax]%
-\baselineskip=0pt\lineskip=0pt\lineskiplimit=0pt\jot=0pt}
+    \baselineskip=0pt\lineskip=0pt\lineskiplimit=0pt\jot=0pt}
 
 
 % equation and subequation forms to use to setup hyperref's \@currentHref
@@ -3634,77 +3644,77 @@ Using a default centering column instead}%
 % to document catcode changes.
 \def\@@IEEEeqnarray[#1]{\begingroup\IEEEnormalcatcodes\@@@IEEEeqnarray[#1]}
 \def\@@@IEEEeqnarray[#1]#2{\endgroup
-   % default to showing the equation number or not based on whether or not
-   % the star form was involked
-   \if@IEEEeqnumpersist\global\@eqnswtrue
-   \else% not the star form
-   \global\@eqnswfalse
-   \fi% if star form
-   % provide a basic hyperref \theHequation if this has not already been setup (hyperref not loaded, or no section counter)
-   \@ifundefined{theHequation}{\def\theHequation{\arabic{equation}}}{}\relax
-   % provide dummy hyperref commands in case hyperref is not loaded
-   \providecommand{\Hy@raisedlink}[1]{}\relax
-   \providecommand{\hyper@anchorstart}[1]{}\relax
-   \providecommand{\hyper@anchorend}{}\relax
-   \providecommand{\@currentHref}{}\relax
-   \@IEEEeqnumpreadvfalse% reset eqnpreadv flag
-   \@IEEEsubeqnumpreadvfalse% reset subeqnpreadv flag
-   \@IEEEeqnarrayISinnerfalse% not yet within the lines of the halign
-   \@IEEEeqnarraystrutsize{0pt}{0pt}[\relax]% turn off struts by default
-   \@IEEEeqnarrayusemasterstruttrue% use master strut till user asks otherwise
-   \IEEEvisiblestrutsfalse% diagnostic mode defaults to off
-   % no extra space unless the user specifically requests it
-   \lineskip=0pt\relax
-   \lineskiplimit=0pt\relax
-   \baselineskip=\normalbaselineskip\relax%
-   \jot=\IEEEnormaljot\relax%
-   \mathsurround\z@\relax% no extra spacing around math
-   \@advanceIEEEeqncolcnttrue% advance the col counter for each col the user uses, 
-                             % used in \IEEEeqnarraymulticol and in the preamble build
-   %V1.8 Here we preadvance to the next equation number.
-   % If the user later wants a continued subequation, we can roll back.
-   \global\@IEEEsubeqnnumrollback=\c@IEEEsubequation%
-   \stepcounter{equation}\@IEEEeqnumpreadvtrue% advance equation counter before first line
-   \setcounter{IEEEsubequation}{0}% no subequation yet
-   \let\@IEEEcurrentlabelsave\@currentlabel% save current label as we later change it globally
-   \let\@IEEEcurrentHrefsave\@currentHref% save current href label as we later change it globally
-   \def\@currentlabel{\p@equation\theequation}% redefine the ref label
-   \def\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
-   \IEEEeqnarraydecl\relax% allow a way for the user to make global overrides
-   #1\relax% allow user to override defaults
-   \let\\\@IEEEeqnarraycr% replace newline with one that can put in eqn. numbers
-   \global\@IEEEeqncolcnt\z@% col. count = 0 for first line
-   \@IEEEbuildpreamble{#2}\relax% build the preamble and put it into \@IEEEtrantmptoksA 
-   % put in the column for the equation number
-   \ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi% col separator for those after the first
-   \toks0={##}%
-   % advance the \@IEEEeqncolcnt for the isolation col, this helps with error checking
-   \@IEEEappendtoksA{\global\advance\@IEEEeqncolcnt by 1\relax}%
-   % add the isolation column
-   \@IEEEappendtoksA{\tabskip\z@skip\bgroup\the\toks0\egroup}%
-   % advance the \@IEEEeqncolcnt for the equation number col, this helps with error checking
-   \@IEEEappendtoksA{&\global\advance\@IEEEeqncolcnt by 1\relax}%
-   % add the equation number col to the preamble
-   \@IEEEappendtoksA{\tabskip\z@skip\hb@xt@\z@\bgroup\hss\the\toks0\egroup}%
-   % note \@IEEEeqnnumcols does not count the equation col or isolation col
-   % set the starting tabskip glue as determined by the preamble build
-   \tabskip=\@IEEEBPstartglue\relax
-   % begin the display alignment
-   \@IEEEeqnarrayISinnertrue% commands are now within the lines
-   $$\everycr{}\halign to\displaywidth\bgroup
-   % "exspand" the preamble
-   \span\the\@IEEEtrantmptoksA\cr}
+% default to showing the equation number or not based on whether or not
+% the star form was involked
+\if@IEEEeqnumpersist\global\@eqnswtrue
+\else% not the star form
+    \global\@eqnswfalse
+\fi% if star form
+% provide a basic hyperref \theHequation if this has not already been setup (hyperref not loaded, or no section counter)
+\@ifundefined{theHequation}{\def\theHequation{\arabic{equation}}}{}\relax
+% provide dummy hyperref commands in case hyperref is not loaded
+\providecommand{\Hy@raisedlink}[1]{}\relax
+\providecommand{\hyper@anchorstart}[1]{}\relax
+\providecommand{\hyper@anchorend}{}\relax
+\providecommand{\@currentHref}{}\relax
+\@IEEEeqnumpreadvfalse% reset eqnpreadv flag
+\@IEEEsubeqnumpreadvfalse% reset subeqnpreadv flag
+\@IEEEeqnarrayISinnerfalse% not yet within the lines of the halign
+\@IEEEeqnarraystrutsize{0pt}{0pt}[\relax]% turn off struts by default
+\@IEEEeqnarrayusemasterstruttrue% use master strut till user asks otherwise
+\IEEEvisiblestrutsfalse% diagnostic mode defaults to off
+% no extra space unless the user specifically requests it
+\lineskip=0pt\relax
+\lineskiplimit=0pt\relax
+\baselineskip=\normalbaselineskip\relax%
+\jot=\IEEEnormaljot\relax%
+\mathsurround\z@\relax% no extra spacing around math
+\@advanceIEEEeqncolcnttrue% advance the col counter for each col the user uses, 
+% used in \IEEEeqnarraymulticol and in the preamble build
+%V1.8 Here we preadvance to the next equation number.
+% If the user later wants a continued subequation, we can roll back.
+\global\@IEEEsubeqnnumrollback=\c@IEEEsubequation%
+\stepcounter{equation}\@IEEEeqnumpreadvtrue% advance equation counter before first line
+\setcounter{IEEEsubequation}{0}% no subequation yet
+\let\@IEEEcurrentlabelsave\@currentlabel% save current label as we later change it globally
+\let\@IEEEcurrentHrefsave\@currentHref% save current href label as we later change it globally
+\def\@currentlabel{\p@equation\theequation}% redefine the ref label
+\def\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
+\IEEEeqnarraydecl\relax% allow a way for the user to make global overrides
+#1\relax% allow user to override defaults
+\let\\\@IEEEeqnarraycr% replace newline with one that can put in eqn. numbers
+\global\@IEEEeqncolcnt\z@% col. count = 0 for first line
+\@IEEEbuildpreamble{#2}\relax% build the preamble and put it into \@IEEEtrantmptoksA 
+% put in the column for the equation number
+\ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi% col separator for those after the first
+\toks0={##}%
+% advance the \@IEEEeqncolcnt for the isolation col, this helps with error checking
+\@IEEEappendtoksA{\global\advance\@IEEEeqncolcnt by 1\relax}%
+% add the isolation column
+\@IEEEappendtoksA{\tabskip\z@skip\bgroup\the\toks0\egroup}%
+% advance the \@IEEEeqncolcnt for the equation number col, this helps with error checking
+\@IEEEappendtoksA{&\global\advance\@IEEEeqncolcnt by 1\relax}%
+% add the equation number col to the preamble
+\@IEEEappendtoksA{\tabskip\z@skip\hb@xt@\z@\bgroup\hss\the\toks0\egroup}%
+% note \@IEEEeqnnumcols does not count the equation col or isolation col
+% set the starting tabskip glue as determined by the preamble build
+\tabskip=\@IEEEBPstartglue\relax
+% begin the display alignment
+\@IEEEeqnarrayISinnertrue% commands are now within the lines
+$$\everycr{}\halign to\displaywidth\bgroup
+% "exspand" the preamble
+\span\the\@IEEEtrantmptoksA\cr}
 
 % enter isolation/strut column (or the next column if the user did not use
 % every column), record the strut status, complete the columns, do the strut if needed,
 % restore counters (to backout any equation setup for a next line that was never used)
 % to their correct values and exit
 \def\end@IEEEeqnarray{\@IEEEeqnarrayglobalizestrutstatus&\@@IEEEeqnarraycr\egroup
-\if@IEEEsubeqnumpreadv\global\advance\c@IEEEsubequation\m@ne\fi
-\if@IEEEeqnumpreadv\global\advance\c@equation\m@ne\global\c@IEEEsubequation=\@IEEEsubeqnnumrollback\fi
-\global\let\@currentlabel\@IEEEcurrentlabelsave% restore current label
-\global\let\@currentHref\@IEEEcurrentHrefsave% restore current href label
-$$\@ignoretrue}
+    \if@IEEEsubeqnumpreadv\global\advance\c@IEEEsubequation\m@ne\fi
+    \if@IEEEeqnumpreadv\global\advance\c@equation\m@ne\global\c@IEEEsubequation=\@IEEEsubeqnnumrollback\fi
+    \global\let\@currentlabel\@IEEEcurrentlabelsave% restore current label
+    \global\let\@currentHref\@IEEEcurrentHrefsave% restore current href label
+    $$\@ignoretrue}
 
 
 % IEEEeqnarray uses a modifed \\ instead of the plain \cr to
@@ -3714,35 +3724,35 @@ $$\@ignoretrue}
 % altering spacing of its math, so we use & to advance to the next column
 % as there is an isolation/strut column after the user's columns
 \def\@IEEEeqnarraycr{\@IEEEeqnarrayglobalizestrutstatus&% save strut status and advance to next column
-   {\ifnum0=`}\fi
-   \@ifstar{%
-      \global\@eqpen\@M\@IEEEeqnarrayYCR
-   }{%
-      \global\@eqpen\interdisplaylinepenalty \@IEEEeqnarrayYCR
-   }%
-}
-
-\def\@IEEEeqnarrayYCR{\@testopt\@IEEEeqnarrayXCR\z@skip}
-
-\def\@IEEEeqnarrayXCR[#1]{%
-   \ifnum0=`{\fi}%
-   \@@IEEEeqnarraycr
-   \noalign{\penalty\@eqpen\vskip\jot\vskip #1\relax}}%
+    {\ifnum0=`}\fi
+            \@ifstar{%
+                \global\@eqpen\@M\@IEEEeqnarrayYCR
+            }{%
+                \global\@eqpen\interdisplaylinepenalty \@IEEEeqnarrayYCR
+            }%
+        }
+    
+    \def\@IEEEeqnarrayYCR{\@testopt\@IEEEeqnarrayXCR\z@skip}
+    
+    \def\@IEEEeqnarrayXCR[#1]{%
+        \ifnum0=`{\fi}%
+    \@@IEEEeqnarraycr
+    \noalign{\penalty\@eqpen\vskip\jot\vskip #1\relax}}%
 
 \def\@@IEEEeqnarraycr{\@IEEEtrantmptoksA={}% clear token register
     \advance\@IEEEeqncolcnt by -1\relax% adjust col count because of the isolation column
     \ifnum\@IEEEeqncolcnt>\@IEEEeqnnumcols\relax
     \@IEEEclspkgerror{Too many columns within the IEEEeqnarray\MessageBreak
-                          environment}%
-    {Use fewer \string &'s or put more columns in the IEEEeqnarray column\MessageBreak 
-     specifications.}\relax%
+        environment}%
+    {Use fewer \string &'s or put more columns in the IEEEeqnarray column\MessageBreak
+        specifications.}\relax%
     \else
     \loop% add cols if the user did not use them all
     \ifnum\@IEEEeqncolcnt<\@IEEEeqnnumcols\relax
-    \@IEEEappendtoksA{&}%
-    \advance\@IEEEeqncolcnt by 1\relax% update the col count
-    \repeat
-    % this number of &'s will take us the the isolation column
+        \@IEEEappendtoksA{&}%
+        \advance\@IEEEeqncolcnt by 1\relax% update the col count
+        \repeat
+        % this number of &'s will take us the the isolation column
     \fi
     % execute the &'s
     \the\@IEEEtrantmptoksA%
@@ -3751,49 +3761,49 @@ $$\@ignoretrue}
     \@IEEEeqnarraystrutreset% reset the strut system for next line or IEEEeqnarray
     &% and enter the equation number column
     \if@eqnsw% only if we display something
-      \Hy@raisedlink{\hyper@anchorstart{\@currentHref}}% start a hyperref anchor
-      \global\@IEEEeqnumpreadvfalse\relax% displaying an equation number means
-      \global\@IEEEsubeqnumpreadvfalse\relax% the equation counters point to valid equations
-      % V1.8 Here we setup the counters, currentlabel and status for what would be the *next*
-      % equation line as would be the case under the current settings. However, there are two problems.
-      % One problem is that there might not ever be a next line. The second problem is that the user
-      % may later alter the meaning of a line with commands such as \IEEEyessubnumber. So, to handle
-      % these cases we have to record the current values of the (sub)equation counters and revert back
-      % to them if the next line is changed or never comes. The \if@IEEEeqnumpreadv, \if@IEEEsubeqnumpreadv
-      % and \@IEEEsubeqnnumrollback stuff tracks this.
-      % The logic to handle all this is surprisingly complex, but a nice feature of the approach here is
-      % that the equation counters and labels remain valid for what the line would be unless a
-      % \IEEEyessubnumber et al. later changes it. So, any hyperref links are always correct.
-      \ifnum\c@IEEEsubequation>0\relax% handle subequation
-         \theIEEEsubequationdis\relax
-         \if@IEEEsubeqnumpersist% setup for default type of next line
-            \stepcounter{IEEEsubequation}\global\@IEEEsubeqnumpreadvtrue\relax
-            \gdef\@currentlabel{\p@IEEEsubequation\theIEEEsubequation}\relax
-            \gdef\@currentHref{\@IEEEtheHrefsubequation}% setup hyperref label
-         \else
-             % if no subeqnum persist, go ahead and setup for a new equation number
-             \global\@IEEEsubeqnnumrollback=\c@IEEEsubequation
-             \stepcounter{equation}\global\@IEEEeqnumpreadvtrue\relax
-             \setcounter{IEEEsubequation}{0}\gdef\@currentlabel{\p@equation\theequation}\relax
-             \gdef\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
-         \fi
-      \else% display a standard equation number
-        \theequationdis\relax
-        \setcounter{IEEEsubequation}{0}\relax% not really needed
-        \if@IEEEsubeqnumpersist% setup for default type of next line
-           % subequations that follow plain equations carry the same equation number e.g, 5, 5a rather than 5, 6a
-           \stepcounter{IEEEsubequation}\global\@IEEEsubeqnumpreadvtrue\relax
-           \gdef\@currentlabel{\p@IEEEsubequation\theIEEEsubequation}\relax
-           \gdef\@currentHref{\@IEEEtheHrefsubequation}% setup hyperref label
-         \else
-             % if no subeqnum persist, go ahead and setup for a new equation number
-             \global\@IEEEsubeqnnumrollback=\c@IEEEsubequation
-             \stepcounter{equation}\global\@IEEEeqnumpreadvtrue\relax
-             \setcounter{IEEEsubequation}{0}\gdef\@currentlabel{\p@equation\theequation}\relax
-             \gdef\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
-         \fi
-      \fi%
-      \Hy@raisedlink{\hyper@anchorend}% end hyperref anchor
+        \Hy@raisedlink{\hyper@anchorstart{\@currentHref}}% start a hyperref anchor
+        \global\@IEEEeqnumpreadvfalse\relax% displaying an equation number means
+        \global\@IEEEsubeqnumpreadvfalse\relax% the equation counters point to valid equations
+        % V1.8 Here we setup the counters, currentlabel and status for what would be the *next*
+        % equation line as would be the case under the current settings. However, there are two problems.
+        % One problem is that there might not ever be a next line. The second problem is that the user
+        % may later alter the meaning of a line with commands such as \IEEEyessubnumber. So, to handle
+        % these cases we have to record the current values of the (sub)equation counters and revert back
+        % to them if the next line is changed or never comes. The \if@IEEEeqnumpreadv, \if@IEEEsubeqnumpreadv
+        % and \@IEEEsubeqnnumrollback stuff tracks this.
+        % The logic to handle all this is surprisingly complex, but a nice feature of the approach here is
+        % that the equation counters and labels remain valid for what the line would be unless a
+        % \IEEEyessubnumber et al. later changes it. So, any hyperref links are always correct.
+        \ifnum\c@IEEEsubequation>0\relax% handle subequation
+            \theIEEEsubequationdis\relax
+            \if@IEEEsubeqnumpersist% setup for default type of next line
+                \stepcounter{IEEEsubequation}\global\@IEEEsubeqnumpreadvtrue\relax
+                \gdef\@currentlabel{\p@IEEEsubequation\theIEEEsubequation}\relax
+                \gdef\@currentHref{\@IEEEtheHrefsubequation}% setup hyperref label
+            \else
+                % if no subeqnum persist, go ahead and setup for a new equation number
+                \global\@IEEEsubeqnnumrollback=\c@IEEEsubequation
+                \stepcounter{equation}\global\@IEEEeqnumpreadvtrue\relax
+                \setcounter{IEEEsubequation}{0}\gdef\@currentlabel{\p@equation\theequation}\relax
+                \gdef\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
+            \fi
+        \else% display a standard equation number
+            \theequationdis\relax
+            \setcounter{IEEEsubequation}{0}\relax% not really needed
+            \if@IEEEsubeqnumpersist% setup for default type of next line
+                % subequations that follow plain equations carry the same equation number e.g, 5, 5a rather than 5, 6a
+                \stepcounter{IEEEsubequation}\global\@IEEEsubeqnumpreadvtrue\relax
+                \gdef\@currentlabel{\p@IEEEsubequation\theIEEEsubequation}\relax
+                \gdef\@currentHref{\@IEEEtheHrefsubequation}% setup hyperref label
+            \else
+                % if no subeqnum persist, go ahead and setup for a new equation number
+                \global\@IEEEsubeqnnumrollback=\c@IEEEsubequation
+                \stepcounter{equation}\global\@IEEEeqnumpreadvtrue\relax
+                \setcounter{IEEEsubequation}{0}\gdef\@currentlabel{\p@equation\theequation}\relax
+                \gdef\@currentHref{\@IEEEtheHrefequation}% setup hyperref label
+            \fi
+        \fi%
+        \Hy@raisedlink{\hyper@anchorend}% end hyperref anchor
     \fi% fi only if we display something
     % reset the flags to indicate the default preferences of the display of equation numbers
     \if@IEEEeqnumpersist\global\@eqnswtrue\else\global\@eqnswfalse\fi
@@ -3832,11 +3842,11 @@ $$\@ignoretrue}
 \@namedef{endIEEEeqnarrayboxt*}{\end@IEEEeqnarraybox}
 
 \def\IEEEeqnarraybox{\@IEEEeqnarrayboxnojotfalse\ifmmode\@IEEEeqnarrayboxHBOXSWfalse\else\@IEEEeqnarrayboxHBOXSWtrue\fi%
-\@IEEEeqnarraybox}
+    \@IEEEeqnarraybox}
 \def\endIEEEeqnarraybox{\end@IEEEeqnarraybox}
 
 \@namedef{IEEEeqnarraybox*}{\@IEEEeqnarrayboxnojottrue\ifmmode\@IEEEeqnarrayboxHBOXSWfalse\else\@IEEEeqnarrayboxHBOXSWtrue\fi%
-\@IEEEeqnarraybox}
+    \@IEEEeqnarraybox}
 \@namedef{endIEEEeqnarraybox*}{\end@IEEEeqnarraybox}
 
 % flag to indicate if the \IEEEeqnarraybox needs to put things into an hbox{$ $} 
@@ -3853,65 +3863,65 @@ $$\@ignoretrue}
 
 % #1 = decl; #2 = t,b,c; #3 = width, #4 = col specs
 \def\@@@@IEEEeqnarraybox[#1][#2][#3]#4{\endgroup\@IEEEeqnarrayISinnerfalse % not yet within the lines of the halign
-   \@IEEEeqnarraymasterstrutsave% save current master strut values
-   \@IEEEeqnarraystrutsize{0pt}{0pt}[\relax]% turn off struts by default
-   \@IEEEeqnarrayusemasterstruttrue% use master strut till user asks otherwise
-   \IEEEvisiblestrutsfalse% diagnostic mode defaults to off
-   % no extra space unless the user specifically requests it
-   \lineskip=0pt\relax%
-   \lineskiplimit=0pt\relax%
-   \baselineskip=\normalbaselineskip\relax%
-   \jot=\IEEEnormaljot\relax%
-   \mathsurround\z@\relax% no extra spacing around math
-   % the default end glues are zero for an \IEEEeqnarraybox
-   \edef\@IEEEeqnarraycolSEPdefaultstart{\@IEEEeqnarraycolSEPzero}% default start glue
-   \edef\@IEEEeqnarraycolSEPdefaultend{\@IEEEeqnarraycolSEPzero}% default end glue
-   \edef\@IEEEeqnarraycolSEPdefaultmid{\@IEEEeqnarraycolSEPzero}% default inter-column glue
-   \@advanceIEEEeqncolcntfalse% do not advance the col counter for each col the user uses, 
-                              % used in \IEEEeqnarraymulticol and in the preamble build
-   \IEEEeqnarrayboxdecl\relax% allow a way for the user to make global overrides
-   #1\relax% allow user to override defaults
-   \let\\\@IEEEeqnarrayboxcr% replace newline with one that allows optional spacing
-   \@IEEEbuildpreamble{#4}\relax% build the preamble and put it into \@IEEEtrantmptoksA
-   % add an isolation column to the preamble to stop \\'s {} from getting into the last col
-   \ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi% col separator for those after the first
-   \toks0={##}%
-   % add the isolation column to the preamble
-   \@IEEEappendtoksA{\tabskip\z@skip\bgroup\the\toks0\egroup}% 
-   % set the starting tabskip glue as determined by the preamble build
-   \tabskip=\@IEEEBPstartglue\relax
-   % begin the alignment
-   \everycr{}%
-   % use only the very first token to determine the positioning
-   \@IEEEextracttoken{#2}\relax
-   \ifx\@IEEEextractedtokensdiscarded\@empty\else
-     \typeout{** WARNING: IEEEeqnarraybox position specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
-   \fi
-   % \@IEEEextractedtoken has the first token, the rest are ignored
-   % if we need to put things into and hbox and go into math mode, do so now
-   \if@IEEEeqnarrayboxHBOXSW \leavevmode \hbox \bgroup $\fi%
-   % use the appropriate vbox type
-   \if\@IEEEextractedtoken t\relax\vtop\else\if\@IEEEextractedtoken c\relax%
-   \vcenter\else\vbox\fi\fi\bgroup%
-   \@IEEEeqnarrayISinnertrue% commands are now within the lines
-   \ifx#3\relax\halign\else\halign to #3\relax\fi%
-   \bgroup
-   % "exspand" the preamble
-   \span\the\@IEEEtrantmptoksA\cr}
+\@IEEEeqnarraymasterstrutsave% save current master strut values
+\@IEEEeqnarraystrutsize{0pt}{0pt}[\relax]% turn off struts by default
+\@IEEEeqnarrayusemasterstruttrue% use master strut till user asks otherwise
+\IEEEvisiblestrutsfalse% diagnostic mode defaults to off
+% no extra space unless the user specifically requests it
+\lineskip=0pt\relax%
+\lineskiplimit=0pt\relax%
+\baselineskip=\normalbaselineskip\relax%
+\jot=\IEEEnormaljot\relax%
+\mathsurround\z@\relax% no extra spacing around math
+% the default end glues are zero for an \IEEEeqnarraybox
+\edef\@IEEEeqnarraycolSEPdefaultstart{\@IEEEeqnarraycolSEPzero}% default start glue
+\edef\@IEEEeqnarraycolSEPdefaultend{\@IEEEeqnarraycolSEPzero}% default end glue
+\edef\@IEEEeqnarraycolSEPdefaultmid{\@IEEEeqnarraycolSEPzero}% default inter-column glue
+\@advanceIEEEeqncolcntfalse% do not advance the col counter for each col the user uses, 
+% used in \IEEEeqnarraymulticol and in the preamble build
+\IEEEeqnarrayboxdecl\relax% allow a way for the user to make global overrides
+#1\relax% allow user to override defaults
+\let\\\@IEEEeqnarrayboxcr% replace newline with one that allows optional spacing
+\@IEEEbuildpreamble{#4}\relax% build the preamble and put it into \@IEEEtrantmptoksA
+% add an isolation column to the preamble to stop \\'s {} from getting into the last col
+\ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi% col separator for those after the first
+\toks0={##}%
+% add the isolation column to the preamble
+\@IEEEappendtoksA{\tabskip\z@skip\bgroup\the\toks0\egroup}% 
+% set the starting tabskip glue as determined by the preamble build
+\tabskip=\@IEEEBPstartglue\relax
+% begin the alignment
+\everycr{}%
+% use only the very first token to determine the positioning
+\@IEEEextracttoken{#2}\relax
+\ifx\@IEEEextractedtokensdiscarded\@empty\else
+    \typeout{** WARNING: IEEEeqnarraybox position specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+\fi
+% \@IEEEextractedtoken has the first token, the rest are ignored
+% if we need to put things into and hbox and go into math mode, do so now
+\if@IEEEeqnarrayboxHBOXSW \leavevmode \hbox \bgroup $\fi%
+% use the appropriate vbox type
+\if\@IEEEextractedtoken t\relax\vtop\else\if\@IEEEextractedtoken c\relax%
+        \vcenter\else\vbox\fi\fi\bgroup%
+\@IEEEeqnarrayISinnertrue% commands are now within the lines
+\ifx#3\relax\halign\else\halign to #3\relax\fi%
+\bgroup
+% "exspand" the preamble
+\span\the\@IEEEtrantmptoksA\cr}
 
 % carry strut status and enter the isolation/strut column, 
 % exit from math mode if needed, and exit
 \def\end@IEEEeqnarraybox{\@IEEEeqnarrayglobalizestrutstatus% carry strut status
-&% enter isolation/strut column
-\@IEEEeqnarrayinsertstrut% do strut if needed
-\@IEEEeqnarraymasterstrutrestore% restore the previous master strut values
-% reset the strut system for next IEEEeqnarray
-% (sets local strut values back to previous master strut values)
-\@IEEEeqnarraystrutreset%
-% ensure last line, exit from halign, close vbox
-\crcr\egroup\egroup%
-% exit from math mode and close hbox if needed
-\if@IEEEeqnarrayboxHBOXSW $\egroup\fi}
+    &% enter isolation/strut column
+    \@IEEEeqnarrayinsertstrut% do strut if needed
+    \@IEEEeqnarraymasterstrutrestore% restore the previous master strut values
+    % reset the strut system for next IEEEeqnarray
+    % (sets local strut values back to previous master strut values)
+    \@IEEEeqnarraystrutreset%
+    % ensure last line, exit from halign, close vbox
+    \crcr\egroup\egroup%
+    % exit from math mode and close hbox if needed
+    \if@IEEEeqnarrayboxHBOXSW $\egroup\fi}
 
 
 
@@ -3923,19 +3933,19 @@ $$\@ignoretrue}
 % altering spacing of its math, so we use & to advance to the isolation/strut column
 % carry strut status into isolation/strut column
 \def\@IEEEeqnarrayboxcr{\@IEEEeqnarrayglobalizestrutstatus% carry strut status
-&% enter isolation/strut column
-\@IEEEeqnarrayinsertstrut% do strut if needed
-% reset the strut system for next line or IEEEeqnarray
-\@IEEEeqnarraystrutreset%
-{\ifnum0=`}\fi%
-\@ifstar{\@IEEEeqnarrayboxYCR}{\@IEEEeqnarrayboxYCR}}
-
-% test and setup the optional argument to \\[]
-\def\@IEEEeqnarrayboxYCR{\@testopt\@IEEEeqnarrayboxXCR\z@skip}
-
-% IEEEeqnarraybox does not automatically increase line spacing by \jot
-\def\@IEEEeqnarrayboxXCR[#1]{\ifnum0=`{\fi}%
-\cr\noalign{\if@IEEEeqnarrayboxnojot\else\vskip\jot\fi\vskip#1\relax}}
+    &% enter isolation/strut column
+    \@IEEEeqnarrayinsertstrut% do strut if needed
+    % reset the strut system for next line or IEEEeqnarray
+    \@IEEEeqnarraystrutreset%
+    {\ifnum0=`}\fi%
+        \@ifstar{\@IEEEeqnarrayboxYCR}{\@IEEEeqnarrayboxYCR}}
+    
+    % test and setup the optional argument to \\[]
+    \def\@IEEEeqnarrayboxYCR{\@testopt\@IEEEeqnarrayboxXCR\z@skip}
+    
+    % IEEEeqnarraybox does not automatically increase line spacing by \jot
+    \def\@IEEEeqnarrayboxXCR[#1]{\ifnum0=`{\fi}%
+    \cr\noalign{\if@IEEEeqnarrayboxnojot\else\vskip\jot\fi\vskip#1\relax}}
 
 
 
@@ -3968,26 +3978,26 @@ $$\@ignoretrue}
 % usage: \@@IEEEbuildpreamble{current column}{next column}
 % parses and builds the halign preamble
 \def\@@IEEEbuildpreamble#1#2{\let\@@nextIEEEbuildpreamble=\@@IEEEbuildpreamble%
-% use only the very first token to check the end
-\@IEEEextracttokengroups{#1}\relax
-\ifx\@IEEEextractedfirsttoken\ignorespaces\let\@@nextIEEEbuildpreamble=\@@IEEEfinishpreamble\else%
-% identify current and next token type
-\@IEEEgetcoltype{#1}{\@IEEEBPcurtype}{1}% current, error on invalid
-\@IEEEgetcoltype{#2}{\@IEEEBPnexttype}{0}% next, no error on invalid next
-% if curtype is a glue, get the glue def
-\if\@IEEEBPcurtype g\@IEEEgetcurglue{#1}{\@IEEEBPcurglue}\fi%
-% if curtype is a column, get the column def and set the current column name
-\if\@IEEEBPcurtype c\@IEEEgetcurcol{#1}\fi%
-% if curtype is a numeral, acquire the user defined glue
-\if\@IEEEBPcurtype n\@IEEEprocessNcol{#1}\fi%
-% process the acquired glue 
-\if\@IEEEBPcurtype g\@IEEEprocessGcol\fi%
-% process the acquired col 
-\if\@IEEEBPcurtype c\@IEEEprocessCcol\fi%
-% ready prevtype for next col spec.
-\let\@IEEEBPprevtype=\@IEEEBPcurtype%
-% be sure and put back the future token(s) as a group
-\fi\@@nextIEEEbuildpreamble{#2}}
+    % use only the very first token to check the end
+    \@IEEEextracttokengroups{#1}\relax
+    \ifx\@IEEEextractedfirsttoken\ignorespaces\let\@@nextIEEEbuildpreamble=\@@IEEEfinishpreamble\else%
+        % identify current and next token type
+        \@IEEEgetcoltype{#1}{\@IEEEBPcurtype}{1}% current, error on invalid
+        \@IEEEgetcoltype{#2}{\@IEEEBPnexttype}{0}% next, no error on invalid next
+        % if curtype is a glue, get the glue def
+        \if\@IEEEBPcurtype g\@IEEEgetcurglue{#1}{\@IEEEBPcurglue}\fi%
+        % if curtype is a column, get the column def and set the current column name
+        \if\@IEEEBPcurtype c\@IEEEgetcurcol{#1}\fi%
+        % if curtype is a numeral, acquire the user defined glue
+        \if\@IEEEBPcurtype n\@IEEEprocessNcol{#1}\fi%
+        % process the acquired glue 
+        \if\@IEEEBPcurtype g\@IEEEprocessGcol\fi%
+        % process the acquired col 
+        \if\@IEEEBPcurtype c\@IEEEprocessCcol\fi%
+        % ready prevtype for next col spec.
+        \let\@IEEEBPprevtype=\@IEEEBPcurtype%
+        % be sure and put back the future token(s) as a group
+    \fi\@@nextIEEEbuildpreamble{#2}}
 
 
 % usage: \@@IEEEfinishpreamble{discarded}
@@ -3995,11 +4005,11 @@ $$\@ignoretrue}
 % warn about zero cols, and if prevtype type = u, put in end tabskip glue
 % argument is not used
 \def\@@IEEEfinishpreamble#1{\ifnum\@IEEEeqnnumcols<1\relax
-\@IEEEclspkgerror{No column specifiers declared for IEEEeqnarray}%
-{At least one column type must be declared for each IEEEeqnarray.}%
-\fi%num cols less than 1
-%if last type undefined, set default end tabskip glue
-\if\@IEEEBPprevtype u\@IEEEappendtoksA{\tabskip=\@IEEEBPendglue}\fi}
+        \@IEEEclspkgerror{No column specifiers declared for IEEEeqnarray}%
+        {At least one column type must be declared for each IEEEeqnarray.}%
+    \fi%num cols less than 1
+    %if last type undefined, set default end tabskip glue
+    \if\@IEEEBPprevtype u\@IEEEappendtoksA{\tabskip=\@IEEEBPendglue}\fi}
 
 
 % usage: \@IEEEgetcoltype{col specifier}{\output}{error more}
@@ -4012,28 +4022,28 @@ $$\@ignoretrue}
 % u = undefined
 % error mode: 0 = no error message, 1 = error on invalid char
 \def\@IEEEgetcoltype#1#2#3{%
-% use only the very first token to determine the type
-\@IEEEextracttoken{#1}\relax
-% \@IEEEextractedtoken has the first token, the rest are discarded
-\let#2=u\relax% assume invalid until know otherwise
-\ifx\@IEEEextractedtoken\ignorespaces\let#2=e\else
-\ifcat\@IEEEextractedtoken\relax\else% screen out control sequences
-\if0\@IEEEextractedtoken\let#2=n\else
-\if1\@IEEEextractedtoken\let#2=n\else
-\if2\@IEEEextractedtoken\let#2=n\else
-\if3\@IEEEextractedtoken\let#2=n\else
-\if4\@IEEEextractedtoken\let#2=n\else
-\if5\@IEEEextractedtoken\let#2=n\else
-\if6\@IEEEextractedtoken\let#2=n\else
-\if7\@IEEEextractedtoken\let#2=n\else
-\if8\@IEEEextractedtoken\let#2=n\else
-\if9\@IEEEextractedtoken\let#2=n\else
-\ifcat,\@IEEEextractedtoken\let#2=g\relax
-\else\ifcat a\@IEEEextractedtoken\let#2=c\relax\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
-\if#2u\relax
-\if0\noexpand#3\relax\else\@IEEEclspkgerror{Invalid character in column specifications}%
-{Only letters, numerals and certain other symbols are allowed \MessageBreak
-as IEEEeqnarray column specifiers.}\fi\fi}
+    % use only the very first token to determine the type
+    \@IEEEextracttoken{#1}\relax
+    % \@IEEEextractedtoken has the first token, the rest are discarded
+    \let#2=u\relax% assume invalid until know otherwise
+    \ifx\@IEEEextractedtoken\ignorespaces\let#2=e\else
+    \ifcat\@IEEEextractedtoken\relax\else% screen out control sequences
+    \if0\@IEEEextractedtoken\let#2=n\else
+    \if1\@IEEEextractedtoken\let#2=n\else
+    \if2\@IEEEextractedtoken\let#2=n\else
+    \if3\@IEEEextractedtoken\let#2=n\else
+    \if4\@IEEEextractedtoken\let#2=n\else
+    \if5\@IEEEextractedtoken\let#2=n\else
+    \if6\@IEEEextractedtoken\let#2=n\else
+    \if7\@IEEEextractedtoken\let#2=n\else
+    \if8\@IEEEextractedtoken\let#2=n\else
+    \if9\@IEEEextractedtoken\let#2=n\else
+    \ifcat,\@IEEEextractedtoken\let#2=g\relax
+    \else\ifcat a\@IEEEextractedtoken\let#2=c\relax\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
+    \if#2u\relax
+        \if0\noexpand#3\relax\else\@IEEEclspkgerror{Invalid character in column specifications}%
+            {Only letters, numerals and certain other symbols are allowed \MessageBreak
+                as IEEEeqnarray column specifiers.}\fi\fi}
 
 
 % usage: \@IEEEgetcurcol{col specifier}
@@ -4041,11 +4051,11 @@ as IEEEeqnarray column specifiers.}\fi\fi}
 % and return its name in \@IEEEBPcurcolname
 % if column specifier is invalid, use the default column @IEEEdefault
 \def\@IEEEgetcurcol#1{\expandafter\ifx\csname @IEEEeqnarraycolDEF#1\endcsname\@IEEEeqnarraycolisdefined%
-\def\@IEEEBPcurcolname{#1}\else% invalid column name
-\@IEEEclspkgerror{Invalid column type "#1" in column specifications.\MessageBreak
-Using a default centering column instead}%
-{You must define IEEEeqnarray column types before use.}%
-\def\@IEEEBPcurcolname{@IEEEdefault}\fi}
+        \def\@IEEEBPcurcolname{#1}\else% invalid column name
+        \@IEEEclspkgerror{Invalid column type "#1" in column specifications.\MessageBreak
+            Using a default centering column instead}%
+        {You must define IEEEeqnarray column types before use.}%
+        \def\@IEEEBPcurcolname{@IEEEdefault}\fi}
 
 
 % usage: \@IEEEgetcurglue{glue specifier}{\output}
@@ -4070,7 +4080,7 @@ Using a default centering column instead}%
 % use only the very first token to determine the type
 \@IEEEextracttoken{#1}\relax
 \ifx\@IEEEextractedtokensdiscarded\@empty\else
-  \typeout{** WARNING: IEEEeqnarray predefined inter-column glue type specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
+    \typeout{** WARNING: IEEEeqnarray predefined inter-column glue type specifiers after the first in `\@IEEEextracttokenarg' ignored (line \the\inputlineno).}\relax
 \fi
 % get the math font 1em value
 % LaTeX2e's NFSS2 does not preload the fonts, but \IEEEeqnarray needs
@@ -4098,10 +4108,10 @@ Using a default centering column instead}%
 \if-\@IEEEextractedtoken\edef#2{\@IEEEeqnarraycolSEPzero}\else
 \edef#2{\@IEEEeqnarraycolSEPzero}%
 \@IEEEclspkgerror{Invalid predefined inter-column glue type "#1" in\MessageBreak
-column specifications. Using a default value of\MessageBreak
-0pt instead}%
-{Only !,:;'"./?*+ and - are valid predefined glue types in the\MessageBreak 
-IEEEeqnarray column specifications.}\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
+    column specifications. Using a default value of\MessageBreak
+    0pt instead}%
+{Only !,:;'"./?*+ and - are valid predefined glue types in the\MessageBreak
+    IEEEeqnarray column specifications.}\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
 
 
 % usage: \@IEEEprocessNcol{digit}
@@ -4109,113 +4119,113 @@ IEEEeqnarray column specifications.}\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
 % and look up the corresponding user defined glue value
 % can transform current type from n to g or a as the user defined glue is acquired
 \def\@IEEEprocessNcol#1{\if\@IEEEBPprevtype g%
-\@IEEEclspkgerror{Back-to-back inter-column glue specifiers in column\MessageBreak
-specifications. Ignoring consecutive glue specifiers\MessageBreak
-after the first}%
-{You cannot have two or more glue types next to each other\MessageBreak 
-in the IEEEeqnarray column specifications.}%
-\let\@IEEEBPcurtype=a% abort this glue, future digits will be discarded
-\@IEEEBPcurnum=0\relax%
-\else% if we previously aborted a glue
-\if\@IEEEBPprevtype a\@IEEEBPcurnum=0\let\@IEEEBPcurtype=a%maintain digit abortion
-\else%acquire this number
-% save the previous type before the numerical digits started
-\if\@IEEEBPprevtype n\else\let\@IEEEBPprevsavedtype=\@IEEEBPprevtype\fi%
-\multiply\@IEEEBPcurnum by 10\relax%
-\advance\@IEEEBPcurnum by #1\relax% add in number, \relax is needed to stop TeX's number scan
-\if\@IEEEBPnexttype n\else%close acquisition
-\expandafter\ifx\csname @IEEEeqnarraycolSEPDEF\expandafter\romannumeral\number\@IEEEBPcurnum\endcsname\@IEEEeqnarraycolisdefined%
-\edef\@IEEEBPcurglue{\csname @IEEEeqnarraycolSEP\expandafter\romannumeral\number\@IEEEBPcurnum\endcsname}%
-\else%user glue not defined
-\@IEEEclspkgerror{Invalid user defined inter-column glue type "\number\@IEEEBPcurnum" in\MessageBreak
-column specifications. Using a default value of\MessageBreak
-0pt instead}%
-{You must define all IEEEeqnarray numerical inter-column glue types via\MessageBreak
-\string\IEEEeqnarraydefcolsep \space before they are used in column specifications.}%
-\edef\@IEEEBPcurglue{\@IEEEeqnarraycolSEPzero}%
-\fi% glue defined or not
-\let\@IEEEBPcurtype=g% change the type to reflect the acquired glue
-\let\@IEEEBPprevtype=\@IEEEBPprevsavedtype% restore the prev type before this number glue
-\@IEEEBPcurnum=0\relax%ready for next acquisition
-\fi%close acquisition, get glue
-\fi%discard or acquire number
-\fi%prevtype glue or not
+        \@IEEEclspkgerror{Back-to-back inter-column glue specifiers in column\MessageBreak
+            specifications. Ignoring consecutive glue specifiers\MessageBreak
+            after the first}%
+        {You cannot have two or more glue types next to each other\MessageBreak
+            in the IEEEeqnarray column specifications.}%
+        \let\@IEEEBPcurtype=a% abort this glue, future digits will be discarded
+        \@IEEEBPcurnum=0\relax%
+    \else% if we previously aborted a glue
+        \if\@IEEEBPprevtype a\@IEEEBPcurnum=0\let\@IEEEBPcurtype=a%maintain digit abortion
+        \else%acquire this number
+            % save the previous type before the numerical digits started
+            \if\@IEEEBPprevtype n\else\let\@IEEEBPprevsavedtype=\@IEEEBPprevtype\fi%
+            \multiply\@IEEEBPcurnum by 10\relax%
+            \advance\@IEEEBPcurnum by #1\relax% add in number, \relax is needed to stop TeX's number scan
+            \if\@IEEEBPnexttype n\else%close acquisition
+                \expandafter\ifx\csname @IEEEeqnarraycolSEPDEF\expandafter\romannumeral\number\@IEEEBPcurnum\endcsname\@IEEEeqnarraycolisdefined%
+                    \edef\@IEEEBPcurglue{\csname @IEEEeqnarraycolSEP\expandafter\romannumeral\number\@IEEEBPcurnum\endcsname}%
+                \else%user glue not defined
+                    \@IEEEclspkgerror{Invalid user defined inter-column glue type "\number\@IEEEBPcurnum" in\MessageBreak
+                        column specifications. Using a default value of\MessageBreak
+                        0pt instead}%
+                    {You must define all IEEEeqnarray numerical inter-column glue types via\MessageBreak
+                        \string\IEEEeqnarraydefcolsep \space before they are used in column specifications.}%
+                    \edef\@IEEEBPcurglue{\@IEEEeqnarraycolSEPzero}%
+                \fi% glue defined or not
+                \let\@IEEEBPcurtype=g% change the type to reflect the acquired glue
+                \let\@IEEEBPprevtype=\@IEEEBPprevsavedtype% restore the prev type before this number glue
+                \@IEEEBPcurnum=0\relax%ready for next acquisition
+            \fi%close acquisition, get glue
+        \fi%discard or acquire number
+    \fi%prevtype glue or not
 }
 
 
 % process an acquired glue
 % add any acquired column/glue pair to the preamble
 \def\@IEEEprocessGcol{\if\@IEEEBPprevtype a\let\@IEEEBPcurtype=a%maintain previous glue abortions
-\else
-% if this is the start glue, save it, but do nothing else 
-% as this is not used in the preamble, but before
-\if\@IEEEBPprevtype s\edef\@IEEEBPstartglue{\@IEEEBPcurglue}%
-\else%not the start glue
-\if\@IEEEBPprevtype g%ignore if back to back glues
-\@IEEEclspkgerror{Back-to-back inter-column glue specifiers in column\MessageBreak
-specifications. Ignoring consecutive glue specifiers\MessageBreak
-after the first}%
-{You cannot have two or more glue types next to each other\MessageBreak 
-in the IEEEeqnarray column specifications.}%
-\let\@IEEEBPcurtype=a% abort this glue
-\else% not a back to back glue
-\if\@IEEEBPprevtype c\relax% if the previoustype was a col, add column/glue pair to preamble
-\ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi
-\toks0={##}%
-% make preamble advance col counter if this environment needs this
-\if@advanceIEEEeqncolcnt\@IEEEappendtoksA{\global\advance\@IEEEeqncolcnt by 1\relax}\fi
-% insert the column defintion into the preamble, being careful not to expand
-% the column definition
-\@IEEEappendtoksA{\tabskip=\@IEEEBPcurglue}%
-\@IEEEappendNOEXPANDtoksA{\begingroup\csname @IEEEeqnarraycolPRE}%
-\@IEEEappendtoksA{\@IEEEBPcurcolname}%
-\@IEEEappendNOEXPANDtoksA{\endcsname}%
-\@IEEEappendtoksA{\the\toks0}%
-\@IEEEappendNOEXPANDtoksA{\relax\relax\relax\relax\relax%
-\relax\relax\relax\relax\relax\csname @IEEEeqnarraycolPOST}%
-\@IEEEappendtoksA{\@IEEEBPcurcolname}%
-\@IEEEappendNOEXPANDtoksA{\endcsname\relax\relax\relax\relax\relax%
-\relax\relax\relax\relax\relax\endgroup}%
-\advance\@IEEEeqnnumcols by 1\relax%one more column in the preamble
-\else% error: non-start glue with no pending column
-\@IEEEclspkgerror{Inter-column glue specifier without a prior column\MessageBreak
-type in the column specifications. Ignoring this glue\MessageBreak 
-specifier}%
-{Except for the first and last positions, glue can be placed only\MessageBreak
-between column types.}%
-\let\@IEEEBPcurtype=a% abort this glue
-\fi% previous was a column
-\fi% back-to-back glues
-\fi% is start column glue
-\fi% prev type not a
+    \else
+        % if this is the start glue, save it, but do nothing else 
+        % as this is not used in the preamble, but before
+        \if\@IEEEBPprevtype s\edef\@IEEEBPstartglue{\@IEEEBPcurglue}%
+        \else%not the start glue
+            \if\@IEEEBPprevtype g%ignore if back to back glues
+                \@IEEEclspkgerror{Back-to-back inter-column glue specifiers in column\MessageBreak
+                    specifications. Ignoring consecutive glue specifiers\MessageBreak
+                    after the first}%
+                {You cannot have two or more glue types next to each other\MessageBreak
+                    in the IEEEeqnarray column specifications.}%
+                \let\@IEEEBPcurtype=a% abort this glue
+            \else% not a back to back glue
+                \if\@IEEEBPprevtype c\relax% if the previoustype was a col, add column/glue pair to preamble
+                    \ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi
+                    \toks0={##}%
+                    % make preamble advance col counter if this environment needs this
+                    \if@advanceIEEEeqncolcnt\@IEEEappendtoksA{\global\advance\@IEEEeqncolcnt by 1\relax}\fi
+                    % insert the column defintion into the preamble, being careful not to expand
+                    % the column definition
+                    \@IEEEappendtoksA{\tabskip=\@IEEEBPcurglue}%
+                    \@IEEEappendNOEXPANDtoksA{\begingroup\csname @IEEEeqnarraycolPRE}%
+                    \@IEEEappendtoksA{\@IEEEBPcurcolname}%
+                    \@IEEEappendNOEXPANDtoksA{\endcsname}%
+                    \@IEEEappendtoksA{\the\toks0}%
+                    \@IEEEappendNOEXPANDtoksA{\relax\relax\relax\relax\relax%
+                        \relax\relax\relax\relax\relax\csname @IEEEeqnarraycolPOST}%
+                    \@IEEEappendtoksA{\@IEEEBPcurcolname}%
+                    \@IEEEappendNOEXPANDtoksA{\endcsname\relax\relax\relax\relax\relax%
+                        \relax\relax\relax\relax\relax\endgroup}%
+                    \advance\@IEEEeqnnumcols by 1\relax%one more column in the preamble
+                \else% error: non-start glue with no pending column
+                    \@IEEEclspkgerror{Inter-column glue specifier without a prior column\MessageBreak
+                        type in the column specifications. Ignoring this glue\MessageBreak 
+                        specifier}%
+                    {Except for the first and last positions, glue can be placed only\MessageBreak
+                        between column types.}%
+                    \let\@IEEEBPcurtype=a% abort this glue
+                \fi% previous was a column
+            \fi% back-to-back glues
+        \fi% is start column glue
+    \fi% prev type not a
 }
 
 
 % process an acquired letter referenced column and, if necessary, add it to the preamble
 \def\@IEEEprocessCcol{\if\@IEEEBPnexttype g\else
-\if\@IEEEBPnexttype n\else
-% we have a column followed by something other than a glue (or numeral glue)
-% so we must add this column to the preamble now
-\ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi%col separator for those after the first
-\if\@IEEEBPnexttype e\@IEEEappendtoksA{\tabskip=\@IEEEBPendglue\relax}\else%put in end glue
-\@IEEEappendtoksA{\tabskip=\@IEEEeqnarraycolSEPdefaultmid\relax}\fi% or default mid glue
-\toks0={##}%
-% make preamble advance col counter if this environment needs this
-\if@advanceIEEEeqncolcnt\@IEEEappendtoksA{\global\advance\@IEEEeqncolcnt by 1\relax}\fi
-% insert the column definition into the preamble, being careful not to expand
-% the column definition
-\@IEEEappendNOEXPANDtoksA{\begingroup\csname @IEEEeqnarraycolPRE}%
-\@IEEEappendtoksA{\@IEEEBPcurcolname}%
-\@IEEEappendNOEXPANDtoksA{\endcsname}%
-\@IEEEappendtoksA{\the\toks0}%
-\@IEEEappendNOEXPANDtoksA{\relax\relax\relax\relax\relax%
-\relax\relax\relax\relax\relax\csname @IEEEeqnarraycolPOST}%
-\@IEEEappendtoksA{\@IEEEBPcurcolname}%
-\@IEEEappendNOEXPANDtoksA{\endcsname\relax\relax\relax\relax\relax%
-\relax\relax\relax\relax\relax\endgroup}%
-\advance\@IEEEeqnnumcols by 1\relax%one more column in the preamble
-\fi%next type not numeral
-\fi%next type not glue
+        \if\@IEEEBPnexttype n\else
+            % we have a column followed by something other than a glue (or numeral glue)
+            % so we must add this column to the preamble now
+            \ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi%col separator for those after the first
+            \if\@IEEEBPnexttype e\@IEEEappendtoksA{\tabskip=\@IEEEBPendglue\relax}\else%put in end glue
+                \@IEEEappendtoksA{\tabskip=\@IEEEeqnarraycolSEPdefaultmid\relax}\fi% or default mid glue
+            \toks0={##}%
+            % make preamble advance col counter if this environment needs this
+            \if@advanceIEEEeqncolcnt\@IEEEappendtoksA{\global\advance\@IEEEeqncolcnt by 1\relax}\fi
+            % insert the column definition into the preamble, being careful not to expand
+            % the column definition
+            \@IEEEappendNOEXPANDtoksA{\begingroup\csname @IEEEeqnarraycolPRE}%
+            \@IEEEappendtoksA{\@IEEEBPcurcolname}%
+            \@IEEEappendNOEXPANDtoksA{\endcsname}%
+            \@IEEEappendtoksA{\the\toks0}%
+            \@IEEEappendNOEXPANDtoksA{\relax\relax\relax\relax\relax%
+                \relax\relax\relax\relax\relax\csname @IEEEeqnarraycolPOST}%
+            \@IEEEappendtoksA{\@IEEEBPcurcolname}%
+            \@IEEEappendNOEXPANDtoksA{\endcsname\relax\relax\relax\relax\relax%
+                \relax\relax\relax\relax\relax\endgroup}%
+            \advance\@IEEEeqnnumcols by 1\relax%one more column in the preamble
+        \fi%next type not numeral
+    \fi%next type not glue
 }
 
 
@@ -4235,119 +4245,119 @@ between column types.}%
 %
 % compsoc uses sans-serif headers and footers
 \ifCLASSOPTIONcompsoc
-  \def\@IEEEheaderstyle{\normalfont\sffamily\scriptsize}
-  \def\@IEEEfooterstyle{\normalfont\sffamily\scriptsize}
+    \def\@IEEEheaderstyle{\normalfont\sffamily\scriptsize}
+    \def\@IEEEfooterstyle{\normalfont\sffamily\scriptsize}
 \fi
 
 
 % standard page style, ps@headings
 \def\ps@headings{% default to standard twoside headers, no footers
-% will change later if the mode requires otherwise
-\def\@oddhead{\hbox{}\@IEEEheaderstyle\rightmark\hfil\thepage}\relax
-\def\@evenhead{\@IEEEheaderstyle\thepage\hfil\leftmark\hbox{}}\relax
-\let\@oddfoot\@empty
-\let\@evenfoot\@empty
-\ifCLASSOPTIONtechnote
-  % technote twoside
-  \def\@oddhead{\hbox{}\@IEEEheaderstyle\leftmark\hfil\thepage}\relax
-  \def\@evenhead{\@IEEEheaderstyle\thepage\hfil\leftmark\hbox{}}\relax
-\fi
-\ifCLASSOPTIONdraftcls
-  % draft footers
-  \def\@oddfoot{\@IEEEfooterstyle\@date\hfil DRAFT}\relax
-  \def\@evenfoot{\@IEEEfooterstyle DRAFT\hfil\@date}\relax
-\fi
-% oneside
-\if@twoside\else
-  % standard one side headers
-  \def\@oddhead{\hbox{}\@IEEEheaderstyle\leftmark\hfil\thepage}\relax
-  \let\@evenhead\@empty
-  \ifCLASSOPTIONdraftcls
-    % oneside draft footers
-    \def\@oddfoot{\@IEEEfooterstyle\@date\hfil DRAFT}\relax
+    % will change later if the mode requires otherwise
+    \def\@oddhead{\hbox{}\@IEEEheaderstyle\rightmark\hfil\thepage}\relax
+    \def\@evenhead{\@IEEEheaderstyle\thepage\hfil\leftmark\hbox{}}\relax
+    \let\@oddfoot\@empty
     \let\@evenfoot\@empty
-  \fi
-\fi
-% turn off headers for conferences
-\ifCLASSOPTIONconference
-  \let\@oddhead\@empty
-  \let\@evenhead\@empty
-\fi
-% turn off footers for draftclsnofoot
-\ifCLASSOPTIONdraftclsnofoot
-  \let\@oddfoot\@empty
-  \let\@evenfoot\@empty
-\fi}
+    \ifCLASSOPTIONtechnote
+        % technote twoside
+        \def\@oddhead{\hbox{}\@IEEEheaderstyle\leftmark\hfil\thepage}\relax
+        \def\@evenhead{\@IEEEheaderstyle\thepage\hfil\leftmark\hbox{}}\relax
+    \fi
+    \ifCLASSOPTIONdraftcls
+        % draft footers
+        \def\@oddfoot{\@IEEEfooterstyle\@date\hfil DRAFT}\relax
+        \def\@evenfoot{\@IEEEfooterstyle DRAFT\hfil\@date}\relax
+    \fi
+    % oneside
+    \if@twoside\else
+        % standard one side headers
+        \def\@oddhead{\hbox{}\@IEEEheaderstyle\leftmark\hfil\thepage}\relax
+        \let\@evenhead\@empty
+        \ifCLASSOPTIONdraftcls
+            % oneside draft footers
+            \def\@oddfoot{\@IEEEfooterstyle\@date\hfil DRAFT}\relax
+            \let\@evenfoot\@empty
+        \fi
+    \fi
+    % turn off headers for conferences
+    \ifCLASSOPTIONconference
+        \let\@oddhead\@empty
+        \let\@evenhead\@empty
+    \fi
+    % turn off footers for draftclsnofoot
+    \ifCLASSOPTIONdraftclsnofoot
+        \let\@oddfoot\@empty
+        \let\@evenfoot\@empty
+    \fi}
 
 
 % title page style, ps@IEEEtitlepagestyle
 \def\ps@IEEEtitlepagestyle{% default title page headers, no footers
-\def\@oddhead{\hbox{}\@IEEEheaderstyle\leftmark\hfil\thepage}\relax
-\def\@evenhead{\@IEEEheaderstyle\thepage\hfil\leftmark\hbox{}}\relax
-\let\@oddfoot\@empty
-\let\@evenfoot\@empty
-% will change later if the mode requires otherwise
-\ifCLASSOPTIONdraftcls
-  % draft footers
-  \ifCLASSOPTIONdraftclsnofoot\else
-    % but only if not draftclsnofoot
-    \def\@oddfoot{\@IEEEfooterstyle\@date\hfil DRAFT}\relax
-    \def\@evenfoot{\@IEEEfooterstyle DRAFT\hfil\@date}\relax
-  \fi
-\else
-  % all nondraft mode footers
-  \if@IEEEusingpubid
-    % for title pages that are using a pubid
-    % do not repeat pubid on the title page if using a peer review cover page
-    \ifCLASSOPTIONpeerreview\else
-      % for noncompsoc papers, the pubid uses footnotesize and
-      % is at the same vertical position as where the last baseline would normally be
-      \def\@oddfoot{\hbox{}\hss\@IEEEfooterstyle\footnotesize\raisebox{\footskip}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
-      \def\@evenfoot{\hbox{}\hss\@IEEEfooterstyle\footnotesize\raisebox{\footskip}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
-      \ifCLASSOPTIONcompsoc
-        % for compsoc papers, the pubid is at the same vertical position as the normal footer
-        \def\@oddfoot{\hbox{}\hss\@IEEEfooterstyle\raisebox{0pt}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
-        \def\@evenfoot{\hbox{}\hss\@IEEEfooterstyle\raisebox{0pt}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
-      \fi
+    \def\@oddhead{\hbox{}\@IEEEheaderstyle\leftmark\hfil\thepage}\relax
+    \def\@evenhead{\@IEEEheaderstyle\thepage\hfil\leftmark\hbox{}}\relax
+    \let\@oddfoot\@empty
+    \let\@evenfoot\@empty
+    % will change later if the mode requires otherwise
+    \ifCLASSOPTIONdraftcls
+        % draft footers
+        \ifCLASSOPTIONdraftclsnofoot\else
+            % but only if not draftclsnofoot
+            \def\@oddfoot{\@IEEEfooterstyle\@date\hfil DRAFT}\relax
+            \def\@evenfoot{\@IEEEfooterstyle DRAFT\hfil\@date}\relax
+        \fi
+    \else
+        % all nondraft mode footers
+        \if@IEEEusingpubid
+            % for title pages that are using a pubid
+            % do not repeat pubid on the title page if using a peer review cover page
+            \ifCLASSOPTIONpeerreview\else
+                % for noncompsoc papers, the pubid uses footnotesize and
+                % is at the same vertical position as where the last baseline would normally be
+                \def\@oddfoot{\hbox{}\hss\@IEEEfooterstyle\footnotesize\raisebox{\footskip}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+                \def\@evenfoot{\hbox{}\hss\@IEEEfooterstyle\footnotesize\raisebox{\footskip}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+                \ifCLASSOPTIONcompsoc
+                    % for compsoc papers, the pubid is at the same vertical position as the normal footer
+                    \def\@oddfoot{\hbox{}\hss\@IEEEfooterstyle\raisebox{0pt}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+                    \def\@evenfoot{\hbox{}\hss\@IEEEfooterstyle\raisebox{0pt}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+                \fi
+            \fi
+        \fi
     \fi
-  \fi
-\fi
-% turn off headers for conferences
-\ifCLASSOPTIONconference
-  \let\@oddhead\@empty
-  \let\@evenhead\@empty
-\fi}
+    % turn off headers for conferences
+    \ifCLASSOPTIONconference
+        \let\@oddhead\@empty
+        \let\@evenhead\@empty
+    \fi}
 
 
 % peer review cover page style, ps@IEEEpeerreviewcoverpagestyle
 \def\ps@IEEEpeerreviewcoverpagestyle{% default peer review cover no headers, no footers
-\let\@oddhead\@empty
-\let\@evenhead\@empty
-\let\@oddfoot\@empty
-\let\@evenfoot\@empty
-% will change later if the mode requires otherwise
-\ifCLASSOPTIONdraftcls
-  % draft footers
-  \ifCLASSOPTIONdraftclsnofoot\else
-    % but only if not draftclsnofoot
-    \def\@oddfoot{\@IEEEfooterstyle\@date\hfil DRAFT}\relax
-    \def\@evenfoot{\@IEEEfooterstyle DRAFT\hfil\@date}\relax
-  \fi
-\else
-  % all nondraft mode footers
-  \if@IEEEusingpubid
-    % for peer review cover pages that are using a pubid
-    % for noncompsoc papers, the pubid uses footnotesize and
-    % is at the same vertical position as where the last baseline would normally be
-    \def\@oddfoot{\hbox{}\hss\@IEEEfooterstyle\footnotesize\raisebox{\footskip}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
-    \def\@evenfoot{\hbox{}\hss\@IEEEfooterstyle\footnotesize\raisebox{\footskip}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
-    \ifCLASSOPTIONcompsoc
-      % for compsoc papers, the pubid is at the same vertical position as the normal footer
-      \def\@oddfoot{\hbox{}\hss\@IEEEfooterstyle\raisebox{0pt}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
-      \def\@evenfoot{\hbox{}\hss\@IEEEfooterstyle\raisebox{0pt}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
-    \fi
-  \fi
-\fi}
+    \let\@oddhead\@empty
+    \let\@evenhead\@empty
+    \let\@oddfoot\@empty
+    \let\@evenfoot\@empty
+    % will change later if the mode requires otherwise
+    \ifCLASSOPTIONdraftcls
+        % draft footers
+        \ifCLASSOPTIONdraftclsnofoot\else
+            % but only if not draftclsnofoot
+            \def\@oddfoot{\@IEEEfooterstyle\@date\hfil DRAFT}\relax
+            \def\@evenfoot{\@IEEEfooterstyle DRAFT\hfil\@date}\relax
+        \fi
+    \else
+        % all nondraft mode footers
+        \if@IEEEusingpubid
+            % for peer review cover pages that are using a pubid
+            % for noncompsoc papers, the pubid uses footnotesize and
+            % is at the same vertical position as where the last baseline would normally be
+            \def\@oddfoot{\hbox{}\hss\@IEEEfooterstyle\footnotesize\raisebox{\footskip}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+            \def\@evenfoot{\hbox{}\hss\@IEEEfooterstyle\footnotesize\raisebox{\footskip}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+            \ifCLASSOPTIONcompsoc
+                % for compsoc papers, the pubid is at the same vertical position as the normal footer
+                \def\@oddfoot{\hbox{}\hss\@IEEEfooterstyle\raisebox{0pt}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+                \def\@evenfoot{\hbox{}\hss\@IEEEfooterstyle\raisebox{0pt}[0pt][0pt]{\@IEEEpubid}\hss\hbox{}}\relax
+            \fi
+        \fi
+    \fi}
 
 
 
@@ -4358,11 +4368,11 @@ between column types.}%
 %% arguments to \markboth.
 %% V1.7b add \protect to work with Babel
 \def\markboth#1#2{\def\leftmark{\MakeUppercase{\protect#1}}%
-\def\rightmark{\MakeUppercase{\protect#2}}}
+    \def\rightmark{\MakeUppercase{\protect#2}}}
 
 \def\today{\ifcase\month\or
-    January\or February\or March\or April\or May\or June\or
-    July\or August\or September\or October\or November\or December\fi
+        January\or February\or March\or April\or May\or June\or
+        July\or August\or September\or October\or November\or December\fi
     \space\number\day, \number\year}
 
 
@@ -4375,16 +4385,16 @@ between column types.}%
 % 
 % Modify Latex2e \@citex to separate citations with "], ["
 \def\@citex[#1]#2{%
-  \let\@citea\@empty
-  \@cite{\@for\@citeb:=#2\do
-    {\@citea\def\@citea{], [}%
-     \edef\@citeb{\expandafter\@firstofone\@citeb\@empty}%
-     \if@filesw\immediate\write\@auxout{\string\citation{\@citeb}}\fi
-     \@ifundefined{b@\@citeb}{\mbox{\reset@font\bfseries ?}%
-       \G@refundefinedtrue
-       \@latex@warning
-         {Citation `\@citeb' on page \thepage \space undefined}}%
-       {\hbox{\csname b@\@citeb\endcsname}}}}{#1}}
+\let\@citea\@empty
+\@cite{\@for\@citeb:=#2\do
+{\@citea\def\@citea{], [}%
+\edef\@citeb{\expandafter\@firstofone\@citeb\@empty}%
+\if@filesw\immediate\write\@auxout{\string\citation{\@citeb}}\fi
+\@ifundefined{b@\@citeb}{\mbox{\reset@font\bfseries ?}%
+    \G@refundefinedtrue
+    \@latex@warning
+    {Citation `\@citeb' on page \thepage \space undefined}}%
+{\hbox{\csname b@\@citeb\endcsname}}}}{#1}}
 
 % V1.6 we create hooks for the optional use of Donald Arseneau's
 % cite.sty package. cite.sty is "smart" and will notice that the
@@ -4411,18 +4421,18 @@ between column types.}%
 % V1.8b article.cls is now providing these too
 % we do not use \@mkboth, nor alter the page style
 \newenvironment{theindex}
-               {\if@twocolumn
-                  \@restonecolfalse
-                \else
-                  \@restonecoltrue
-                \fi
-                \twocolumn[\section*{\indexname}]%
-                \parindent\z@
-                \parskip\z@ \@plus .3\p@\relax
-                \columnseprule \z@
-                \columnsep 35\p@
-                \let\item\@idxitem}
-               {\if@restonecol\onecolumn\else\clearpage\fi}
+{\if@twocolumn
+        \@restonecolfalse
+    \else
+        \@restonecoltrue
+    \fi
+    \twocolumn[\section*{\indexname}]%
+    \parindent\z@
+    \parskip\z@ \@plus .3\p@\relax
+    \columnseprule \z@
+    \columnsep 35\p@
+    \let\item\@idxitem}
+{\if@restonecol\onecolumn\else\clearpage\fi}
 \newcommand\@idxitem{\par\hangindent 40\p@}
 \newcommand\subitem{\@idxitem \hspace*{20\p@}}
 \newcommand\subsubitem{\@idxitem \hspace*{30\p@}}
@@ -4437,28 +4447,28 @@ between column types.}%
 % \bstctlcite[@auxoutsec]{BSTcontrolhak}
 \def\bstctlcite{\@ifnextchar[{\@bstctlcite}{\@bstctlcite[@auxout]}}
 \def\@bstctlcite[#1]#2{\@bsphack
-  \@for\@citeb:=#2\do{%
-    \edef\@citeb{\expandafter\@firstofone\@citeb}%
-    \if@filesw\immediate\write\csname #1\endcsname{\string\citation{\@citeb}}\fi}%
-  \@esphack}
+    \@for\@citeb:=#2\do{%
+        \edef\@citeb{\expandafter\@firstofone\@citeb}%
+        \if@filesw\immediate\write\csname #1\endcsname{\string\citation{\@citeb}}\fi}%
+    \@esphack}
 
 % \IEEEnoauxwrite{} allows for citations that do not add to or affect
 % the order of the existing citation list. Can be useful for \cite
 % within \thanks{}.
 \DeclareRobustCommand{\IEEEnoauxwrite}[1]{\relax
-\if@filesw
-\@fileswfalse
-#1\relax\relax\relax\relax\relax
-\@fileswtrue
-\else
-#1\relax\relax\relax\relax\relax
-\fi}
+    \if@filesw
+        \@fileswfalse
+        #1\relax\relax\relax\relax\relax
+        \@fileswtrue
+    \else
+        #1\relax\relax\relax\relax\relax
+    \fi}
 
 % V1.6 provide a way for a user to execute a command just before 
 % a given reference number - used to insert a \newpage to balance
 % the columns on the last page
 \edef\@IEEEtriggerrefnum{0}   % the default of zero means that
-                              % the command is not executed
+% the command is not executed
 \def\@IEEEtriggercmd{\newpage}
 
 % allow the user to alter the triggered command
@@ -4467,12 +4477,12 @@ between column types.}%
 % allow user a way to specify the reference number just before the
 % command is executed
 \def\IEEEtriggeratref#1{\@IEEEtrantmpcountA=#1%
-\edef\@IEEEtriggerrefnum{\the\@IEEEtrantmpcountA}}%
+    \edef\@IEEEtriggerrefnum{\the\@IEEEtrantmpcountA}}%
 
 % trigger command at the given reference
 \def\@IEEEbibitemprefix{\@IEEEtrantmpcountA=\@IEEEtriggerrefnum\relax%
-\advance\@IEEEtrantmpcountA by -1\relax%
-\ifnum\c@enumiv=\@IEEEtrantmpcountA\relax\@IEEEtriggercmd\relax\fi}
+    \advance\@IEEEtrantmpcountA by -1\relax%
+    \ifnum\c@enumiv=\@IEEEtrantmpcountA\relax\@IEEEtriggercmd\relax\fi}
 
 
 \def\@biblabel#1{[#1]}
@@ -4492,25 +4502,25 @@ between column types.}%
     \footnotesize\vskip 0.3\baselineskip plus 0.1\baselineskip minus 0.1\baselineskip%
     \list{\@biblabel{\@arabic\c@enumiv}}%
     {\settowidth\labelwidth{\@biblabel{#1}}%
-    \leftmargin\labelwidth
-    \advance\leftmargin\labelsep\relax
-    \itemsep \IEEEbibitemsep\relax
-    \usecounter{enumiv}%
-    \let\p@enumiv\@empty
-    \renewcommand\theenumiv{\@arabic\c@enumiv}}%
+        \leftmargin\labelwidth
+        \advance\leftmargin\labelsep\relax
+        \itemsep \IEEEbibitemsep\relax
+        \usecounter{enumiv}%
+        \let\p@enumiv\@empty
+        \renewcommand\theenumiv{\@arabic\c@enumiv}}%
     \let\@IEEElatexbibitem\bibitem%
     \def\bibitem{\@IEEEbibitemprefix\@IEEElatexbibitem}%
-\def\newblock{\hskip .11em plus .33em minus .07em}%
-% originally:
-%   \sloppy\clubpenalty4000\widowpenalty4000%
-% by adding the \interlinepenalty here, we make it more
-% difficult, but not impossible, for LaTeX to break within a reference.
-% The IEEE almost never breaks a reference (but they do it more often with
-% technotes). You may get an underfull vbox warning around the bibliography, 
-% but the final result will be much more like what the IEEE will publish. 
-% MDS 11/2000
-\ifCLASSOPTIONtechnote\sloppy\clubpenalty4000\widowpenalty4000\interlinepenalty100%
-\else\sloppy\clubpenalty4000\widowpenalty4000\interlinepenalty500\fi%
+    \def\newblock{\hskip .11em plus .33em minus .07em}%
+    % originally:
+    %   \sloppy\clubpenalty4000\widowpenalty4000%
+    % by adding the \interlinepenalty here, we make it more
+    % difficult, but not impossible, for LaTeX to break within a reference.
+    % The IEEE almost never breaks a reference (but they do it more often with
+    % technotes). You may get an underfull vbox warning around the bibliography, 
+    % but the final result will be much more like what the IEEE will publish. 
+    % MDS 11/2000
+    \ifCLASSOPTIONtechnote\sloppy\clubpenalty4000\widowpenalty4000\interlinepenalty100%
+    \else\sloppy\clubpenalty4000\widowpenalty4000\interlinepenalty500\fi%
     \sfcode`\.=1000\relax}
 \let\endthebibliography=\endlist
 
@@ -4523,7 +4533,7 @@ between column types.}%
 % \IEEEmembership is used to produce the sublargesize italic font used to indicate author 
 % IEEE membership. compsoc uses a large size sans slant font
 \def\IEEEmembership#1{{\@IEEEnotcompsoconly{\sublargesize}\normalfont\@IEEEcompsoconly{\sffamily}\textit{#1}}}
- 
+
 
 % \IEEEauthorrefmark{} produces a footnote type symbol to indicate author affiliation.
 % When given an argument of 1 to 9, \IEEEauthorrefmark{} follows the standard LaTeX footnote
@@ -4538,11 +4548,12 @@ between column types.}%
 % V1.7 make this a robust command
 % V1.8 transmag uses an arabic author affiliation symbol
 \ifCLASSOPTIONtransmag
-\DeclareRobustCommand*{\IEEEauthorrefmark}[1]{\raisebox{0pt}[0pt][0pt]{\textsuperscript{\footnotesize #1}}}
+    \DeclareRobustCommand*{\IEEEauthorrefmark}[1]{\raisebox{0pt}[0pt][0pt]{\textsuperscript{\footnotesize #1}}}
 \else
-\DeclareRobustCommand*{\IEEEauthorrefmark}[1]{\raisebox{0pt}[0pt][0pt]{\textsuperscript{\footnotesize\ensuremath{\ifcase#1\or *\or \dagger\or \ddagger\or%
-    \mathsection\or \mathparagraph\or \|\or **\or \dagger\dagger%
-    \or \ddagger\ddagger \else\textsuperscript{\expandafter\romannumeral#1}\fi}}}}
+    \DeclareRobustCommand*{\IEEEauthorrefmark}[1]{\raisebox{0pt}[0pt][0pt]{\textsuperscript{\footnotesize\ensuremath{\ifcase#1\or *\or \dagger\or \ddagger\or%
+                        \mathsection\or \mathparagraph\or \|\or **\or \dagger\dagger%
+                    \or \ddagger\ddagger \else\textsuperscript{\expandafter\romannumeral#1}\fi}}}}
+
 \fi
 
 
@@ -4570,8 +4581,8 @@ between column types.}%
 \def\@IEEEauthorblockNtopspace{0.0ex}
 \def\@IEEEauthorblockAtopspace{0.0ex}
 \ifCLASSOPTIONtransmag
-% transmag uses one line of space above first affiliation block
-\def\@IEEEauthorblockAtopspace{1\@IEEEnormalsizeunitybaselineskip}
+    % transmag uses one line of space above first affiliation block
+    \def\@IEEEauthorblockAtopspace{1\@IEEEnormalsizeunitybaselineskip}
 \fi
 
 % baseline spacing within name and affiliation blocks (confmode)
@@ -4598,19 +4609,19 @@ between column types.}%
 
 % saves the current font attributes
 \def\@IEEEcurfontSAVE{\global\let\@IEEESAVESTATEfontsize\f@size%
-\global\let\@IEEESAVESTATEfontbaselineskip\f@baselineskip%
-\global\let\@IEEESAVESTATEfontencoding\f@encoding%
-\global\let\@IEEESAVESTATEfontfamily\f@family%
-\global\let\@IEEESAVESTATEfontseries\f@series%
-\global\let\@IEEESAVESTATEfontshape\f@shape}
+    \global\let\@IEEESAVESTATEfontbaselineskip\f@baselineskip%
+    \global\let\@IEEESAVESTATEfontencoding\f@encoding%
+    \global\let\@IEEESAVESTATEfontfamily\f@family%
+    \global\let\@IEEESAVESTATEfontseries\f@series%
+    \global\let\@IEEESAVESTATEfontshape\f@shape}
 
 % restores the saved font attributes
 \def\@IEEEcurfontRESTORE{\fontsize{\@IEEESAVESTATEfontsize}{\@IEEESAVESTATEfontbaselineskip}%
-\fontencoding{\@IEEESAVESTATEfontencoding}%
-\fontfamily{\@IEEESAVESTATEfontfamily}%
-\fontseries{\@IEEESAVESTATEfontseries}%
-\fontshape{\@IEEESAVESTATEfontshape}%
-\selectfont}
+    \fontencoding{\@IEEESAVESTATEfontencoding}%
+    \fontfamily{\@IEEESAVESTATEfontfamily}%
+    \fontseries{\@IEEESAVESTATEfontseries}%
+    \fontshape{\@IEEESAVESTATEfontshape}%
+    \selectfont}
 
 
 % variable to indicate if the current block is the first block in the column
@@ -4624,8 +4635,8 @@ between column types.}%
 % baseline portion at 70% of the total length.
 % Responds to changes in the document's \baselinestretch
 \def\@IEEEauthorstrutrule{\@IEEEtrantmpdimenA\@IEEEauthorblockXinterlinespace%
-\@IEEEtrantmpdimenA=\baselinestretch\@IEEEtrantmpdimenA%
-\rule[-0.3\@IEEEtrantmpdimenA]{0pt}{\@IEEEtrantmpdimenA}}
+    \@IEEEtrantmpdimenA=\baselinestretch\@IEEEtrantmpdimenA%
+    \rule[-0.3\@IEEEtrantmpdimenA]{0pt}{\@IEEEtrantmpdimenA}}
 
 
 % blocks to hold the authors' names and affilations. 
@@ -4634,37 +4645,37 @@ between column types.}%
 % use real definitions in conference mode
 % name block
 \def\IEEEauthorblockN#1{\relax\@IEEEauthorblockNstyle% set the default text style
-\gdef\@IEEEauthorblockXinterlinespace{0pt}% disable strut for spacer row
-% the \expandafter hides the \cr in conditional tex, see the array.sty docs
-% for details, probably not needed here as the \cr is in a macro
-% do a spacer row if needed
-\if@IEEEprevauthorblockincol\expandafter\@IEEEauthorblockNtopspaceline\fi
-\global\@IEEEprevauthorblockincoltrue% we now have a block in this column
-%restore the correct strut value
-\gdef\@IEEEauthorblockXinterlinespace{\@IEEEauthorblockNinterlinespace}%
-% input the author names
-#1%
-% end the row if the user did not already
-\crcr}
+    \gdef\@IEEEauthorblockXinterlinespace{0pt}% disable strut for spacer row
+    % the \expandafter hides the \cr in conditional tex, see the array.sty docs
+    % for details, probably not needed here as the \cr is in a macro
+    % do a spacer row if needed
+    \if@IEEEprevauthorblockincol\expandafter\@IEEEauthorblockNtopspaceline\fi
+    \global\@IEEEprevauthorblockincoltrue% we now have a block in this column
+    %restore the correct strut value
+    \gdef\@IEEEauthorblockXinterlinespace{\@IEEEauthorblockNinterlinespace}%
+    % input the author names
+    #1%
+    % end the row if the user did not already
+    \crcr}
 % spacer row for names
 \def\@IEEEauthorblockNtopspaceline{\cr\noalign{\vskip\@IEEEauthorblockNtopspace}}
 %
 % affiliation block
 \def\IEEEauthorblockA#1{\relax\@IEEEauthorblockAstyle% set the default text style
-\gdef\@IEEEauthorblockXinterlinespace{0pt}%disable strut for spacer row
-% the \expandafter hides the \cr in conditional tex, see the array.sty docs
-% for details, probably not needed here as the \cr is in a macro
-% do a spacer row if needed
-\if@IEEEprevauthorblockincol\expandafter\@IEEEauthorblockAtopspaceline\fi
-\global\@IEEEprevauthorblockincoltrue% we now have a block in this column
-%restore the correct strut value
-\gdef\@IEEEauthorblockXinterlinespace{\@IEEEauthorblockAinterlinespace}%
-% input the author affiliations
-#1%
-% end the row if the user did not already
-\crcr
-% V1.8 transmag does not use any additional affiliation spacing after the first author
-\ifCLASSOPTIONtransmag\gdef\@IEEEauthorblockAtopspace{0pt}\fi}
+    \gdef\@IEEEauthorblockXinterlinespace{0pt}%disable strut for spacer row
+    % the \expandafter hides the \cr in conditional tex, see the array.sty docs
+    % for details, probably not needed here as the \cr is in a macro
+    % do a spacer row if needed
+    \if@IEEEprevauthorblockincol\expandafter\@IEEEauthorblockAtopspaceline\fi
+    \global\@IEEEprevauthorblockincoltrue% we now have a block in this column
+    %restore the correct strut value
+    \gdef\@IEEEauthorblockXinterlinespace{\@IEEEauthorblockAinterlinespace}%
+    % input the author affiliations
+    #1%
+    % end the row if the user did not already
+    \crcr
+    % V1.8 transmag does not use any additional affiliation spacing after the first author
+    \ifCLASSOPTIONtransmag\gdef\@IEEEauthorblockAtopspace{0pt}\fi}
 
 % spacer row for affiliations
 \def\@IEEEauthorblockAtopspaceline{\cr\noalign{\vskip\@IEEEauthorblockAtopspace}}
@@ -4674,44 +4685,44 @@ between column types.}%
 % than conference or peerreviewca. For such cases, we provide dummy blocks.
 \ifCLASSOPTIONconference
 \else
-   \ifCLASSOPTIONpeerreviewca\else
-      % not conference, peerreviewca or transmag mode
-      \ifCLASSOPTIONtransmag\else
-         \def\IEEEauthorblockN#1{#1}%
-         \def\IEEEauthorblockA#1{#1}%
-      \fi
-   \fi
+    \ifCLASSOPTIONpeerreviewca\else
+        % not conference, peerreviewca or transmag mode
+        \ifCLASSOPTIONtransmag\else
+            \def\IEEEauthorblockN#1{#1}%
+            \def\IEEEauthorblockA#1{#1}%
+        \fi
+    \fi
 \fi
 
 
 
 % we provide our own halign so as not to have to depend on tabular
 \def\@IEEEauthorhalign{\@IEEEauthordefaulttextstyle% default text style
-   \lineskip=0pt\relax% disable line spacing
-   \lineskiplimit=0pt\relax%
-   \baselineskip=0pt\relax%
-   \@IEEEcurfontSAVE% save the current font
-   \mathsurround\z@\relax% no extra spacing around math
-   \let\\\@IEEEauthorhaligncr% replace newline with halign friendly one
-   \tabskip=0pt\relax% no column spacing
-   \everycr{}% ensure no problems here
-   \@IEEEprevauthorblockincolfalse% no author blocks yet
-   \def\@IEEEauthorblockXinterlinespace{2.7ex}% default interline space
-   \vtop\bgroup%vtop box
-   \halign\bgroup&\relax\hfil\@IEEEcurfontRESTORE\relax ##\relax
-   \hfil\@IEEEcurfontSAVE\@IEEEauthorstrutrule\cr}
+    \lineskip=0pt\relax% disable line spacing
+    \lineskiplimit=0pt\relax%
+    \baselineskip=0pt\relax%
+    \@IEEEcurfontSAVE% save the current font
+    \mathsurround\z@\relax% no extra spacing around math
+    \let\\\@IEEEauthorhaligncr% replace newline with halign friendly one
+    \tabskip=0pt\relax% no column spacing
+    \everycr{}% ensure no problems here
+    \@IEEEprevauthorblockincolfalse% no author blocks yet
+    \def\@IEEEauthorblockXinterlinespace{2.7ex}% default interline space
+    \vtop\bgroup%vtop box
+    \halign\bgroup&\relax\hfil\@IEEEcurfontRESTORE\relax ##\relax
+    \hfil\@IEEEcurfontSAVE\@IEEEauthorstrutrule\cr}
 
 % ensure last line, exit from halign, close vbox
 \def\end@IEEEauthorhalign{\crcr\egroup\egroup}
 
 % handle bogus star form
 \def\@IEEEauthorhaligncr{{\ifnum0=`}\fi\@ifstar{\@@IEEEauthorhaligncr}{\@@IEEEauthorhaligncr}}
-
-% test and setup the optional argument to \\[]
-\def\@@IEEEauthorhaligncr{\@testopt\@@@IEEEauthorhaligncr\z@skip}
-
-% end the line and do the optional spacer
-\def\@@@IEEEauthorhaligncr[#1]{\ifnum0=`{\fi}\cr\noalign{\vskip#1\relax}}
+    
+    % test and setup the optional argument to \\[]
+    \def\@@IEEEauthorhaligncr{\@testopt\@@@IEEEauthorhaligncr\z@skip}
+    
+    % end the line and do the optional spacer
+    \def\@@@IEEEauthorhaligncr[#1]{\ifnum0=`{\fi}\cr\noalign{\vskip#1\relax}}
 
 
 
@@ -4725,24 +4736,24 @@ between column types.}%
 \def\and{\relax} % provide a bogus \and that we will then override
 
 \renewcommand{\and}[1][\relax]{\if@IEEEWARNand\typeout{** WARNING: \noexpand\and is valid only
-                               when in conference or peerreviewca}\typeout{modes (line \the\inputlineno).}\fi\global\@IEEEWARNandfalse}
+            when in conference or peerreviewca}\typeout{modes (line \the\inputlineno).}\fi\global\@IEEEWARNandfalse}
 
 \ifCLASSOPTIONconference%
-\renewcommand{\and}[1][\hfill]{\end{@IEEEauthorhalign}#1\begin{@IEEEauthorhalign}}%
-\fi
-\ifCLASSOPTIONpeerreviewca
-\renewcommand{\and}[1][\hfill]{\end{@IEEEauthorhalign}#1\begin{@IEEEauthorhalign}}%
-\fi
-% V1.8 transmag uses conference author format
-\ifCLASSOPTIONtransmag
-\renewcommand{\and}[1][\hfill]{\end{@IEEEauthorhalign}#1\begin{@IEEEauthorhalign}}%
+    \renewcommand{\and}[1][\hfill]{\end{@IEEEauthorhalign}#1\begin{@IEEEauthorhalign}}%
+            \fi
+            \ifCLASSOPTIONpeerreviewca
+            \renewcommand{\and}[1][\hfill]{\end{@IEEEauthorhalign}#1\begin{@IEEEauthorhalign}}%
+            \fi
+            % V1.8 transmag uses conference author format
+            \ifCLASSOPTIONtransmag
+            \renewcommand{\and}[1][\hfill]{\end{@IEEEauthorhalign}#1\begin{@IEEEauthorhalign}}%
 \fi
 
 % page clearing command
 % based on LaTeX2e's \cleardoublepage, but allows different page styles
 % for the inserted blank pages
 \def\@IEEEcleardoublepage#1{\clearpage\if@twoside\ifodd\c@page\else
-\hbox{}\thispagestyle{#1}\newpage\if@twocolumn\hbox{}\thispagestyle{#1}\newpage\fi\fi\fi}
+            \hbox{}\thispagestyle{#1}\newpage\if@twocolumn\hbox{}\thispagestyle{#1}\newpage\fi\fi\fi}
 
 % V1.8b hooks to allow adjustment of space above title
 \def\IEEEtitletopspace{0.5\baselineskip}
@@ -4751,123 +4762,123 @@ between column types.}%
 
 % user command to invoke the title page
 \def\maketitle{\par%
-  \begingroup%
-  \normalfont%
-  \def\thefootnote{}%  the \thanks{} mark type is empty
-  \def\footnotemark{}% and kill space from \thanks within author
-  \let\@makefnmark\relax% V1.7, must *really* kill footnotemark to remove all \textsuperscript spacing as well.
-  \footnotesize%       equal spacing between thanks lines
-  \footnotesep 0.7\baselineskip%see global setting of \footnotesep for more info
-  % V1.7 disable \thanks note indention for compsoc
-  \@IEEEcompsoconly{\long\def\@makefntext##1{\parindent 1em\noindent\hbox{\@makefnmark}##1}}%
-  \normalsize%
-  \ifCLASSOPTIONpeerreview
-     \newpage\global\@topnum\z@ \@maketitle\@IEEEstatictitlevskip\@IEEEaftertitletext%
-     \thispagestyle{IEEEpeerreviewcoverpagestyle}\@thanks%
-  \else
-     \if@twocolumn%
-        \ifCLASSOPTIONtechnote%
-           \newpage\global\@topnum\z@ \@maketitle\@IEEEstatictitlevskip\@IEEEaftertitletext%
+    \begingroup%
+    \normalfont%
+    \def\thefootnote{}%  the \thanks{} mark type is empty
+    \def\footnotemark{}% and kill space from \thanks within author
+    \let\@makefnmark\relax% V1.7, must *really* kill footnotemark to remove all \textsuperscript spacing as well.
+    \footnotesize%       equal spacing between thanks lines
+    \footnotesep 0.7\baselineskip%see global setting of \footnotesep for more info
+    % V1.7 disable \thanks note indention for compsoc
+    \@IEEEcompsoconly{\long\def\@makefntext##1{\parindent 1em\noindent\hbox{\@makefnmark}##1}}%
+    \normalsize%
+    \ifCLASSOPTIONpeerreview
+        \newpage\global\@topnum\z@ \@maketitle\@IEEEstatictitlevskip\@IEEEaftertitletext%
+        \thispagestyle{IEEEpeerreviewcoverpagestyle}\@thanks%
+    \else
+        \if@twocolumn%
+            \ifCLASSOPTIONtechnote%
+                \newpage\global\@topnum\z@ \@maketitle\@IEEEstatictitlevskip\@IEEEaftertitletext%
+            \else
+                \twocolumn[{\IEEEquantizevspace{\@maketitle}[\IEEEquantizedisabletitlecmds]{0pt}[-\topskip]{\baselineskip}{\@IEEENORMtitlevspace}{\@IEEEMINtitlevspace}\@IEEEaftertitletext}]%
+            \fi
         \else
-           \twocolumn[{\IEEEquantizevspace{\@maketitle}[\IEEEquantizedisabletitlecmds]{0pt}[-\topskip]{\baselineskip}{\@IEEENORMtitlevspace}{\@IEEEMINtitlevspace}\@IEEEaftertitletext}]%
+            \newpage\global\@topnum\z@ \@maketitle\@IEEEstatictitlevskip\@IEEEaftertitletext%
         \fi
-     \else
-        \newpage\global\@topnum\z@ \@maketitle\@IEEEstatictitlevskip\@IEEEaftertitletext%
-     \fi
-     \thispagestyle{IEEEtitlepagestyle}\@thanks%
-  \fi
-  % pullup page for pubid if used.
-  \if@IEEEusingpubid
-     \enlargethispage{-\@IEEEpubidpullup}%
-  \fi 
-  \endgroup
-  \setcounter{footnote}{0}\let\maketitle\relax\let\@maketitle\relax
-  \gdef\@thanks{}%
-  % v1.6b do not clear these as we will need the title again for peer review papers
-  % \gdef\@author{}\gdef\@title{}%
-  \let\thanks\relax}
+        \thispagestyle{IEEEtitlepagestyle}\@thanks%
+    \fi
+    % pullup page for pubid if used.
+    \if@IEEEusingpubid
+        \enlargethispage{-\@IEEEpubidpullup}%
+    \fi
+    \endgroup
+    \setcounter{footnote}{0}\let\maketitle\relax\let\@maketitle\relax
+    \gdef\@thanks{}%
+    % v1.6b do not clear these as we will need the title again for peer review papers
+    % \gdef\@author{}\gdef\@title{}%
+    \let\thanks\relax}
 
 
 % V1.8 parbox to format \@IEEEtitleabstractindextext
 \long\def\@IEEEtitleabstractindextextbox#1{\parbox{1\textwidth}{#1}}
 % V1.8 compsoc is partial width
 \ifCLASSOPTIONcompsoc
-% comparison with proofs suggests it's in the range of 92.1-92.3%
-\long\def\@IEEEtitleabstractindextextbox#1{\parbox{0.922\textwidth}{\@IEEEcompsocnotconfonly{\rightskip\@flushglue\leftskip\z@skip}#1}}
+    % comparison with proofs suggests it's in the range of 92.1-92.3%
+    \long\def\@IEEEtitleabstractindextextbox#1{\parbox{0.922\textwidth}{\@IEEEcompsocnotconfonly{\rightskip\@flushglue\leftskip\z@skip}#1}}
 \fi
 
 % formats the Title, authors names, affiliations and special paper notice
 % THIS IS A CONTROLLED SPACING COMMAND! Do not allow blank lines or unintentional
 % spaces to enter the definition - use % at the end of each line
 \def\@maketitle{\newpage
-\bgroup\par\vskip\IEEEtitletopspace\vskip\IEEEtitletopspaceextra\centering%
-\ifCLASSOPTIONtechnote% technotes, V1.8a abstract and index terms are not treated differently for compsoc technotes
-   {\bfseries\large\@IEEEcompsoconly{\Large\sffamily}\@title\par}\vskip 1.3em{\lineskip .5em\@IEEEcompsoconly{\large\sffamily}\@author
-   \@IEEEspecialpapernotice\par}\relax
-\else% not a technote
-   \vskip0.2em{\Huge\ifCLASSOPTIONtransmag\bfseries\LARGE\fi\@IEEEcompsoconly{\sffamily}\@IEEEcompsocconfonly{\normalfont\normalsize\vskip 2\@IEEEnormalsizeunitybaselineskip
-   \bfseries\Large}\@IEEEcompsocnotconfonly{\vskip 0.75\@IEEEnormalsizeunitybaselineskip}\@title\par}\relax
-   \@IEEEcompsocnotconfonly{\vskip 0.5\@IEEEnormalsizeunitybaselineskip}\vskip1.0em\par%
-   % V1.6 handle \author differently if in conference mode
-   \ifCLASSOPTIONconference%
-      {\@IEEEspecialpapernotice\mbox{}\vskip\@IEEEauthorblockconfadjspace%
-       \mbox{}\hfill\begin{@IEEEauthorhalign}\@author\end{@IEEEauthorhalign}\hfill\mbox{}\par}\relax
-   \else% peerreviewca, peerreview or journal
-      \ifCLASSOPTIONpeerreviewca
-         % peerreviewca handles author names just like conference mode
-         {\@IEEEcompsoconly{\sffamily}\@IEEEspecialpapernotice\mbox{}\vskip\@IEEEauthorblockconfadjspace%
-          \mbox{}\hfill\begin{@IEEEauthorhalign}\@author\end{@IEEEauthorhalign}\hfill\mbox{}\par
-          {\@IEEEcompsoconly{\vskip 1.5em\relax
-           \@IEEEtitleabstractindextextbox{\@IEEEtitleabstractindextext}\par\noindent\hfill
-           \IEEEcompsocdiamondline\hfill\hbox{}\par}}}\relax
-      \else% journal, peerreview or transmag
-         \ifCLASSOPTIONtransmag
-            % transmag also handles author names just like conference mode
-            % it also uses \@IEEEtitleabstractindextex, but with one line less
-            % space above, and one more below
-           {\@IEEEspecialpapernotice\mbox{}\vskip\@IEEEauthorblockconfadjspace%
-            \mbox{}\hfill\begin{@IEEEauthorhalign}\@author\end{@IEEEauthorhalign}\hfill\mbox{}\par
-           {\vspace{0.5\baselineskip}\relax\@IEEEtitleabstractindextextbox{\@IEEEtitleabstractindextext}\vspace{-1\baselineskip}\par}}\relax
-         \else% journal or peerreview
-           {\lineskip.5em\@IEEEcompsoconly{\sffamily}\sublargesize\@author\@IEEEspecialpapernotice\par
-           {\@IEEEcompsoconly{\vskip 1.5em\relax
-            \@IEEEtitleabstractindextextbox{\@IEEEtitleabstractindextext}\par\noindent\hfill
-            \IEEEcompsocdiamondline\hfill\hbox{}\par}}}\relax
-         \fi
-      \fi
-   \fi
-\fi\par\addvspace{0.5\baselineskip}\egroup}
+    \bgroup\par\vskip\IEEEtitletopspace\vskip\IEEEtitletopspaceextra\centering%
+    \ifCLASSOPTIONtechnote% technotes, V1.8a abstract and index terms are not treated differently for compsoc technotes
+        {\bfseries\large\@IEEEcompsoconly{\Large\sffamily}\@title\par}\vskip 1.3em{\lineskip .5em\@IEEEcompsoconly{\large\sffamily}\@author
+                \@IEEEspecialpapernotice\par}\relax
+    \else% not a technote
+        \vskip0.2em{\Huge\ifCLASSOPTIONtransmag\bfseries\LARGE\fi\@IEEEcompsoconly{\sffamily}\@IEEEcompsocconfonly{\normalfont\normalsize\vskip 2\@IEEEnormalsizeunitybaselineskip
+            \bfseries\Large}\@IEEEcompsocnotconfonly{\vskip 0.75\@IEEEnormalsizeunitybaselineskip}\@title\par}\relax
+        \@IEEEcompsocnotconfonly{\vskip 0.5\@IEEEnormalsizeunitybaselineskip}\vskip1.0em\par%
+        % V1.6 handle \author differently if in conference mode
+        \ifCLASSOPTIONconference%
+            {\@IEEEspecialpapernotice\mbox{}\vskip\@IEEEauthorblockconfadjspace%
+            \mbox{}\hfill\begin{@IEEEauthorhalign}\@author\end{@IEEEauthorhalign}\hfill\mbox{}\par}\relax
+        \else% peerreviewca, peerreview or journal
+            \ifCLASSOPTIONpeerreviewca
+                % peerreviewca handles author names just like conference mode
+                {\@IEEEcompsoconly{\sffamily}\@IEEEspecialpapernotice\mbox{}\vskip\@IEEEauthorblockconfadjspace%
+                \mbox{}\hfill\begin{@IEEEauthorhalign}\@author\end{@IEEEauthorhalign}\hfill\mbox{}\par
+                {\@IEEEcompsoconly{\vskip 1.5em\relax
+                        \@IEEEtitleabstractindextextbox{\@IEEEtitleabstractindextext}\par\noindent\hfill
+                        \IEEEcompsocdiamondline\hfill\hbox{}\par}}}\relax
+            \else% journal, peerreview or transmag
+                \ifCLASSOPTIONtransmag
+                    % transmag also handles author names just like conference mode
+                    % it also uses \@IEEEtitleabstractindextex, but with one line less
+                    % space above, and one more below
+                    {\@IEEEspecialpapernotice\mbox{}\vskip\@IEEEauthorblockconfadjspace%
+                    \mbox{}\hfill\begin{@IEEEauthorhalign}\@author\end{@IEEEauthorhalign}\hfill\mbox{}\par
+                    {\vspace{0.5\baselineskip}\relax\@IEEEtitleabstractindextextbox{\@IEEEtitleabstractindextext}\vspace{-1\baselineskip}\par}}\relax
+                \else% journal or peerreview
+                    {\lineskip.5em\@IEEEcompsoconly{\sffamily}\sublargesize\@author\@IEEEspecialpapernotice\par
+                        {\@IEEEcompsoconly{\vskip 1.5em\relax
+                                \@IEEEtitleabstractindextextbox{\@IEEEtitleabstractindextext}\par\noindent\hfill
+                                \IEEEcompsocdiamondline\hfill\hbox{}\par}}}\relax
+                \fi
+            \fi
+        \fi
+    \fi\par\addvspace{0.5\baselineskip}\egroup}
 
 
 % V1.7 Computer Society "diamond line" which follows index terms for nonconference papers
 % V1.8a full width diamond line for single column use
 \def\@IEEEcompsocdiamondlinei{\vrule depth 0pt height 0.5pt width 4cm\nobreak\hspace{7.5pt}\nobreak
-\raisebox{-3.5pt}{\fontfamily{pzd}\fontencoding{U}\fontseries{m}\fontshape{n}\fontsize{11}{12}\selectfont\char70}\nobreak
-\hspace{7.5pt}\nobreak\vrule depth 0pt height 0.5pt width 4cm\relax}
+    \raisebox{-3.5pt}{\fontfamily{pzd}\fontencoding{U}\fontseries{m}\fontshape{n}\fontsize{11}{12}\selectfont\char70}\nobreak
+    \hspace{7.5pt}\nobreak\vrule depth 0pt height 0.5pt width 4cm\relax}
 % V1.8a narrower width diamond line for double column use
 \def\@IEEEcompsocdiamondlineii{\vrule depth 0pt height 0.5pt width 2.5cm\nobreak\hspace{7.5pt}\nobreak
-\raisebox{-3.5pt}{\fontfamily{pzd}\fontencoding{U}\fontseries{m}\fontshape{n}\fontsize{11}{12}\selectfont\char70}\nobreak
-\hspace{7.5pt}\nobreak\vrule depth 0pt height 0.5pt width 2.5cm\relax}
+    \raisebox{-3.5pt}{\fontfamily{pzd}\fontencoding{U}\fontseries{m}\fontshape{n}\fontsize{11}{12}\selectfont\char70}\nobreak
+    \hspace{7.5pt}\nobreak\vrule depth 0pt height 0.5pt width 2.5cm\relax}
 % V1.8a bare core without rules to base a last resort on for very narrow linewidths
 \def\@IEEEcompsocdiamondlineiii{\mbox{}\nobreak\hspace{7.5pt}\nobreak
-\raisebox{-3.5pt}{\fontfamily{pzd}\fontencoding{U}\fontseries{m}\fontshape{n}\fontsize{11}{12}\selectfont\char70}\nobreak
-\hspace{7.5pt}\nobreak\mbox{}\relax}
+    \raisebox{-3.5pt}{\fontfamily{pzd}\fontencoding{U}\fontseries{m}\fontshape{n}\fontsize{11}{12}\selectfont\char70}\nobreak
+    \hspace{7.5pt}\nobreak\mbox{}\relax}
 
 % V1.8a allow \IEEEcompsocdiamondline to adjust for different linewidths.
 % Use \@IEEEcompsocdiamondlinei if its width is less than 0.66\linewidth (0.487 nominal for single column)
 % if not, fall back to \@IEEEcompsocdiamondlineii if its width is less than 0.75\linewidth (0.659 nominal for double column)
 % if all else fails, try to make a custom diamondline based on the abnormally narrow linewidth
 \def\IEEEcompsocdiamondline{\settowidth{\@IEEEtrantmpdimenA}{\@IEEEcompsocdiamondlinei}\relax
-\ifdim\@IEEEtrantmpdimenA<0.66\linewidth\relax\@IEEEcompsocdiamondlinei\relax
-\else
-\settowidth{\@IEEEtrantmpdimenA}{\@IEEEcompsocdiamondlineii}\relax
-\ifdim\@IEEEtrantmpdimenA<0.75\linewidth\relax\@IEEEcompsocdiamondlineii\relax
-\else
-\settowidth{\@IEEEtrantmpdimenA}{\@IEEEcompsocdiamondlineiii}\relax
-\@IEEEtrantmpdimenB=\linewidth\relax
-\addtolength{\@IEEEtrantmpdimenB}{-1\@IEEEtrantmpdimenA}\relax
-\vrule depth 0pt height 0.5pt width 0.33\@IEEEtrantmpdimenB\@IEEEcompsocdiamondlineiii\vrule depth 0pt height 0.5pt width 0.33\@IEEEtrantmpdimenB\relax
-\fi\fi}
+    \ifdim\@IEEEtrantmpdimenA<0.66\linewidth\relax\@IEEEcompsocdiamondlinei\relax
+    \else
+        \settowidth{\@IEEEtrantmpdimenA}{\@IEEEcompsocdiamondlineii}\relax
+        \ifdim\@IEEEtrantmpdimenA<0.75\linewidth\relax\@IEEEcompsocdiamondlineii\relax
+        \else
+            \settowidth{\@IEEEtrantmpdimenA}{\@IEEEcompsocdiamondlineiii}\relax
+            \@IEEEtrantmpdimenB=\linewidth\relax
+            \addtolength{\@IEEEtrantmpdimenB}{-1\@IEEEtrantmpdimenA}\relax
+            \vrule depth 0pt height 0.5pt width 0.33\@IEEEtrantmpdimenB\@IEEEcompsocdiamondlineiii\vrule depth 0pt height 0.5pt width 0.33\@IEEEtrantmpdimenB\relax
+        \fi\fi}
 
 
 % V1.7 standard LateX2e \thanks, but with \itshape under compsoc. Also make it a \long\def
@@ -4878,7 +4889,7 @@ between column types.}%
 \long\def\thanks#1{\footnotemark
     \protected@xdef\@thanks{\@thanks
         \protect\footnotetext[\the\c@footnote]{\@IEEEcompsoconly{\itshape
-        \protect\@IEEEtriggeroneshotfootnoterule\relax}\ignorespaces#1}}}
+                \protect\@IEEEtriggeroneshotfootnoterule\relax}\ignorespaces#1}}}
 \let\@thanks\@empty
 
 
@@ -4889,8 +4900,8 @@ between column types.}%
 % in addition to setting up IEEEitemize, we need to remove a baselineskip space above and
 % below it because \list's \pars introduce blank lines because of the footnote struts.
 \def\@IEEEsetupcompsocitemizelist{\def\labelitemi{$\bullet$}%
-\setlength{\IEEElabelindent}{0pt}\setlength{\labelsep}{1.2em}\setlength{\parskip}{0pt}%
-\setlength{\partopsep}{0pt}\setlength{\topsep}{0.5\baselineskip}\vspace{-1\baselineskip}\relax}
+    \setlength{\IEEElabelindent}{0pt}\setlength{\labelsep}{1.2em}\setlength{\parskip}{0pt}%
+    \setlength{\partopsep}{0pt}\setlength{\topsep}{0.5\baselineskip}\vspace{-1\baselineskip}\relax}
 
 
 % flag for fake non-compsoc \IEEEcompsocthanksitem - prevents line break on very first item
@@ -4902,33 +4913,34 @@ between column types.}%
 \long\def\IEEEcompsocitemizethanks#1{\relax\@IEEEbreakcompsocthanksitemfalse\footnotemark
     \protected@xdef\@thanks{\@thanks
         \protect\footnotetext[\the\c@footnote]{\itshape\protect\@IEEEtriggeroneshotfootnoterule
-        {\let\IEEEiedlistdecl\relax\protect\begin{IEEEitemize}[\protect\@IEEEsetupcompsocitemizelist]\ignorespaces#1\relax
-        \protect\end{IEEEitemize}}\protect\vspace{-1\baselineskip}}}}
+            {\let\IEEEiedlistdecl\relax\protect\begin{IEEEitemize}[\protect\@IEEEsetupcompsocitemizelist]\ignorespaces#1\relax
+                    \protect\end{IEEEitemize}}\protect\vspace{-1\baselineskip}}}}
 \DeclareRobustCommand*{\IEEEcompsocthanksitem}{\item}
 \else
 % non-compsoc, allow for dual compilation via rerouting to normal \thanks
 \long\def\IEEEcompsocitemizethanks#1{\thanks{#1}}
 % redirect to "pseudo-par" \hfil\break\indent after swallowing [] from \IEEEcompsocthanksitem[]
 \DeclareRobustCommand{\IEEEcompsocthanksitem}{\@ifnextchar [{\@IEEEthanksswallowoptionalarg}%
-{\@IEEEthanksswallowoptionalarg[\relax]}}
-% be sure and break only after first item, be sure and ignore spaces after optional argument
-\def\@IEEEthanksswallowoptionalarg[#1]{\relax\if@IEEEbreakcompsocthanksitem\hfil\break
-\indent\fi\@IEEEbreakcompsocthanksitemtrue\ignorespaces}
+        {\@IEEEthanksswallowoptionalarg[\relax]}}
+    % be sure and break only after first item, be sure and ignore spaces after optional argument
+    \def\@IEEEthanksswallowoptionalarg[#1]{\relax\if@IEEEbreakcompsocthanksitem\hfil\break
+            \indent\fi\@IEEEbreakcompsocthanksitemtrue\ignorespaces}
 \fi
 
 
 % V1.6b define the \IEEEpeerreviewmaketitle as needed
 \ifCLASSOPTIONpeerreview
-\def\IEEEpeerreviewmaketitle{\@IEEEcleardoublepage{empty}%
-\ifCLASSOPTIONtwocolumn
-\twocolumn[{\IEEEquantizevspace{\@IEEEpeerreviewmaketitle}[\IEEEquantizedisabletitlecmds]{0pt}[-\topskip]{\baselineskip}{\@IEEENORMtitlevspace}{\@IEEEMINtitlevspace}}]
-\else
-\newpage\@IEEEpeerreviewmaketitle\@IEEEstatictitlevskip
-\fi
-\thispagestyle{IEEEtitlepagestyle}}
+    \def\IEEEpeerreviewmaketitle{\@IEEEcleardoublepage{empty}%
+        \ifCLASSOPTIONtwocolumn
+            \twocolumn[{\IEEEquantizevspace{\@IEEEpeerreviewmaketitle}[\IEEEquantizedisabletitlecmds]{0pt}[-\topskip]{\baselineskip}{\@IEEENORMtitlevspace}{\@IEEEMINtitlevspace}}]
+        \else
+            \newpage\@IEEEpeerreviewmaketitle\@IEEEstatictitlevskip
+        \fi
+        \thispagestyle{IEEEtitlepagestyle}}
 \else
-% \IEEEpeerreviewmaketitle does nothing if peer review option has not been selected
-\def\IEEEpeerreviewmaketitle{\relax}
+    % \IEEEpeerreviewmaketitle does nothing if peer review option has not been selected
+    \def\IEEEpeerreviewmaketitle{\relax}
+
 \fi
 
 % peerreview formats the repeated title like the title in journal papers.
@@ -4943,23 +4955,23 @@ between column types.}%
 % used for single column text, or when the title appears in the first column
 % of two column text (technotes). 
 \def\@IEEEstatictitlevskip{{\normalfont\normalsize
-% adjust spacing to next text
-% v1.6b handle peer review papers
-\ifCLASSOPTIONpeerreview
-% for peer review papers, the same value is used for both title pages
-% regardless of the other paper modes
-   \vskip 1\baselineskip plus 0.375\baselineskip minus 0.1875\baselineskip
-\else
-   \ifCLASSOPTIONconference% conference
-      \vskip 1\baselineskip plus 0.375\baselineskip minus 0.1875\baselineskip%
-   \else%
-      \ifCLASSOPTIONtechnote% technote
-         \vskip 1\baselineskip plus 0.375\baselineskip minus 0.1875\baselineskip%
-      \else% journal uses more space
-         \vskip 2.5\baselineskip plus 0.75\baselineskip minus 0.375\baselineskip%
-      \fi
-   \fi
-\fi}}
+            % adjust spacing to next text
+            % v1.6b handle peer review papers
+            \ifCLASSOPTIONpeerreview
+                % for peer review papers, the same value is used for both title pages
+                % regardless of the other paper modes
+                \vskip 1\baselineskip plus 0.375\baselineskip minus 0.1875\baselineskip
+            \else
+                \ifCLASSOPTIONconference% conference
+                    \vskip 1\baselineskip plus 0.375\baselineskip minus 0.1875\baselineskip%
+                \else%
+                    \ifCLASSOPTIONtechnote% technote
+                        \vskip 1\baselineskip plus 0.375\baselineskip minus 0.1875\baselineskip%
+                    \else% journal uses more space
+                        \vskip 2.5\baselineskip plus 0.75\baselineskip minus 0.375\baselineskip%
+                    \fi
+                \fi
+            \fi}}
 
 
 % set the nominal and minimum values for the quantized title spacer
@@ -4970,12 +4982,12 @@ between column types.}%
 \def\@IEEEMINtitlevspace{2\baselineskip}
 % conferences and technotes need tighter spacing
 \ifCLASSOPTIONconference% conference
-  \def\@IEEENORMtitlevspace{1\baselineskip}
-  \def\@IEEEMINtitlevspace{0.75\baselineskip}
+    \def\@IEEENORMtitlevspace{1\baselineskip}
+    \def\@IEEEMINtitlevspace{0.75\baselineskip}
 \fi
 \ifCLASSOPTIONtechnote% technote
-  \def\@IEEENORMtitlevspace{1\baselineskip}
-  \def\@IEEEMINtitlevspace{0.75\baselineskip}
+    \def\@IEEENORMtitlevspace{1\baselineskip}
+    \def\@IEEEMINtitlevspace{0.75\baselineskip}
 \fi
 
 
@@ -5102,111 +5114,111 @@ between column types.}%
 
 % main routine
 \def\@ixIEEEquantizevspace#1#2#3{\relax
-\edef\@IEEEquantizeunitheight{#1}\relax
-\edef\@IEEEquantizenomvspace{#2}\relax
-\edef\@IEEEquantizeminvspace{#3}\relax
-% \@IEEEquantizeoutdimenreg
-% \@IEEEquantizeobject
-% \@IEEEquantizeobjectdecl
-% \@IEEEquantizetopbaselineskip
-% \@IEEEquantizeoffset
-% \@IEEEquantizeprevdepth
-% \@IEEEquantizelineskiplimit
-% \@IEEEquantizelineskip
-% \@IEEEquantizeunitheight
-% \@IEEEquantizenomvspace
-% \@IEEEquantizeminvspace
-% get overall height of object
-\setbox\@IEEEquantizeboxA\vbox{\begingroup\@IEEEquantizeobjectdecl\@IEEEquantizeobject\relax\endgroup}\relax
-\@IEEEquantizeheightA\ht\@IEEEquantizeboxA\relax
-% get height of first line of object
-\setbox\@IEEEquantizeboxA\vtop{\begingroup\@IEEEquantizeobjectdecl\@IEEEquantizeobject\relax\endgroup}\relax
-\@IEEEquantizeheightB\ht\@IEEEquantizeboxA\relax
-\ifdim\@IEEEquantizeprevdepth>-1000pt\relax % prevdepth > -1000pf means full baselineskip\lineskip rules in effect
-% lineskip spacing rule takes effect if height of top line > baselineskip - prevdepth - lineskiplimit, 
-% otherwise the baselineskip rule is in effect and the height of the first line does not matter at all.
-\@IEEEquantizeheightC=\@IEEEquantizetopbaselineskip\relax
-\advance\@IEEEquantizeheightC-\@IEEEquantizeprevdepth\relax
-\advance\@IEEEquantizeheightC-\@IEEEquantizelineskiplimit\relax % this works even though \@IEEEquantizelineskiplimit is a macro because TeX allows --10pt notation
-\ifdim\@IEEEquantizeheightB>\@IEEEquantizeheightC\relax
-% lineskip spacing rule is in effect i.e., the object is going to be shifted down relative to the
-% baselineskip set position by its top line height (already a part of the total height) + prevdepth + lineskip - baselineskip
-\advance\@IEEEquantizeheightA\@IEEEquantizeprevdepth\relax
-\advance\@IEEEquantizeheightA\@IEEEquantizelineskip\relax
-\advance\@IEEEquantizeheightA-\@IEEEquantizetopbaselineskip\relax
-\else
-% height of first line <= \@IEEEquantizetopbaselineskip - \@IEEEquantizeprevdepth - \@IEEEquantizelineskiplimit
-% standard baselineskip rules are in effect, so don't consider height of first line
-\advance\@IEEEquantizeheightA-\@IEEEquantizeheightB\relax
-\fi
-%
-\else % prevdepth <= -1000pt, simplier \topskip type rules in effect
-\ifdim\@IEEEquantizeheightB>\@IEEEquantizetopbaselineskip
-% height of top line (already included in the total height) in excess of
-% baselineskip is the amount it will be downshifted
-\advance\@IEEEquantizeheightA-\@IEEEquantizetopbaselineskip\relax
-\else
-% height of first line is irrelevant, remove it
-\advance\@IEEEquantizeheightA-\@IEEEquantizeheightB\relax
-\fi
-\fi % prevdepth <= -1000pt
-%
-% adjust height for any manual offset
-\advance\@IEEEquantizeheightA\@IEEEquantizeoffset\relax
-% add in nominal spacer
-\advance\@IEEEquantizeheightA\@IEEEquantizenomvspace\relax
-% check for nonzero unitheight
-\@IEEEquantizeheightB=\@IEEEquantizeunitheight\relax
-\ifnum\@IEEEquantizeheightB=0\relax
-\@IEEEclspkgerror{IEEEquantizevspace unit height cannot be zero. Assuming 10pt.}%
-{Division by zero is not allowed.}
-\@IEEEquantizeheightB=10pt\relax
-\fi
-% get integer number of lines
-\@IEEEquantizemultiple=\@IEEEquantizeheightA\relax
-\divide\@IEEEquantizemultiple\@IEEEquantizeheightB\relax
-% set A to contain the excess height over the \@IEEEquantizemultiple of lines
-% A = height - multiple*unitheight
-\@IEEEquantizeheightC\@IEEEquantizeheightB\relax
-\multiply\@IEEEquantizeheightC\@IEEEquantizemultiple\relax
-\advance\@IEEEquantizeheightA-\@IEEEquantizeheightC\relax
-% set B to contain the height short of \@IEEEquantizemultiple+1 of lines
-% B = unitheight - A
-\advance\@IEEEquantizeheightB-\@IEEEquantizeheightA\relax
-% choose A or B based on which is closer
-\@IEEEquantizeheightC\@IEEEquantizenomvspace\relax
-\ifdim\@IEEEquantizeheightA<\@IEEEquantizeheightB\relax
-% C = nomvspace - A, go with lower
-\advance\@IEEEquantizeheightC-\@IEEEquantizeheightA\relax
-\else
-% C = nomvspace + B, go with upper
-\advance\@IEEEquantizeheightC\@IEEEquantizeheightB\relax
-\fi
-% if violate lower bound, use next integer bound
-\ifdim\@IEEEquantizeheightC<\@IEEEquantizeminvspace\relax
-% A + B = unitheight
-\advance\@IEEEquantizeheightC\@IEEEquantizeheightA\relax
-\advance\@IEEEquantizeheightC\@IEEEquantizeheightB\relax
-\fi
-% export object and spacer outside of group
-\global\let\@IEEEquantizeobjectout\@IEEEquantizeobject\relax
-\global\@IEEEquantizeheightC\@IEEEquantizeheightC\relax
-\endgroup
-\if@IEEEquantizevspaceuseoutdimenreg
-\@IEEEquantizeoutdimenreg=\@IEEEquantizeheightC\relax
-\else
-\@IEEEquantizeobjectout\relax
-\vskip\@IEEEquantizeheightC\relax
-\fi}
+    \edef\@IEEEquantizeunitheight{#1}\relax
+    \edef\@IEEEquantizenomvspace{#2}\relax
+    \edef\@IEEEquantizeminvspace{#3}\relax
+    % \@IEEEquantizeoutdimenreg
+    % \@IEEEquantizeobject
+    % \@IEEEquantizeobjectdecl
+    % \@IEEEquantizetopbaselineskip
+    % \@IEEEquantizeoffset
+    % \@IEEEquantizeprevdepth
+    % \@IEEEquantizelineskiplimit
+    % \@IEEEquantizelineskip
+    % \@IEEEquantizeunitheight
+    % \@IEEEquantizenomvspace
+    % \@IEEEquantizeminvspace
+    % get overall height of object
+    \setbox\@IEEEquantizeboxA\vbox{\begingroup\@IEEEquantizeobjectdecl\@IEEEquantizeobject\relax\endgroup}\relax
+    \@IEEEquantizeheightA\ht\@IEEEquantizeboxA\relax
+    % get height of first line of object
+    \setbox\@IEEEquantizeboxA\vtop{\begingroup\@IEEEquantizeobjectdecl\@IEEEquantizeobject\relax\endgroup}\relax
+    \@IEEEquantizeheightB\ht\@IEEEquantizeboxA\relax
+    \ifdim\@IEEEquantizeprevdepth>-1000pt\relax % prevdepth > -1000pf means full baselineskip\lineskip rules in effect
+        % lineskip spacing rule takes effect if height of top line > baselineskip - prevdepth - lineskiplimit, 
+        % otherwise the baselineskip rule is in effect and the height of the first line does not matter at all.
+        \@IEEEquantizeheightC=\@IEEEquantizetopbaselineskip\relax
+        \advance\@IEEEquantizeheightC-\@IEEEquantizeprevdepth\relax
+        \advance\@IEEEquantizeheightC-\@IEEEquantizelineskiplimit\relax % this works even though \@IEEEquantizelineskiplimit is a macro because TeX allows --10pt notation
+        \ifdim\@IEEEquantizeheightB>\@IEEEquantizeheightC\relax
+            % lineskip spacing rule is in effect i.e., the object is going to be shifted down relative to the
+            % baselineskip set position by its top line height (already a part of the total height) + prevdepth + lineskip - baselineskip
+            \advance\@IEEEquantizeheightA\@IEEEquantizeprevdepth\relax
+            \advance\@IEEEquantizeheightA\@IEEEquantizelineskip\relax
+            \advance\@IEEEquantizeheightA-\@IEEEquantizetopbaselineskip\relax
+        \else
+            % height of first line <= \@IEEEquantizetopbaselineskip - \@IEEEquantizeprevdepth - \@IEEEquantizelineskiplimit
+            % standard baselineskip rules are in effect, so don't consider height of first line
+            \advance\@IEEEquantizeheightA-\@IEEEquantizeheightB\relax
+        \fi
+        %
+    \else % prevdepth <= -1000pt, simplier \topskip type rules in effect
+        \ifdim\@IEEEquantizeheightB>\@IEEEquantizetopbaselineskip
+            % height of top line (already included in the total height) in excess of
+            % baselineskip is the amount it will be downshifted
+            \advance\@IEEEquantizeheightA-\@IEEEquantizetopbaselineskip\relax
+        \else
+            % height of first line is irrelevant, remove it
+            \advance\@IEEEquantizeheightA-\@IEEEquantizeheightB\relax
+        \fi
+    \fi % prevdepth <= -1000pt
+    %
+    % adjust height for any manual offset
+    \advance\@IEEEquantizeheightA\@IEEEquantizeoffset\relax
+    % add in nominal spacer
+    \advance\@IEEEquantizeheightA\@IEEEquantizenomvspace\relax
+    % check for nonzero unitheight
+    \@IEEEquantizeheightB=\@IEEEquantizeunitheight\relax
+    \ifnum\@IEEEquantizeheightB=0\relax
+        \@IEEEclspkgerror{IEEEquantizevspace unit height cannot be zero. Assuming 10pt.}%
+        {Division by zero is not allowed.}
+        \@IEEEquantizeheightB=10pt\relax
+    \fi
+    % get integer number of lines
+    \@IEEEquantizemultiple=\@IEEEquantizeheightA\relax
+    \divide\@IEEEquantizemultiple\@IEEEquantizeheightB\relax
+    % set A to contain the excess height over the \@IEEEquantizemultiple of lines
+    % A = height - multiple*unitheight
+    \@IEEEquantizeheightC\@IEEEquantizeheightB\relax
+    \multiply\@IEEEquantizeheightC\@IEEEquantizemultiple\relax
+    \advance\@IEEEquantizeheightA-\@IEEEquantizeheightC\relax
+    % set B to contain the height short of \@IEEEquantizemultiple+1 of lines
+    % B = unitheight - A
+    \advance\@IEEEquantizeheightB-\@IEEEquantizeheightA\relax
+    % choose A or B based on which is closer
+    \@IEEEquantizeheightC\@IEEEquantizenomvspace\relax
+    \ifdim\@IEEEquantizeheightA<\@IEEEquantizeheightB\relax
+        % C = nomvspace - A, go with lower
+        \advance\@IEEEquantizeheightC-\@IEEEquantizeheightA\relax
+    \else
+        % C = nomvspace + B, go with upper
+        \advance\@IEEEquantizeheightC\@IEEEquantizeheightB\relax
+    \fi
+    % if violate lower bound, use next integer bound
+    \ifdim\@IEEEquantizeheightC<\@IEEEquantizeminvspace\relax
+        % A + B = unitheight
+        \advance\@IEEEquantizeheightC\@IEEEquantizeheightA\relax
+        \advance\@IEEEquantizeheightC\@IEEEquantizeheightB\relax
+    \fi
+    % export object and spacer outside of group
+    \global\let\@IEEEquantizeobjectout\@IEEEquantizeobject\relax
+    \global\@IEEEquantizeheightC\@IEEEquantizeheightC\relax
+    \endgroup
+    \if@IEEEquantizevspaceuseoutdimenreg
+        \@IEEEquantizeoutdimenreg=\@IEEEquantizeheightC\relax
+    \else
+        \@IEEEquantizeobjectout\relax
+        \vskip\@IEEEquantizeheightC\relax
+    \fi}
 
 
 % user command to disable all global assignments, possible use within object decl
 \def\IEEEquantizedisableglobal{\let\global\relax
-\let\gdef\def
-\let\xdef\edef}
+    \let\gdef\def
+    \let\xdef\edef}
 % user command to allow for the disabling of \thanks and other commands, possible use within object decl
 \def\IEEEquantizedisabletitlecmds{\long\def\thanks##1{\relax}\relax
-\long\def\IEEEcompsocitemizethanks##1{\relax}\def\newpage{\relax}}
+    \long\def\IEEEcompsocitemizethanks##1{\relax}\def\newpage{\relax}}
 
 
 
@@ -5230,30 +5242,30 @@ between column types.}%
 % not in compsoc or transmag journal mode - this way abstract and keywords
 % can still be placed in their conventional position if not in those modes.
 \def\IEEEdisplaynontitleabstractindextext{%
-% display for all conference formats
-\ifCLASSOPTIONconference\@IEEEtitleabstractindextext\relax
-\else% non-conferences
-  % V1.8a display for all technotes
-  \ifCLASSOPTIONtechnote\@IEEEtitleabstractindextext\relax
-  % V1.8a add diamond line after abstract and index terms for compsoc technotes
-  \@IEEEcompsoconly{\noindent\hfill\IEEEcompsocdiamondline\hfill\hbox{}\par}\relax
-  \else % non-conferences and non-technotes
-    \ifCLASSOPTIONcompsoc% display if not compsoc and not transmag
-    \else
-      \ifCLASSOPTIONtransmag
-      \else% not compsoc journal nor transmag journal
-         \@IEEEtitleabstractindextext\relax
-      \fi
-    \fi
-  \fi
-\fi}
+    % display for all conference formats
+    \ifCLASSOPTIONconference\@IEEEtitleabstractindextext\relax
+    \else% non-conferences
+        % V1.8a display for all technotes
+        \ifCLASSOPTIONtechnote\@IEEEtitleabstractindextext\relax
+            % V1.8a add diamond line after abstract and index terms for compsoc technotes
+            \@IEEEcompsoconly{\noindent\hfill\IEEEcompsocdiamondline\hfill\hbox{}\par}\relax
+        \else % non-conferences and non-technotes
+            \ifCLASSOPTIONcompsoc% display if not compsoc and not transmag
+            \else
+                \ifCLASSOPTIONtransmag
+                \else% not compsoc journal nor transmag journal
+                    \@IEEEtitleabstractindextext\relax
+                \fi
+            \fi
+        \fi
+    \fi}
 
 
 % command to allow alteration of baselinestretch, but only if the current
 % baselineskip is unity. Used to tweak the compsoc abstract and keywords line spacing.
 \def\@IEEEtweakunitybaselinestretch#1{{\def\baselinestretch{1}\selectfont
-\global\@tempskipa\baselineskip}\ifnum\@tempskipa=\baselineskip%
-\def\baselinestretch{#1}\selectfont\fi\relax}
+            \global\@tempskipa\baselineskip}\ifnum\@tempskipa=\baselineskip%
+        \def\baselinestretch{#1}\selectfont\fi\relax}
 
 
 % abstract and keywords are in \small, except 
@@ -5262,7 +5274,7 @@ between column types.}%
 % becomes a rather awkward 8.5pt
 \def\@IEEEabskeysecsize{\small}
 \ifx\CLASSOPTIONpt\@IEEEptsizenine
- \def\@IEEEabskeysecsize{\footnotesize}
+    \def\@IEEEabskeysecsize{\footnotesize}
 \fi
 
 % compsoc journals use \footnotesize, compsoc conferences use normalsize
@@ -5274,9 +5286,9 @@ between column types.}%
 % so that spacing is more tightly controlled.
 \def\abstract{\normalfont
     \if@twocolumn
-      \@IEEEabskeysecsize\bfseries\textit{\abstractname}---\relax
+        \@IEEEabskeysecsize\bfseries\textit{\abstractname}---\relax
     \else
-      \bgroup\par\addvspace{0.5\baselineskip}\centering\vspace{-1.78ex}\@IEEEabskeysecsize\textbf{\abstractname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize
+        \bgroup\par\addvspace{0.5\baselineskip}\centering\vspace{-1.78ex}\@IEEEabskeysecsize\textbf{\abstractname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize
     \fi\@IEEEgobbleleadPARNLSP}
 % V1.6 The IEEE wants only 1 pica from end of abstract to introduction heading when in 
 % conference mode (the heading already has this much above it)
@@ -5285,9 +5297,9 @@ between column types.}%
 
 \def\IEEEkeywords{\normalfont
     \if@twocolumn
-      \@IEEEabskeysecsize\bfseries\textit{\IEEEkeywordsname}---\relax
+        \@IEEEabskeysecsize\bfseries\textit{\IEEEkeywordsname}---\relax
     \else
-      \bgroup\par\addvspace{0.5\baselineskip}\centering\@IEEEabskeysecsize\textbf{\IEEEkeywordsname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize
+        \bgroup\par\addvspace{0.5\baselineskip}\centering\@IEEEabskeysecsize\textbf{\IEEEkeywordsname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize
     \fi\@IEEEgobbleleadPARNLSP}
 \def\endIEEEkeywords{\relax\ifCLASSOPTIONtechnote\vspace{1.34ex}\else\vspace{0.67ex}\fi
     \par\if@twocolumn\else\endquotation\fi%
@@ -5295,55 +5307,56 @@ between column types.}%
 
 % V1.7 compsoc keywords index terms
 \ifCLASSOPTIONcompsoc
-  \ifCLASSOPTIONconference% compsoc conference
-\def\abstract{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\bfseries
-    \if@twocolumn
-      \@IEEEabskeysecsize\noindent\textit{\abstractname}---\relax
-    \else
-      \bgroup\par\addvspace{0.5\baselineskip}\centering\vspace{-1.78ex}\@IEEEabskeysecsize\textbf{\abstractname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize%
-    \fi\@IEEEgobbleleadPARNLSP}
-\def\IEEEkeywords{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\bfseries
-    \if@twocolumn
-      \@IEEEabskeysecsize\vskip 0.5\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip\noindent
-      \textit{\IEEEkeywordsname}---\relax
-    \else
-      \bgroup\par\addvspace{0.5\baselineskip}\centering\@IEEEabskeysecsize\textbf{\IEEEkeywordsname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize%
-    \fi\@IEEEgobbleleadPARNLSP}
-  \else% compsoc not conference
-\def\abstract{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\sffamily
-    \if@twocolumn
-      \@IEEEabskeysecsize\noindent\textbf{\abstractname}---\relax
-    \else
-      \bgroup\par\addvspace{0.5\baselineskip}\centering\vspace{-1.78ex}\@IEEEabskeysecsize\textbf{\abstractname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize%
-    \fi\@IEEEgobbleleadPARNLSP}
-\def\IEEEkeywords{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\sffamily
-    \if@twocolumn
-      \@IEEEabskeysecsize\vskip 0.5\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip\noindent
-      \textbf{\IEEEkeywordsname}---\relax
-    \else
-      \bgroup\par\addvspace{0.5\baselineskip}\centering\@IEEEabskeysecsize\textbf{\IEEEkeywordsname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize%
-    \fi\@IEEEgobbleleadPARNLSP}
-  \fi
+    \ifCLASSOPTIONconference% compsoc conference
+        \def\abstract{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\bfseries
+            \if@twocolumn
+                \@IEEEabskeysecsize\noindent\textit{\abstractname}---\relax
+            \else
+                \bgroup\par\addvspace{0.5\baselineskip}\centering\vspace{-1.78ex}\@IEEEabskeysecsize\textbf{\abstractname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize%
+            \fi\@IEEEgobbleleadPARNLSP}
+        \def\IEEEkeywords{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\bfseries
+            \if@twocolumn
+                \@IEEEabskeysecsize\vskip 0.5\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip\noindent
+                \textit{\IEEEkeywordsname}---\relax
+            \else
+                \bgroup\par\addvspace{0.5\baselineskip}\centering\@IEEEabskeysecsize\textbf{\IEEEkeywordsname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize%
+            \fi\@IEEEgobbleleadPARNLSP}
+    \else% compsoc not conference
+        \def\abstract{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\sffamily
+            \if@twocolumn
+                \@IEEEabskeysecsize\noindent\textbf{\abstractname}---\relax
+            \else
+                \bgroup\par\addvspace{0.5\baselineskip}\centering\vspace{-1.78ex}\@IEEEabskeysecsize\textbf{\abstractname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize%
+            \fi\@IEEEgobbleleadPARNLSP}
+        \def\IEEEkeywords{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\sffamily
+            \if@twocolumn
+                \@IEEEabskeysecsize\vskip 0.5\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip\noindent
+                \textbf{\IEEEkeywordsname}---\relax
+            \else
+                \bgroup\par\addvspace{0.5\baselineskip}\centering\@IEEEabskeysecsize\textbf{\IEEEkeywordsname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize%
+            \fi\@IEEEgobbleleadPARNLSP}
+
+    \fi
 \fi
 
 % V1.8 transmag keywords index terms
 % no abstract name, use indentation
 \ifCLASSOPTIONtransmag
-\def\abstract{\normalfont\parindent 1em\relax
-    \if@twocolumn
-      \@IEEEabskeysecsize\bfseries\indent
-    \else
-      \bgroup\par\addvspace{0.5\baselineskip}\centering\vspace{-1.78ex}\@IEEEabskeysecsize
-      \textbf{\abstractname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize
-    \fi\@IEEEgobbleleadPARNLSP}
-
-\def\IEEEkeywords{\normalfont\parindent 1em\relax
-    \if@twocolumn
-      \@IEEEabskeysecsize\vspace{1\baselineskip}\bfseries\indent\textit{\IEEEkeywordsname}---\relax
-    \else
-      \bgroup\par\vspace{1\baselineskip}\centering\@IEEEabskeysecsize
-      \textbf{\IEEEkeywordsname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize
-    \fi\@IEEEgobbleleadPARNLSP}
+    \def\abstract{\normalfont\parindent 1em\relax
+        \if@twocolumn
+            \@IEEEabskeysecsize\bfseries\indent
+        \else
+            \bgroup\par\addvspace{0.5\baselineskip}\centering\vspace{-1.78ex}\@IEEEabskeysecsize
+            \textbf{\abstractname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize
+        \fi\@IEEEgobbleleadPARNLSP}
+    
+    \def\IEEEkeywords{\normalfont\parindent 1em\relax
+        \if@twocolumn
+            \@IEEEabskeysecsize\vspace{1\baselineskip}\bfseries\indent\textit{\IEEEkeywordsname}---\relax
+        \else
+            \bgroup\par\vspace{1\baselineskip}\centering\@IEEEabskeysecsize
+            \textbf{\IEEEkeywordsname}\par\addvspace{0.5\baselineskip}\egroup\quotation\@IEEEabskeysecsize
+        \fi\@IEEEgobbleleadPARNLSP}
 \fi
 
 
@@ -5355,94 +5368,95 @@ between column types.}%
 % so that such things in the beginning of an environment will not
 % affect the formatting of the text
 \long\def\@IEEEgobbleleadPARNLSP#1{\let\@IEEEswallowthistoken=0%
-\let\@IEEEgobbleleadPARNLSPtoken#1%
-\let\@IEEEgobbleleadPARtoken=\par%
-\let\@IEEEgobbleleadNLtoken=\\%
-\let\@IEEEgobbleleadSPtoken=\ %
-\def\@IEEEgobbleleadSPMACRO{\ }%
-\ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadPARtoken%
-\let\@IEEEswallowthistoken=1%
-\fi%
-\ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadNLtoken%
-\let\@IEEEswallowthistoken=1%
-\fi%
-\ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadSPtoken%
-\let\@IEEEswallowthistoken=1%
-\fi%
-% a control space will come in as a macro
-% when it is the last one on a line
-\ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadSPMACRO%
-\let\@IEEEswallowthistoken=1%
-\fi%
-% if we have to swallow this token, do so and taste the next one
-% else spit it out and stop gobbling
-\ifx\@IEEEswallowthistoken 1\let\@IEEEnextgobbleleadPARNLSP=\@IEEEgobbleleadPARNLSP\else%
-\let\@IEEEnextgobbleleadPARNLSP=#1\fi%
-\@IEEEnextgobbleleadPARNLSP}%
+    \let\@IEEEgobbleleadPARNLSPtoken#1%
+    \let\@IEEEgobbleleadPARtoken=\par%
+    \let\@IEEEgobbleleadNLtoken=\\%
+    \let\@IEEEgobbleleadSPtoken=\ %
+    \def\@IEEEgobbleleadSPMACRO{\ }%
+    \ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadPARtoken%
+        \let\@IEEEswallowthistoken=1%
+    \fi%
+    \ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadNLtoken%
+        \let\@IEEEswallowthistoken=1%
+    \fi%
+    \ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadSPtoken%
+        \let\@IEEEswallowthistoken=1%
+    \fi%
+    % a control space will come in as a macro
+    % when it is the last one on a line
+    \ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadSPMACRO%
+        \let\@IEEEswallowthistoken=1%
+    \fi%
+    % if we have to swallow this token, do so and taste the next one
+    % else spit it out and stop gobbling
+    \ifx\@IEEEswallowthistoken 1\let\@IEEEnextgobbleleadPARNLSP=\@IEEEgobbleleadPARNLSP\else%
+        \let\@IEEEnextgobbleleadPARNLSP=#1\fi%
+    \@IEEEnextgobbleleadPARNLSP}%
 
 
 
 
 % TITLING OF SECTIONS
 \def\@IEEEsectpunct{:\ \,}  % Punctuation after run-in section heading  (headings which are
-                            % part of the paragraphs), need little bit more than a single space
-                            % spacing from section number to title
+% part of the paragraphs), need little bit more than a single space
+% spacing from section number to title
 % compsoc conferences use regular period/space punctuation
 \ifCLASSOPTIONcompsoc
-\ifCLASSOPTIONconference
-\def\@IEEEsectpunct{.\ }
-\fi\fi
+    \ifCLASSOPTIONconference
+        \def\@IEEEsectpunct{.\ }
+    \fi\fi
 
 
 \def\@seccntformat#1{\csname the#1dis\endcsname\hskip 0.5em\relax}
 
 \ifCLASSOPTIONcompsoc
-% compsoc journals need extra spacing
-\ifCLASSOPTIONconference\else
-\def\@seccntformat#1{\csname the#1dis\endcsname\hskip 1em\relax}
-\fi\fi
+    % compsoc journals need extra spacing
+    \ifCLASSOPTIONconference\else
+        \def\@seccntformat#1{\csname the#1dis\endcsname\hskip 1em\relax}
+
+    \fi\fi
 
 %v1.7 put {} after #6 to allow for some types of user font control
 %and use \@@par rather than \par
 \def\@sect#1#2#3#4#5#6[#7]#8{%
-  \ifnum #2>\c@secnumdepth
-     \let\@svsec\@empty
-  \else
-     \refstepcounter{#1}%
-     % load section label and spacer into \@svsec
-     \protected@edef\@svsec{\@seccntformat{#1}\relax}%
-  \fi%
-  \@tempskipa #5\relax
-  \ifdim \@tempskipa>\z@% tempskipa determines whether is treated as a high
-     \begingroup #6{\relax% or low level heading
-      \noindent % subsections are NOT indented
-       % print top level headings. \@svsec is label, #8 is heading title
-       % The IEEE does not block indent the section title text, it flows like normal
-       {\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\@@par}}%
-     \endgroup
-     \addcontentsline{toc}{#1}{\ifnum #2>\c@secnumdepth\relax\else
-               \protect\numberline{\csname the#1\endcsname}\fi#7}%
-  \else % printout low level headings
-     % svsechd seems to swallow the trailing space, protect it with \mbox{}
-     % got rid of sectionmark stuff
-     \def\@svsechd{#6{\hskip #3\relax\@svsec #8\@IEEEsectpunct\mbox{}}%
-     \addcontentsline{toc}{#1}{\ifnum #2>\c@secnumdepth\relax\else
-               \protect\numberline{\csname the#1\endcsname}\fi#7}}%
-  \fi%skip down
-  \@xsect{#5}}
+    \ifnum #2>\c@secnumdepth
+        \let\@svsec\@empty
+    \else
+        \refstepcounter{#1}%
+        % load section label and spacer into \@svsec
+        \protected@edef\@svsec{\@seccntformat{#1}\relax}%
+    \fi%
+    \@tempskipa #5\relax
+    \ifdim \@tempskipa>\z@% tempskipa determines whether is treated as a high
+        \begingroup #6{\relax% or low level heading
+            \noindent % subsections are NOT indented
+            % print top level headings. \@svsec is label, #8 is heading title
+            % The IEEE does not block indent the section title text, it flows like normal
+            {\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\@@par}}%
+        \endgroup
+        \addcontentsline{toc}{#1}{\ifnum #2>\c@secnumdepth\relax\else
+                \protect\numberline{\csname the#1\endcsname}\fi#7}%
+    \else % printout low level headings
+        % svsechd seems to swallow the trailing space, protect it with \mbox{}
+        % got rid of sectionmark stuff
+        \def\@svsechd{#6{\hskip #3\relax\@svsec #8\@IEEEsectpunct\mbox{}}%
+            \addcontentsline{toc}{#1}{\ifnum #2>\c@secnumdepth\relax\else
+                    \protect\numberline{\csname the#1\endcsname}\fi#7}}%
+    \fi%skip down
+    \@xsect{#5}}
 
 
 % section* handler
 %v1.7 put {} after #4 to allow for some types of user font control
 %and use \@@par rather than \par
 \def\@ssect#1#2#3#4#5{\@tempskipa #3\relax
-  \ifdim \@tempskipa>\z@
-     %\begingroup #4\@hangfrom{\hskip #1}{\interlinepenalty \@M #5\par}\endgroup
-     % The IEEE does not block indent the section title text, it flows like normal
-     \begingroup \noindent #4{\relax{\hskip #1}{\interlinepenalty \@M #5\@@par}}\endgroup
-  % svsechd swallows the trailing space, protect it with \mbox{}
-  \else \def\@svsechd{#4{\hskip #1\relax #5\@IEEEsectpunct\mbox{}}}\fi
-  \@xsect{#3}}
+    \ifdim \@tempskipa>\z@
+        %\begingroup #4\@hangfrom{\hskip #1}{\interlinepenalty \@M #5\par}\endgroup
+        % The IEEE does not block indent the section title text, it flows like normal
+        \begingroup \noindent #4{\relax{\hskip #1}{\interlinepenalty \@M #5\@@par}}\endgroup
+        % svsechd swallows the trailing space, protect it with \mbox{}
+    \else \def\@svsechd{#4{\hskip #1\relax #5\@IEEEsectpunct\mbox{}}}\fi
+    \@xsect{#3}}
 
 
 %% SECTION heading spacing and font
@@ -5463,58 +5477,58 @@ between column types.}%
 % heading by up to about 0.5in. However, it may not be a good
 % idea to let LaTeX have quite this much rubber.
 \ifCLASSOPTIONconference%
-% The IEEE wants section heading spacing to decrease for conference mode
-\def\section{\@startsection{section}{1}{\z@}{1.5ex plus 1.5ex minus 0.5ex}%
-{0.7ex plus 1ex minus 0ex}{\normalfont\normalsize\centering\scshape}}%
-\def\subsection{\@startsection{subsection}{2}{\z@}{1.5ex plus 1.5ex minus 0.5ex}%
-{0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\itshape}}%
+    % The IEEE wants section heading spacing to decrease for conference mode
+    \def\section{\@startsection{section}{1}{\z@}{1.5ex plus 1.5ex minus 0.5ex}%
+      {0.7ex plus 1ex minus 0ex}{\normalfont\normalsize\centering\scshape}}%
+    \def\subsection{\@startsection{subsection}{2}{\z@}{1.5ex plus 1.5ex minus 0.5ex}%
+        {0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\itshape}}%
 \else % for journals
-\def\section{\@startsection{section}{1}{\z@}{3.0ex plus 1.5ex minus 1.5ex}% V1.6 3.0ex from 3.5ex
-{0.7ex plus 1ex minus 0ex}{\normalfont\normalsize\centering\scshape}}%
-\def\subsection{\@startsection{subsection}{2}{\z@}{3.5ex plus 1.5ex minus 1.5ex}%
-{0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\itshape}}%
+    \def\section{\@startsection{section}{1}{\z@}{3.0ex plus 1.5ex minus 1.5ex}% V1.6 3.0ex from 3.5ex
+      {0.7ex plus 1ex minus 0ex}{\normalfont\normalsize\centering\scshape}}%
+    \def\subsection{\@startsection{subsection}{2}{\z@}{3.5ex plus 1.5ex minus 1.5ex}%
+        {0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\itshape}}%
 \fi
 
 % for both journals and conferences
 % decided to put in a little rubber above the section, might help somebody
 \def\subsubsection{\@startsection{subsubsection}{3}{\parindent}{0ex plus 0.1ex minus 0.1ex}%
-{0ex}{\normalfont\normalsize\itshape}}%
+    {0ex}{\normalfont\normalsize\itshape}}%
 \def\paragraph{\@startsection{paragraph}{4}{2\parindent}{0ex plus 0.1ex minus 0.1ex}%
-{0ex}{\normalfont\normalsize\itshape}}%
+    {0ex}{\normalfont\normalsize\itshape}}%
 
 
 % compsoc
 \ifCLASSOPTIONcompsoc
-\ifCLASSOPTIONconference
-% compsoc conference
-\def\section{\@startsection{section}{1}{\z@}{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}%
-{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}{\normalfont\large\bfseries}}%
-\def\subsection{\@startsection{subsection}{2}{\z@}{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}%
-{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}{\normalfont\sublargesize\bfseries}}%
-\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}%
-{0ex}{\normalfont\normalsize\bfseries}}%
-\def\paragraph{\@startsection{paragraph}{4}{2\parindent}{0ex plus 0.1ex minus 0.1ex}%
-{0ex}{\normalfont\normalsize}}%
-\else% compsoc journals
-% use negative top separation as compsoc journals do not indent paragraphs after section titles
-\def\section{\@startsection{section}{1}{\z@}{-3.5ex plus -2ex minus -1.5ex}%
-{0.7ex plus 1ex minus 0ex}{\normalfont\sublargesize\sffamily\bfseries\scshape}}%
-% Note that subsection and smaller may not be correct for the Computer Society,
-% I have to look up an example.
-\def\subsection{\@startsection{subsection}{2}{\z@}{-3.5ex plus -1.5ex minus -1.5ex}%
-{0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\sffamily\bfseries}}%
-\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-2.5ex plus -1ex minus -1ex}%
-{0.5ex plus 0.5ex minus 0ex}{\normalfont\normalsize\sffamily\itshape}}%
-\def\paragraph{\@startsection{paragraph}{4}{2\parindent}{-0ex plus -0.1ex minus -0.1ex}%
-{0ex}{\normalfont\normalsize}}%
-\fi\fi
+    \ifCLASSOPTIONconference
+        % compsoc conference
+        \def\section{\@startsection{section}{1}{\z@}{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}%
+          {1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}{\normalfont\large\bfseries}}%
+        \def\subsection{\@startsection{subsection}{2}{\z@}{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}%
+            {1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}{\normalfont\sublargesize\bfseries}}%
+        \def\subsubsection{\@startsection{subsubsection}{3}{\z@}{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}%
+            {0ex}{\normalfont\normalsize\bfseries}}%
+        \def\paragraph{\@startsection{paragraph}{4}{2\parindent}{0ex plus 0.1ex minus 0.1ex}%
+            {0ex}{\normalfont\normalsize}}%
+    \else% compsoc journals
+        % use negative top separation as compsoc journals do not indent paragraphs after section titles
+        \def\section{\@startsection{section}{1}{\z@}{-3.5ex plus -2ex minus -1.5ex}%
+          {0.7ex plus 1ex minus 0ex}{\normalfont\sublargesize\sffamily\bfseries\scshape}}%
+        % Note that subsection and smaller may not be correct for the Computer Society,
+        % I have to look up an example.
+        \def\subsection{\@startsection{subsection}{2}{\z@}{-3.5ex plus -1.5ex minus -1.5ex}%
+            {0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\sffamily\bfseries}}%
+        \def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-2.5ex plus -1ex minus -1ex}%
+            {0.5ex plus 0.5ex minus 0ex}{\normalfont\normalsize\sffamily\itshape}}%
+        \def\paragraph{\@startsection{paragraph}{4}{2\parindent}{-0ex plus -0.1ex minus -0.1ex}%
+            {0ex}{\normalfont\normalsize}}%
+    \fi\fi
 
 % transmag
 \ifCLASSOPTIONtransmag
-\def\subsection{\@startsection{subsection}{2}{0.75\parindent}{3.5ex plus 1.5ex minus 1.5ex}%
-{0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\itshape}}%
-\def\subsubsection{\@startsection{subsubsection}{3}{1.25\parindent}{0.1ex plus 0.1ex minus 0.1ex}%
-{0.1ex}{\normalfont\normalsize\itshape}}%
+    \def\subsection{\@startsection{subsection}{2}{0.75\parindent}{3.5ex plus 1.5ex minus 1.5ex}%
+        {0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\itshape}}%
+    \def\subsubsection{\@startsection{subsubsection}{3}{1.25\parindent}{0.1ex plus 0.1ex minus 0.1ex}%
+        {0.1ex}{\normalfont\normalsize\itshape}}%
 \fi
 
 
@@ -5534,9 +5548,9 @@ between column types.}%
 % V1.6 some journals use an open box instead that will just fit around a closed one
 \def\IEEEQEDopen{{\setlength{\fboxsep}{0pt}\setlength{\fboxrule}{0.2pt}\fbox{\rule[0pt]{0pt}{1.3ex}\rule[0pt]{1.3ex}{0pt}}}}
 \ifCLASSOPTIONcompsoc
-\def\IEEEQED{\IEEEQEDopen}   % default to open for compsoc
+    \def\IEEEQED{\IEEEQEDopen}   % default to open for compsoc
 \else
-\def\IEEEQED{\IEEEQEDclosed} % otherwise default to closed
+    \def\IEEEQED{\IEEEQEDclosed} % otherwise default to closed
 \fi
 
 %V1.8 flag to indicate if QED symbol is to be shown
@@ -5549,7 +5563,7 @@ between column types.}%
 \def\endIEEEproof{\if@IEEEQEDshow\hspace*{\fill}\nobreakspace\IEEEQED\fi\par}
 % qedhere for equation environments, similar to AMS \qedhere
 \def\IEEEQEDhereeqn{\global\@IEEEQEDshowfalse\eqno\let\eqno\relax\let\leqno\relax
-                    \let\veqno\relax\hbox{\IEEEQED}}
+    \let\veqno\relax\hbox{\IEEEQED}}
 % IEEE style qedhere for IEEEeqnarray and other environments
 \def\IEEEQEDhere{\global\@IEEEQEDshowfalse\IEEEQED}
 % command to disable QED at end of IEEEproof
@@ -5560,37 +5574,38 @@ between column types.}%
 \newdimen\@IEEEtmpitemindent
 
 \ifCLASSOPTIONcompsoc
-% V1.8a compsoc uses bold theorem titles, a period instead of a colon, vertical spacing, and hanging indentation
-% V1.8 allow long theorem names to break across lines.
-% Thanks to Miquel Payaro for reporting this.
-\def\@begintheorem#1#2{\@IEEEtmpitemindent\itemindent\relax
-    \topsep 0.2\@IEEEnormalsizeunitybaselineskip plus 0.26\@IEEEnormalsizeunitybaselineskip minus 0.05\@IEEEnormalsizeunitybaselineskip
-    \rmfamily\trivlist\hangindent\parindent%
-    \item[]\textit{\bfseries\noindent #1\ #2.} \itemindent\@IEEEtmpitemindent\relax}
-\def\@opargbegintheorem#1#2#3{\@IEEEtmpitemindent\itemindent\relax
-\topsep 0.2\@IEEEnormalsizeunitybaselineskip plus 0.26\@IEEEnormalsizeunitybaselineskip minus 0.05\@IEEEnormalsizeunitybaselineskip
-\rmfamily\trivlist\hangindent\parindent%
-% V1.6 The IEEE is back to using () around theorem names which are also in italics
-% Thanks to Christian Peel for reporting this.
-    \item[]\textit{\bfseries\noindent #1\ #2\ (#3).} \itemindent\@IEEEtmpitemindent\relax}
-% V1.7 remove bogus \unskip that caused equations in theorems to collide with
-% lines below.
-\def\@endtheorem{\endtrivlist\vskip 0.25\@IEEEnormalsizeunitybaselineskip plus 0.26\@IEEEnormalsizeunitybaselineskip minus 0.05\@IEEEnormalsizeunitybaselineskip}
+    % V1.8a compsoc uses bold theorem titles, a period instead of a colon, vertical spacing, and hanging indentation
+    % V1.8 allow long theorem names to break across lines.
+    % Thanks to Miquel Payaro for reporting this.
+    \def\@begintheorem#1#2{\@IEEEtmpitemindent\itemindent\relax
+        \topsep 0.2\@IEEEnormalsizeunitybaselineskip plus 0.26\@IEEEnormalsizeunitybaselineskip minus 0.05\@IEEEnormalsizeunitybaselineskip
+        \rmfamily\trivlist\hangindent\parindent%
+        \item[]\textit{\bfseries\noindent #1\ #2.} \itemindent\@IEEEtmpitemindent\relax}
+    \def\@opargbegintheorem#1#2#3{\@IEEEtmpitemindent\itemindent\relax
+        \topsep 0.2\@IEEEnormalsizeunitybaselineskip plus 0.26\@IEEEnormalsizeunitybaselineskip minus 0.05\@IEEEnormalsizeunitybaselineskip
+        \rmfamily\trivlist\hangindent\parindent%
+        % V1.6 The IEEE is back to using () around theorem names which are also in italics
+        % Thanks to Christian Peel for reporting this.
+        \item[]\textit{\bfseries\noindent #1\ #2\ (#3).} \itemindent\@IEEEtmpitemindent\relax}
+    % V1.7 remove bogus \unskip that caused equations in theorems to collide with
+    % lines below.
+    \def\@endtheorem{\endtrivlist\vskip 0.25\@IEEEnormalsizeunitybaselineskip plus 0.26\@IEEEnormalsizeunitybaselineskip minus 0.05\@IEEEnormalsizeunitybaselineskip}
 \else
-%
-% noncompsoc
-% 
-% V1.8 allow long theorem names to break across lines.
-% Thanks to Miquel Payaro for reporting this.
-\def\@begintheorem#1#2{\@IEEEtmpitemindent\itemindent\relax\topsep 0pt\rmfamily\trivlist%
-    \item[]\textit{\indent #1\ #2:} \itemindent\@IEEEtmpitemindent\relax}
-\def\@opargbegintheorem#1#2#3{\@IEEEtmpitemindent\itemindent\relax\topsep 0pt\rmfamily \trivlist%
-% V1.6 The IEEE is back to using () around theorem names which are also in italics
-% Thanks to Christian Peel for reporting this.
-    \item[]\textit{\indent #1\ #2\ (#3):} \itemindent\@IEEEtmpitemindent\relax}
-% V1.7 remove bogus \unskip that caused equations in theorems to collide with
-% lines below.
-\def\@endtheorem{\endtrivlist}
+    %
+    % noncompsoc
+    % 
+    % V1.8 allow long theorem names to break across lines.
+    % Thanks to Miquel Payaro for reporting this.
+    \def\@begintheorem#1#2{\@IEEEtmpitemindent\itemindent\relax\topsep 0pt\rmfamily\trivlist%
+        \item[]\textit{\indent #1\ #2:} \itemindent\@IEEEtmpitemindent\relax}
+    \def\@opargbegintheorem#1#2#3{\@IEEEtmpitemindent\itemindent\relax\topsep 0pt\rmfamily \trivlist%
+        % V1.6 The IEEE is back to using () around theorem names which are also in italics
+        % Thanks to Christian Peel for reporting this.
+        \item[]\textit{\indent #1\ #2\ (#3):} \itemindent\@IEEEtmpitemindent\relax}
+    % V1.7 remove bogus \unskip that caused equations in theorems to collide with
+    % lines below.
+    \def\@endtheorem{\endtrivlist}
+
 \fi
 
 
@@ -5611,20 +5626,20 @@ between column types.}%
 % redefine the #1#2[#3] form of newtheorem to use a hook to \@IEEEthmcounterinsection
 % if section in_counter is used
 \def\@xnthm#1#2[#3]{%
-  \expandafter\@ifdefinable\csname #1\endcsname
+    \expandafter\@ifdefinable\csname #1\endcsname
     {\@definecounter{#1}\@newctr{#1}[#3]%
-     \edef\@IEEEstringtmp{#3}
-     \ifx\@IEEEstringtmp\@IEEEstringsection
-     \expandafter\xdef\csname the#1\endcsname{%
-     \noexpand\@IEEEthmcounterinsection{#3}\@thmcountersep
-          \@thmcounter{#1}}%
-     \else
-     \expandafter\xdef\csname the#1\endcsname{%
-       \expandafter\noexpand\csname the#3\endcsname \@thmcountersep
-          \@thmcounter{#1}}%
-     \fi
-     \global\@namedef{#1}{\@thm{#1}{#2}}%
-     \global\@namedef{end#1}{\@endtheorem}}}
+        \edef\@IEEEstringtmp{#3}
+        \ifx\@IEEEstringtmp\@IEEEstringsection
+            \expandafter\xdef\csname the#1\endcsname{%
+                \noexpand\@IEEEthmcounterinsection{#3}\@thmcountersep
+                \@thmcounter{#1}}%
+        \else
+            \expandafter\xdef\csname the#1\endcsname{%
+                \expandafter\noexpand\csname the#3\endcsname \@thmcountersep
+                \@thmcounter{#1}}%
+        \fi
+        \global\@namedef{#1}{\@thm{#1}{#2}}%
+        \global\@namedef{end#1}{\@endtheorem}}}
 
 
 
@@ -5637,11 +5652,12 @@ between column types.}%
 % however, for peerreview the cover sheet is page 0 or page -1
 % (for duplex printing)
 \ifCLASSOPTIONpeerreview
-   \if@twoside
-      \setcounter{page}{-1}
-   \else
-      \setcounter{page}{0}
-   \fi
+    \if@twoside
+        \setcounter{page}{-1}
+    \else
+        \setcounter{page}{0}
+
+    \fi
 \fi
 
 % standard book class behavior - let bottom line float up and down as
@@ -5650,12 +5666,12 @@ between column types.}%
 % if two column - turn on twocolumn, allow word spacings to stretch more and
 % enforce a rigid position for the last lines
 \ifCLASSOPTIONtwocolumn
-% the peer review option delays invoking twocolumn
-   \ifCLASSOPTIONpeerreview\else
-      \twocolumn
-   \fi
-\sloppy 
-\flushbottom
+    % the peer review option delays invoking twocolumn
+    \ifCLASSOPTIONpeerreview\else
+        \twocolumn
+    \fi
+    \sloppy 
+    \flushbottom
 \fi
 
 
@@ -5693,24 +5709,24 @@ between column types.}%
 % we don't save \section here as it may be redefined later by other
 % packages (hyperref.sty, etc.)
 \def\@IEEEsaveoriginalsectiononce{\let\@IEEEappendixsavesection\section
-\let\@IEEEsaveoriginalsectiononce\relax}
+    \let\@IEEEsaveoriginalsectiononce\relax}
 
 % neat trick to grab and process the argument from \section{argument}
 % we process differently if the user invoked \section{} with no
 % argument (title)
 % note we reroute the call to the old \section*
 \def\@IEEEprocessthesectionargument#1{%
-\@ifmtarg{#1}{%
-\@IEEEappendixsavesection*{\appendixname\nobreakspace\thesectiondis}%
-\addcontentsline{toc}{section}{\appendixname\nobreakspace\thesection}}{%
-\@IEEEappendixsavesection*{\appendixname\nobreakspace\thesectiondis\\* #1}%
-\addcontentsline{toc}{section}{\appendixname\nobreakspace\thesection: #1}}}
+    \@ifmtarg{#1}{%
+        \@IEEEappendixsavesection*{\appendixname\nobreakspace\thesectiondis}%
+        \addcontentsline{toc}{section}{\appendixname\nobreakspace\thesection}}{%
+        \@IEEEappendixsavesection*{\appendixname\nobreakspace\thesectiondis\\* #1}%
+        \addcontentsline{toc}{section}{\appendixname\nobreakspace\thesection: #1}}}
 
 % we use this if the user calls \section{} after
 % \appendix-- which has no meaning. So, we ignore the
 % command and its argument. Then, warn the user.
 \def\@IEEEdestroythesectionargument#1{\typeout{** WARNING: Ignoring useless
-\protect\section\space in Appendix (line \the\inputlineno).}}
+        \protect\section\space in Appendix (line \the\inputlineno).}}
 
 
 % remember \thesection forms will be displayed in \ref calls
@@ -5737,14 +5753,14 @@ between column types.}%
     \gdef\@IEEEthmcounterinsection##1{A}
     \refstepcounter{section}% update the \ref counter
     \@ifmtarg{#1}{\@IEEEappendixsavesection*{\appendixname}%
-                  \addcontentsline{toc}{section}{\appendixname}}{%
-             \@IEEEappendixsavesection*{\appendixname\nobreakspace\\* #1}%
-             \addcontentsline{toc}{section}{\appendixname: #1}}%
+        \addcontentsline{toc}{section}{\appendixname}}{%
+        \@IEEEappendixsavesection*{\appendixname\nobreakspace\\* #1}%
+        \addcontentsline{toc}{section}{\appendixname: #1}}%
     % redefine \section command for appendix
     % leave \section* as is
     \def\section{\@ifstar{\@IEEEappendixsavesection*}{%
-                    \@IEEEdestroythesectionargument}}% throw out the argument
-                                                     % of the normal form
+          \@IEEEdestroythesectionargument}}% throw out the argument
+    % of the normal form
 }
 
 
@@ -5762,33 +5778,34 @@ between column types.}%
     \setcounter{subsubsection}{0}%
     \setcounter{paragraph}{0}%
     \ifCLASSOPTIONromanappendices%
-    \gdef\thesection{\Roman{section}}%
-    \gdef\thesectiondis{\Roman{section}}%
-    \@IEEEcompsocconfonly{\gdef\thesectiondis{\Roman{section}.}}%
-    \gdef\@IEEEthmcounterinsection##1{A\arabic{##1}}
+        \gdef\thesection{\Roman{section}}%
+        \gdef\thesectiondis{\Roman{section}}%
+        \@IEEEcompsocconfonly{\gdef\thesectiondis{\Roman{section}.}}%
+        \gdef\@IEEEthmcounterinsection##1{A\arabic{##1}}
     \else%
-    \gdef\thesection{\Alph{section}}%
-    \gdef\thesectiondis{\Alph{section}}%
-    \@IEEEcompsocconfonly{\gdef\thesectiondis{\Alph{section}.}}%
-    \gdef\@IEEEthmcounterinsection##1{\Alph{##1}}
+        \gdef\thesection{\Alph{section}}%
+        \gdef\thesectiondis{\Alph{section}}%
+        \@IEEEcompsocconfonly{\gdef\thesectiondis{\Alph{section}.}}%
+        \gdef\@IEEEthmcounterinsection##1{\Alph{##1}}
+
     \fi%
     \refstepcounter{section}% update the \ref counter
     \setcounter{section}{0}% NEXT \section will be the FIRST appendix
     % redefine \section command for appendices
     % leave \section* as is
     \def\section{\@ifstar{\@IEEEappendixsavesection*}{% process the *-form
-                    \refstepcounter{section}% or is a new section so,
-                    \@IEEEprocessthesectionargument}}% process the argument 
-                                                 % of the normal form
+          \refstepcounter{section}% or is a new section so,
+          \@IEEEprocessthesectionargument}}% process the argument 
+    % of the normal form
 }
 
 
 
 % V1.7 compoc uses nonbold drop cap and small caps word style
 \ifCLASSOPTIONcompsoc
-  \def\IEEEPARstartFONTSTYLE{\mdseries}
-  \def\IEEEPARstartWORDFONTSTYLE{\scshape}
-  \def\IEEEPARstartWORDCAPSTYLE{\relax}
+    \def\IEEEPARstartFONTSTYLE{\mdseries}
+    \def\IEEEPARstartWORDFONTSTYLE{\scshape}
+    \def\IEEEPARstartWORDCAPSTYLE{\relax}
 \fi
 %
 %
@@ -5861,82 +5878,82 @@ between column types.}%
 % The first argument is the first letter of the first word and the second
 % argument is the rest of the first word(s).
 \def\IEEEPARstart#1#2{\par{%
-% if this page does not have enough space, break it and lets start
-% on a new one
-\@IEEEtranneedspace{\IEEEPARstartMINPAGELINES\baselineskip}{\relax}%
-% V1.7 move this up here in case user uses \textbf for \IEEEPARstartFONTSTYLE
-% which uses command \leavevmode which causes an unwanted \indent to be issued
-\noindent
-% calculate the desired height of the big letter
-% it extends from the top of \IEEEPARstartHEIGHTTEXT in the current font
-% down to \IEEEPARstartDROPDEPTH below the current baseline
-\settoheight{\@IEEEtrantmpdimenA}{\IEEEPARstartHEIGHTTEXT}%
-\addtolength{\@IEEEtrantmpdimenA}{\IEEEPARstartDROPDEPTH}%
-% extract the name of the current font in bold
-% and place it in \@IEEEPARstartFONTNAME
-\def\@IEEEPARstartGETFIRSTWORD##1 ##2\relax{##1}%
-{\IEEEPARstartFONTSTYLE{\selectfont\edef\@IEEEPARstartFONTNAMESPACE{\fontname\font\space}%
-\xdef\@IEEEPARstartFONTNAME{\expandafter\@IEEEPARstartGETFIRSTWORD\@IEEEPARstartFONTNAMESPACE\relax}}}%
-% define a font based on this name with a point size equal to the desired
-% height of the drop letter
-\font\@IEEEPARstartsubfont\@IEEEPARstartFONTNAME\space at \@IEEEtrantmpdimenA\relax%
-% save this value as a counter (integer) value (sp points)
-\@IEEEtrantmpcountA=\@IEEEtrantmpdimenA%
-% now get the height of the actual letter produced by this font size
-\settoheight{\@IEEEtrantmpdimenB}{\@IEEEPARstartsubfont\IEEEPARstartCAPSTYLE{#1}}%
-% If something bogus happens like the first argument is empty or the
-% current font is strange, do not allow a zero height.
-\ifdim\@IEEEtrantmpdimenB=0pt\relax%
-\typeout{** WARNING: IEEEPARstart drop letter has zero height! (line \the\inputlineno)}%
-\typeout{ Forcing the drop letter font size to 10pt.}%
-\@IEEEtrantmpdimenB=10pt%
-\fi%
-% and store it as a counter
-\@IEEEtrantmpcountB=\@IEEEtrantmpdimenB%
-% Since a font size doesn't exactly correspond to the height of the capital
-% letters in that font, the actual height of the letter, \@IEEEtrantmpcountB,
-% will be less than that desired, \@IEEEtrantmpcountA
-% we need to raise the font size, \@IEEEtrantmpdimenA 
-% by \@IEEEtrantmpcountA / \@IEEEtrantmpcountB
-% But, TeX doesn't have floating point division, so we have to use integer
-% division. Hence the use of the counters.
-% We need to reduce the denominator so that the loss of the remainder will
-% have minimal affect on the accuracy of the result
-\divide\@IEEEtrantmpcountB by 200%
-\divide\@IEEEtrantmpcountA by \@IEEEtrantmpcountB%
-% Then reequalize things when we use TeX's ability to multiply by
-% floating point values
-\@IEEEtrantmpdimenB=0.005\@IEEEtrantmpdimenA%
-\multiply\@IEEEtrantmpdimenB by \@IEEEtrantmpcountA%
-% \@IEEEPARstartfont is globaly set to the calculated font of the big letter
-% We need to carry this out of the local calculation area to to create the
-% big letter.
-\global\font\@IEEEPARstartfont\@IEEEPARstartFONTNAME\space at \@IEEEtrantmpdimenB%
-% Now set \@IEEEtrantmpdimenA to the width of the big letter
-% We need to carry this out of the local calculation area to set the
-% hanging indent
-\settowidth{\global\@IEEEtrantmpdimenA}{\@IEEEPARstartfont
-\IEEEPARstartCAPSTYLE{#1\IEEEPARstartITLCORRECT}}}%
-% end of the isolated calculation environment
-\global\IEEEPARstartletwidth\@IEEEtrantmpdimenA\relax%
-% add in the extra clearance we want
-\advance\@IEEEtrantmpdimenA by \IEEEPARstartSEP\relax%
-% add in the optional offset
-\advance\@IEEEtrantmpdimenA by \IEEEPARstartHOFFSET\relax%
-% V1.7 don't allow negative offsets to produce negative hanging indents
-\@IEEEtrantmpdimenB\@IEEEtrantmpdimenA
-\ifnum\@IEEEtrantmpdimenB < 0 \@IEEEtrantmpdimenB 0pt\fi
-% \@IEEEtrantmpdimenA has the width of the big letter plus the
-% separation space and \@IEEEPARstartfont is the font we need to use
-% Now, we make the letter and issue the hanging indent command
-% The letter is placed in a box of zero width and height so that other
-% text won't be displaced by it.
-\hangindent\@IEEEtrantmpdimenB\hangafter=-\IEEEPARstartDROPLINES%
-\makebox[0pt][l]{\hspace{-\@IEEEtrantmpdimenA}%
-\raisebox{-\IEEEPARstartDROPDEPTH}[0pt][0pt]{\hspace{\IEEEPARstartHOFFSET}%
-\@IEEEPARstartfont\IEEEPARstartCAPSTYLE{#1\IEEEPARstartITLCORRECT}%
-\hspace{\IEEEPARstartSEP}}}%
-{\IEEEPARstartWORDFONTSTYLE{\IEEEPARstartWORDCAPSTYLE{\selectfont#2}}}}
+        % if this page does not have enough space, break it and lets start
+        % on a new one
+        \@IEEEtranneedspace{\IEEEPARstartMINPAGELINES\baselineskip}{\relax}%
+        % V1.7 move this up here in case user uses \textbf for \IEEEPARstartFONTSTYLE
+        % which uses command \leavevmode which causes an unwanted \indent to be issued
+        \noindent
+        % calculate the desired height of the big letter
+        % it extends from the top of \IEEEPARstartHEIGHTTEXT in the current font
+        % down to \IEEEPARstartDROPDEPTH below the current baseline
+        \settoheight{\@IEEEtrantmpdimenA}{\IEEEPARstartHEIGHTTEXT}%
+        \addtolength{\@IEEEtrantmpdimenA}{\IEEEPARstartDROPDEPTH}%
+        % extract the name of the current font in bold
+        % and place it in \@IEEEPARstartFONTNAME
+        \def\@IEEEPARstartGETFIRSTWORD##1 ##2\relax{##1}%
+        {\IEEEPARstartFONTSTYLE{\selectfont\edef\@IEEEPARstartFONTNAMESPACE{\fontname\font\space}%
+                \xdef\@IEEEPARstartFONTNAME{\expandafter\@IEEEPARstartGETFIRSTWORD\@IEEEPARstartFONTNAMESPACE\relax}}}%
+        % define a font based on this name with a point size equal to the desired
+        % height of the drop letter
+        \font\@IEEEPARstartsubfont\@IEEEPARstartFONTNAME\space at \@IEEEtrantmpdimenA\relax%
+        % save this value as a counter (integer) value (sp points)
+        \@IEEEtrantmpcountA=\@IEEEtrantmpdimenA%
+        % now get the height of the actual letter produced by this font size
+        \settoheight{\@IEEEtrantmpdimenB}{\@IEEEPARstartsubfont\IEEEPARstartCAPSTYLE{#1}}%
+        % If something bogus happens like the first argument is empty or the
+        % current font is strange, do not allow a zero height.
+        \ifdim\@IEEEtrantmpdimenB=0pt\relax%
+            \typeout{** WARNING: IEEEPARstart drop letter has zero height! (line \the\inputlineno)}%
+            \typeout{ Forcing the drop letter font size to 10pt.}%
+            \@IEEEtrantmpdimenB=10pt%
+        \fi%
+        % and store it as a counter
+        \@IEEEtrantmpcountB=\@IEEEtrantmpdimenB%
+        % Since a font size doesn't exactly correspond to the height of the capital
+        % letters in that font, the actual height of the letter, \@IEEEtrantmpcountB,
+        % will be less than that desired, \@IEEEtrantmpcountA
+        % we need to raise the font size, \@IEEEtrantmpdimenA 
+        % by \@IEEEtrantmpcountA / \@IEEEtrantmpcountB
+        % But, TeX doesn't have floating point division, so we have to use integer
+        % division. Hence the use of the counters.
+        % We need to reduce the denominator so that the loss of the remainder will
+        % have minimal affect on the accuracy of the result
+        \divide\@IEEEtrantmpcountB by 200%
+        \divide\@IEEEtrantmpcountA by \@IEEEtrantmpcountB%
+        % Then reequalize things when we use TeX's ability to multiply by
+        % floating point values
+        \@IEEEtrantmpdimenB=0.005\@IEEEtrantmpdimenA%
+        \multiply\@IEEEtrantmpdimenB by \@IEEEtrantmpcountA%
+        % \@IEEEPARstartfont is globaly set to the calculated font of the big letter
+        % We need to carry this out of the local calculation area to to create the
+        % big letter.
+        \global\font\@IEEEPARstartfont\@IEEEPARstartFONTNAME\space at \@IEEEtrantmpdimenB%
+        % Now set \@IEEEtrantmpdimenA to the width of the big letter
+        % We need to carry this out of the local calculation area to set the
+        % hanging indent
+        \settowidth{\global\@IEEEtrantmpdimenA}{\@IEEEPARstartfont
+            \IEEEPARstartCAPSTYLE{#1\IEEEPARstartITLCORRECT}}}%
+    % end of the isolated calculation environment
+    \global\IEEEPARstartletwidth\@IEEEtrantmpdimenA\relax%
+    % add in the extra clearance we want
+    \advance\@IEEEtrantmpdimenA by \IEEEPARstartSEP\relax%
+    % add in the optional offset
+    \advance\@IEEEtrantmpdimenA by \IEEEPARstartHOFFSET\relax%
+    % V1.7 don't allow negative offsets to produce negative hanging indents
+    \@IEEEtrantmpdimenB\@IEEEtrantmpdimenA
+    \ifnum\@IEEEtrantmpdimenB < 0 \@IEEEtrantmpdimenB 0pt\fi
+    % \@IEEEtrantmpdimenA has the width of the big letter plus the
+    % separation space and \@IEEEPARstartfont is the font we need to use
+    % Now, we make the letter and issue the hanging indent command
+    % The letter is placed in a box of zero width and height so that other
+    % text won't be displaced by it.
+    \hangindent\@IEEEtrantmpdimenB\hangafter=-\IEEEPARstartDROPLINES%
+    \makebox[0pt][l]{\hspace{-\@IEEEtrantmpdimenA}%
+        \raisebox{-\IEEEPARstartDROPDEPTH}[0pt][0pt]{\hspace{\IEEEPARstartHOFFSET}%
+            \@IEEEPARstartfont\IEEEPARstartCAPSTYLE{#1\IEEEPARstartITLCORRECT}%
+            \hspace{\IEEEPARstartSEP}}}%
+    {\IEEEPARstartWORDFONTSTYLE{\IEEEPARstartWORDCAPSTYLE{\selectfont#2}}}}
 
 
 
@@ -5955,11 +5972,11 @@ between column types.}%
 % so DO NOT PASS \@IEEEtrantmpdimenB to this routine
 % if you need a dimen register, import with \@IEEEtrantmpdimenA instead
 \def\@IEEEtranneedspace#1#2{\penalty-100\begingroup%shield temp variable
-\@IEEEtrantmpdimenB\pagegoal\advance\@IEEEtrantmpdimenB-\pagetotal% space left
-\ifdim #1>\@IEEEtrantmpdimenB\relax% not enough space left
-\ifdim\@IEEEtrantmpdimenB>\z@\relax #2\fi%
-\newpage%
-\fi\endgroup}
+    \@IEEEtrantmpdimenB\pagegoal\advance\@IEEEtrantmpdimenB-\pagetotal% space left
+    \ifdim #1>\@IEEEtrantmpdimenB\relax% not enough space left
+        \ifdim\@IEEEtrantmpdimenB>\z@\relax #2\fi%
+        \newpage%
+    \fi\endgroup}
 
 
 
@@ -5984,102 +6001,102 @@ between column types.}%
 % area cleared for photo
 \def\@IEEEBIOhangwidth{1.14in}    % width cleared for the biography photo area
 \def\@IEEEBIOhangdepth{1.25in}    % depth cleared for the biography photo area
-                                  % actual depth will be a multiple of 
-                                  % \baselineskip, rounded up
+% actual depth will be a multiple of 
+% \baselineskip, rounded up
 \def\@IEEEBIOskipN{4\baselineskip}% nominal value of the vskip above the biography
 
 \newenvironment{IEEEbiography}[2][]{\normalfont\@IEEEcompsoconly{\sffamily}\footnotesize%
-\unitlength 1in\parskip=0pt\par\parindent 1em\interlinepenalty500%
-% we need enough space to support the hanging indent
-% the nominal value of the spacer
-% and one extra line for good measure
-\@IEEEtrantmpdimenA=\@IEEEBIOhangdepth%
-\advance\@IEEEtrantmpdimenA by \@IEEEBIOskipN%
-\advance\@IEEEtrantmpdimenA by 1\baselineskip%
-% if this page does not have enough space, break it and lets start
-% with a new one
-\@IEEEtranneedspace{\@IEEEtrantmpdimenA}{\relax}%
-% nominal spacer can strech, not shrink use 1fil so user can out stretch with \vfill
-\vskip \@IEEEBIOskipN plus 1fil minus 0\baselineskip%
-% the default box for where the photo goes
-\def\@IEEEtempbiographybox{{\setlength{\fboxsep}{0pt}\framebox{%
-\begin{minipage}[b][\@IEEEBIOphotodepth][c]{\@IEEEBIOphotowidth}\centering PLACE\\ PHOTO\\ HERE \end{minipage}}}}%
-%
-% detect if the optional argument was supplied, this requires the
-% \@ifmtarg command as defined in the appendix section above
-% and if so, override the default box with what they want
-\@ifmtarg{#1}{\relax}{\def\@IEEEtempbiographybox{\mbox{\begin{minipage}[b][\@IEEEBIOphotodepth][c]{\@IEEEBIOphotowidth}%
-\centering%
-#1%
-\end{minipage}}}}% end if optional argument supplied
-% Make an entry into the table of contents only if we have not done so before
-\if@IEEEbiographyTOCentrynotmade%
-% link labels to the biography counter so hyperref will jump
-% to the biography, not the previous section
-\setcounter{IEEEbiography}{-1}%
-\refstepcounter{IEEEbiography}%
-\addcontentsline{toc}{section}{Biographies}%
-\global\@IEEEbiographyTOCentrynotmadefalse%
-\fi%
-% one more biography
-\refstepcounter{IEEEbiography}%
-% Make an entry for this name into the table of contents 
-\addcontentsline{toc}{subsection}{#2}%
-% V1.6 properly handle if a new paragraph should occur while the
-% hanging indent is still active. Do this by redefining \par so
-% that it will not start a new paragraph. (But it will appear to the
-% user as if it did.) Also, strip any leading pars, newlines, or spaces.
-\let\@IEEEBIOORGparCMD=\par% save the original \par command
-\edef\par{\hfil\break\indent}% the new \par will not be a "real" \par
-\settoheight{\@IEEEtrantmpdimenA}{\@IEEEtempbiographybox}% get height of biography box
-\@IEEEtrantmpdimenB=\@IEEEBIOhangdepth%
-\@IEEEtrantmpcountA=\@IEEEtrantmpdimenB% countA has the hang depth
-\divide\@IEEEtrantmpcountA by \baselineskip%  calculates lines needed to produce the hang depth
-\advance\@IEEEtrantmpcountA by 1% ensure we overestimate
-% set the hanging indent
-\hangindent\@IEEEBIOhangwidth%
-\hangafter-\@IEEEtrantmpcountA%
-% reference the top of the photo area to the top of a capital T
-\settoheight{\@IEEEtrantmpdimenB}{\mbox{T}}%
-% set the photo box, give it zero width and height so as not to disturb anything
-\noindent\makebox[0pt][l]{\hspace{-\@IEEEBIOhangwidth}\raisebox{\@IEEEtrantmpdimenB}[0pt][0pt]{%
-\raisebox{-\@IEEEBIOphotodepth}[0pt][0pt]{\@IEEEtempbiographybox}}}%
-% now place the author name and begin the bio text
-\noindent\textbf{#2\ }\@IEEEgobbleleadPARNLSP}{\relax\let\par=\@IEEEBIOORGparCMD\par%
-% 7/2001 V1.5 detect when the biography text is shorter than the photo area
-% and pad the unused area - preventing a collision from the next biography entry
-% MDS
-\ifnum \prevgraf <\@IEEEtrantmpcountA\relax% detect when the biography text is shorter than the photo
-    \advance\@IEEEtrantmpcountA by -\prevgraf% calculate how many lines we need to pad
-    \advance\@IEEEtrantmpcountA by -1\relax% we compensate for the fact that we indented an extra line
-    \@IEEEtrantmpdimenA=\baselineskip% calculate the length of the padding
-    \multiply\@IEEEtrantmpdimenA by \@IEEEtrantmpcountA%
-    \noindent\rule{0pt}{\@IEEEtrantmpdimenA}% insert an invisible support strut
-\fi%
-\par\normalfont}
+    \unitlength 1in\parskip=0pt\par\parindent 1em\interlinepenalty500%
+    % we need enough space to support the hanging indent
+    % the nominal value of the spacer
+    % and one extra line for good measure
+    \@IEEEtrantmpdimenA=\@IEEEBIOhangdepth%
+    \advance\@IEEEtrantmpdimenA by \@IEEEBIOskipN%
+    \advance\@IEEEtrantmpdimenA by 1\baselineskip%
+    % if this page does not have enough space, break it and lets start
+    % with a new one
+    \@IEEEtranneedspace{\@IEEEtrantmpdimenA}{\relax}%
+    % nominal spacer can strech, not shrink use 1fil so user can out stretch with \vfill
+    \vskip \@IEEEBIOskipN plus 1fil minus 0\baselineskip%
+    % the default box for where the photo goes
+    \def\@IEEEtempbiographybox{{\setlength{\fboxsep}{0pt}\framebox{%
+                    \begin{minipage}[b][\@IEEEBIOphotodepth][c]{\@IEEEBIOphotowidth}\centering PLACE\\ PHOTO\\ HERE \end{minipage}}}}%
+    %
+    % detect if the optional argument was supplied, this requires the
+    % \@ifmtarg command as defined in the appendix section above
+    % and if so, override the default box with what they want
+    \@ifmtarg{#1}{\relax}{\def\@IEEEtempbiographybox{\mbox{\begin{minipage}[b][\@IEEEBIOphotodepth][c]{\@IEEEBIOphotowidth}%
+                    \centering%
+                    #1%
+                \end{minipage}}}}% end if optional argument supplied
+    % Make an entry into the table of contents only if we have not done so before
+    \if@IEEEbiographyTOCentrynotmade%
+        % link labels to the biography counter so hyperref will jump
+        % to the biography, not the previous section
+        \setcounter{IEEEbiography}{-1}%
+        \refstepcounter{IEEEbiography}%
+        \addcontentsline{toc}{section}{Biographies}%
+        \global\@IEEEbiographyTOCentrynotmadefalse%
+    \fi%
+    % one more biography
+    \refstepcounter{IEEEbiography}%
+    % Make an entry for this name into the table of contents 
+    \addcontentsline{toc}{subsection}{#2}%
+    % V1.6 properly handle if a new paragraph should occur while the
+    % hanging indent is still active. Do this by redefining \par so
+    % that it will not start a new paragraph. (But it will appear to the
+    % user as if it did.) Also, strip any leading pars, newlines, or spaces.
+    \let\@IEEEBIOORGparCMD=\par% save the original \par command
+    \edef\par{\hfil\break\indent}% the new \par will not be a "real" \par
+    \settoheight{\@IEEEtrantmpdimenA}{\@IEEEtempbiographybox}% get height of biography box
+    \@IEEEtrantmpdimenB=\@IEEEBIOhangdepth%
+    \@IEEEtrantmpcountA=\@IEEEtrantmpdimenB% countA has the hang depth
+    \divide\@IEEEtrantmpcountA by \baselineskip%  calculates lines needed to produce the hang depth
+    \advance\@IEEEtrantmpcountA by 1% ensure we overestimate
+    % set the hanging indent
+    \hangindent\@IEEEBIOhangwidth%
+    \hangafter-\@IEEEtrantmpcountA%
+    % reference the top of the photo area to the top of a capital T
+    \settoheight{\@IEEEtrantmpdimenB}{\mbox{T}}%
+    % set the photo box, give it zero width and height so as not to disturb anything
+    \noindent\makebox[0pt][l]{\hspace{-\@IEEEBIOhangwidth}\raisebox{\@IEEEtrantmpdimenB}[0pt][0pt]{%
+            \raisebox{-\@IEEEBIOphotodepth}[0pt][0pt]{\@IEEEtempbiographybox}}}%
+    % now place the author name and begin the bio text
+    \noindent\textbf{#2\ }\@IEEEgobbleleadPARNLSP}{\relax\let\par=\@IEEEBIOORGparCMD\par%
+    % 7/2001 V1.5 detect when the biography text is shorter than the photo area
+    % and pad the unused area - preventing a collision from the next biography entry
+    % MDS
+    \ifnum \prevgraf <\@IEEEtrantmpcountA\relax% detect when the biography text is shorter than the photo
+        \advance\@IEEEtrantmpcountA by -\prevgraf% calculate how many lines we need to pad
+        \advance\@IEEEtrantmpcountA by -1\relax% we compensate for the fact that we indented an extra line
+        \@IEEEtrantmpdimenA=\baselineskip% calculate the length of the padding
+        \multiply\@IEEEtrantmpdimenA by \@IEEEtrantmpcountA%
+        \noindent\rule{0pt}{\@IEEEtrantmpdimenA}% insert an invisible support strut
+    \fi%
+    \par\normalfont}
 
 
 
 % V1.6
 % added biography without a photo environment
 \newenvironment{IEEEbiographynophoto}[1]{%
-% Make an entry into the table of contents only if we have not done so before
-\if@IEEEbiographyTOCentrynotmade%
-% link labels to the biography counter so hyperref will jump
-% to the biography, not the previous section
-\setcounter{IEEEbiography}{-1}%
-\refstepcounter{IEEEbiography}%
-\addcontentsline{toc}{section}{Biographies}%
-\global\@IEEEbiographyTOCentrynotmadefalse%
-\fi%
-% one more biography
-\refstepcounter{IEEEbiography}%
-% Make an entry for this name into the table of contents 
-\addcontentsline{toc}{subsection}{#1}%
-\normalfont\@IEEEcompsoconly{\sffamily}\footnotesize\interlinepenalty500%
-\vskip 4\baselineskip plus 1fil minus 0\baselineskip%
-\parskip=0pt\par%
-\noindent\textbf{#1\ }\@IEEEgobbleleadPARNLSP}{\relax\par\normalfont}
+    % Make an entry into the table of contents only if we have not done so before
+    \if@IEEEbiographyTOCentrynotmade%
+        % link labels to the biography counter so hyperref will jump
+        % to the biography, not the previous section
+        \setcounter{IEEEbiography}{-1}%
+        \refstepcounter{IEEEbiography}%
+        \addcontentsline{toc}{section}{Biographies}%
+        \global\@IEEEbiographyTOCentrynotmadefalse%
+    \fi%
+    % one more biography
+    \refstepcounter{IEEEbiography}%
+    % Make an entry for this name into the table of contents 
+    \addcontentsline{toc}{subsection}{#1}%
+    \normalfont\@IEEEcompsoconly{\sffamily}\footnotesize\interlinepenalty500%
+    \vskip 4\baselineskip plus 1fil minus 0\baselineskip%
+    \parskip=0pt\par%
+    \noindent\textbf{#1\ }\@IEEEgobbleleadPARNLSP}{\relax\par\normalfont}
 
 
 % provide the user with some old font commands
@@ -6099,14 +6116,14 @@ between column types.}%
 % 
 % holds the special notice text
 \def\@IEEEspecialpapernotice{\relax}
- 
+
 % for special papers, like invited papers, the user can do:
 % \IEEEspecialpapernotice{(Invited Paper)} before \maketitle
 \def\IEEEspecialpapernotice#1{\ifCLASSOPTIONconference%
-\def\@IEEEspecialpapernotice{{\sublargesize\textit{#1}\vspace*{1em}}}%
-\else%
-\def\@IEEEspecialpapernotice{{\\*[1.5ex]\sublargesize\textit{#1}}\vspace*{-2ex}}%
-\fi}
+        \def\@IEEEspecialpapernotice{{\sublargesize\textit{#1}\vspace*{1em}}}%
+    \else%
+        \def\@IEEEspecialpapernotice{{\\*[1.5ex]\sublargesize\textit{#1}}\vspace*{-2ex}}%
+    \fi}
 
 
 
@@ -6126,20 +6143,20 @@ between column types.}%
 % dynamic title spacing will clean up the fraction
 \def\@IEEEpubidpullup{1.3\baselineskip}
 \ifCLASSOPTIONtechnote
-% for technotes it must be an integer of baselineskip as there can be no
-% dynamic title spacing for two column mode technotes (the title is in the
-% in first column) and we should maintain an integer number of lines in the
-% second column
-% There are some examples (such as older issues of "Transactions on
-% Information Theory") in which the IEEE really pulls the text off the ID for
-% technotes - about 0.55in (or 4\baselineskip). We'll use 2\baselineskip
-% and call it even.
-\def\@IEEEpubidpullup{2\baselineskip}
+    % for technotes it must be an integer of baselineskip as there can be no
+    % dynamic title spacing for two column mode technotes (the title is in the
+    % in first column) and we should maintain an integer number of lines in the
+    % second column
+    % There are some examples (such as older issues of "Transactions on
+    % Information Theory") in which the IEEE really pulls the text off the ID for
+    % technotes - about 0.55in (or 4\baselineskip). We'll use 2\baselineskip
+    % and call it even.
+    \def\@IEEEpubidpullup{2\baselineskip}
 \fi
 
 % V1.7 compsoc does not use a pullup
 \ifCLASSOPTIONcompsoc
-\def\@IEEEpubidpullup{0pt}
+    \def\@IEEEpubidpullup{0pt}
 \fi
 
 % holds the ID text
@@ -6165,7 +6182,7 @@ between column types.}%
 % selected because \IEEEpubidadjcol will not be on the cover page.
 % V1.7 do nothing if compsoc
 \def\IEEEpubidadjcol{\ifCLASSOPTIONcompsoc\else\ifCLASSOPTIONpeerreview\else
-\if@twocolumn\if@IEEEusingpubid\enlargethispage{-\@IEEEpubidpullup}\fi\fi\fi\fi}
+            \if@twocolumn\if@IEEEusingpubid\enlargethispage{-\@IEEEpubidpullup}\fi\fi\fi\fi}
 
 % Special thanks to Peter Wilson, Daniel Luecking, and the other
 % gurus at comp.text.tex, for helping me to understand how best to
@@ -6219,58 +6236,58 @@ between column types.}%
 % At any rate, it seems too formal to have a drop letter in a draft
 % paper.
 \ifCLASSOPTIONdraftcls
-\def\IEEEPARstart#1#2{#1#2\if@IEEEWARNIEEEPARstart\typeout{** ATTENTION: \noexpand\IEEEPARstart
- is disabled in draft mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEPARstartfalse}
+    \def\IEEEPARstart#1#2{#1#2\if@IEEEWARNIEEEPARstart\typeout{** ATTENTION: \noexpand\IEEEPARstart
+                is disabled in draft mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEPARstartfalse}
 \fi
 % and for technotes
 \ifCLASSOPTIONtechnote
-\def\IEEEPARstart#1#2{#1#2\if@IEEEWARNIEEEPARstart\typeout{** WARNING: \noexpand\IEEEPARstart
- is locked out for technotes (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEPARstartfalse}
+    \def\IEEEPARstart#1#2{#1#2\if@IEEEWARNIEEEPARstart\typeout{** WARNING: \noexpand\IEEEPARstart
+                is locked out for technotes (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEPARstartfalse}
 \fi
 
 
 % lockout unneeded commands when in conference mode
 \ifCLASSOPTIONconference
-% when locked out, \thanks, \IEEEbiography, \IEEEbiographynophoto, \IEEEpubid,
-% \IEEEmembership and \IEEEaftertitletext will all swallow their given text. 
-% \IEEEPARstart will output a normal character instead
-% warn the user about these commands only once to prevent the console screen
-% from filling up with redundant messages
-\def\thanks#1{\if@IEEEWARNthanks\typeout{** WARNING: \noexpand\thanks
- is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNthanksfalse}
-\def\IEEEPARstart#1#2{#1#2\if@IEEEWARNIEEEPARstart\typeout{** WARNING: \noexpand\IEEEPARstart
- is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEPARstartfalse}
-
-
-% LaTeX treats environments and commands with optional arguments differently.
-% the actual ("internal") command is stored as \\commandname 
-% (accessed via \csname\string\commandname\endcsname )
-% the "external" command \commandname is a macro with code to determine
-% whether or not the optional argument is presented and to provide the 
-% default if it is absent. So, in order to save and restore such a command
-% we would have to save and restore \\commandname as well. But, if LaTeX
-% ever changes the way it names the internal names, the trick would break.
-% Instead let us just define a new environment so that the internal
-% name can be left undisturbed.
-\newenvironment{@IEEEbogusbiography}[2][]{\if@IEEEWARNIEEEbiography\typeout{** WARNING: \noexpand\IEEEbiography
- is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEbiographyfalse%
-\setbox\@IEEEtranrubishbin\vbox\bgroup}{\egroup\relax}
-% and make biography point to our bogus biography
-\let\IEEEbiography=\@IEEEbogusbiography
-\let\endIEEEbiography=\end@IEEEbogusbiography
-
-\renewenvironment{IEEEbiographynophoto}[1]{\if@IEEEWARNIEEEbiographynophoto\typeout{** WARNING: \noexpand\IEEEbiographynophoto
- is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEbiographynophotofalse%
-\setbox\@IEEEtranrubishbin\vbox\bgroup}{\egroup\relax}
-
-\def\IEEEpubid#1{\if@IEEEWARNIEEEpubid\typeout{** WARNING: \noexpand\IEEEpubid 
- is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEpubidfalse}
-\def\IEEEpubidadjcol{\if@IEEEWARNIEEEpubidadjcol\typeout{** WARNING: \noexpand\IEEEpubidadjcol
- is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEpubidadjcolfalse}
-\def\IEEEmembership#1{\if@IEEEWARNIEEEmembership\typeout{** WARNING: \noexpand\IEEEmembership
- is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEmembershipfalse}
-\def\IEEEaftertitletext#1{\if@IEEEWARNIEEEaftertitletext\typeout{** WARNING: \noexpand\IEEEaftertitletext
- is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEaftertitletextfalse}
+    % when locked out, \thanks, \IEEEbiography, \IEEEbiographynophoto, \IEEEpubid,
+    % \IEEEmembership and \IEEEaftertitletext will all swallow their given text. 
+    % \IEEEPARstart will output a normal character instead
+    % warn the user about these commands only once to prevent the console screen
+    % from filling up with redundant messages
+    \def\thanks#1{\if@IEEEWARNthanks\typeout{** WARNING: \noexpand\thanks
+                is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNthanksfalse}
+    \def\IEEEPARstart#1#2{#1#2\if@IEEEWARNIEEEPARstart\typeout{** WARNING: \noexpand\IEEEPARstart
+                is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEPARstartfalse}
+    
+    
+    % LaTeX treats environments and commands with optional arguments differently.
+    % the actual ("internal") command is stored as \\commandname 
+    % (accessed via \csname\string\commandname\endcsname )
+    % the "external" command \commandname is a macro with code to determine
+    % whether or not the optional argument is presented and to provide the 
+    % default if it is absent. So, in order to save and restore such a command
+    % we would have to save and restore \\commandname as well. But, if LaTeX
+    % ever changes the way it names the internal names, the trick would break.
+    % Instead let us just define a new environment so that the internal
+    % name can be left undisturbed.
+    \newenvironment{@IEEEbogusbiography}[2][]{\if@IEEEWARNIEEEbiography\typeout{** WARNING: \noexpand\IEEEbiography
+                is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEbiographyfalse%
+        \setbox\@IEEEtranrubishbin\vbox\bgroup}{\egroup\relax}
+    % and make biography point to our bogus biography
+    \let\IEEEbiography=\@IEEEbogusbiography
+    \let\endIEEEbiography=\end@IEEEbogusbiography
+    
+    \renewenvironment{IEEEbiographynophoto}[1]{\if@IEEEWARNIEEEbiographynophoto\typeout{** WARNING: \noexpand\IEEEbiographynophoto
+                is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEbiographynophotofalse%
+        \setbox\@IEEEtranrubishbin\vbox\bgroup}{\egroup\relax}
+    
+    \def\IEEEpubid#1{\if@IEEEWARNIEEEpubid\typeout{** WARNING: \noexpand\IEEEpubid
+                is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEpubidfalse}
+    \def\IEEEpubidadjcol{\if@IEEEWARNIEEEpubidadjcol\typeout{** WARNING: \noexpand\IEEEpubidadjcol
+                is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEpubidadjcolfalse}
+    \def\IEEEmembership#1{\if@IEEEWARNIEEEmembership\typeout{** WARNING: \noexpand\IEEEmembership
+                is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEmembershipfalse}
+    \def\IEEEaftertitletext#1{\if@IEEEWARNIEEEaftertitletext\typeout{** WARNING: \noexpand\IEEEaftertitletext
+                is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEaftertitletextfalse}
 \fi
 
 
@@ -6297,7 +6314,7 @@ between column types.}%
 
 % hook to allow easy disabling of all legacy warnings
 \def\@IEEElegacywarn#1#2{\typeout{** ATTENTION: \@IEEEbackslash #1 is deprecated (line \the\inputlineno).
-Use \@IEEEbackslash #2 instead.}}
+        Use \@IEEEbackslash #2 instead.}}
 
 
 % provide some legacy IEEEtran commands
diff --git a/article/cutting-planes-article.pdf b/article/cutting-planes-article.pdf
index ef36b5a02ce566c475341e458ae880b6d0f16869..aabb73134eb53db414f6cb9d5df3bc05833ac25e 100644
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index c4dbc23..4df33f3 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -131,8 +131,10 @@ \subsection{Definição de Inequações Pseudo-Booleanas}
     x_1 + x_2 + \overline x_3 \ge 2
 \end{equation}
 
-A implementação em Lean 4 consiste de definir o tipo para coeficientes \textit{Coeff}, onde usamos a estrutura
-\textit{FinVec}, que permite termos uma lista com $n$ elementos, onde cada elemento da lista será um par
+Na implementação em Lean 4 utilizamos o tipo \texttt{Fin 2}, que só permite os valores $0$ ou $1$
+para os pseudo-booleanos.
+Para os coeficientes \texttt{Coeff}, usamos a estrutura \texttt{FinVec},
+que permite termos uma lista com $n$ elementos, onde cada elemento da lista será um par
 de dois números naturais, o tipo $\mathbb{N}$ em Lean:
 \begin{minted}[fontsize=\small]{lean4}
 abbrev Coeff (n : ℕ) := Fin n → (ℕ × ℕ)
@@ -294,7 +296,68 @@ \subsection{Prova da Regra Divisão}
 
 % * Prova da Regra Addition
 \subsection{Prova da Regra Adição}
-Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+Adição foi a última prova demonstrada, e se tornou a mais difícil, pois, além de somar duas inequações,
+dois literais de polaridades opostas se aniquilam, o que chamamos aqui de \textit{Redução}:
+
+\begin{equation}
+    \frac
+    {{\sum_i{a_i l_i} \ge A}\qquad {\sum_i{b_i l_i} \ge B}}
+    {\sum_i{(a_i + b_i) l_i} \ge (A+B)}
+\end{equation}
+
+Primeiro implementamos uma regra \textit{Addition'}, que realiza a adição diretamente sem a redução:
+
+\begin{minted}[fontsize=\small]{lean4}
+theorem Addition'
+  (xs : Fin n → Fin 2)
+  (as : Coeff n) (A : ℕ) (ha : PBIneq as xs A)
+  (bs : Coeff n) (B : ℕ) (hb : PBIneq bs xs B)
+  : PBIneq (as + bs) xs (A + B)
+\end{minted}
+
+Os teoremas da \textit{mathlib4} deram bom suporte à prova, o que \texttt{Finset.sum\_add\_distrib} resolveu em um passo.
+
+Um lema usado para seguir adiante foi \texttt{ite\_eq\_bmul},
+que nos permite transitar da notação \texttt{if-then-else}
+para a multiplicação dos termos pseudo-booleanos:
+
+\begin{minted}[fontsize=\small]{lean4}
+lemma ite_eq_bmul (x y : ℕ) (b : Fin 2)
+  : (if b = 1 then x else y)
+  = (x * b + y * (1 - b))
+\end{minted}
+
+Em seguida, implementamos a regra \textit{Reduction}, que toma uma inequação e aniquila coeficientes
+onde ambos elementos do par são maiores que $0$.
+Quando isso acontece, essa diferença "\textit{slack}" deve ser subtraída da constante:
+
+\begin{minted}[fontsize=\small]{lean4}
+def ReductionProp (xs : Fin n → Fin 2)
+    (ks : Coeff n) (K : ℕ) : Prop :=
+  let pos := λ i => ks i |>.1
+  let neg := λ i => ks i |>.2
+  let slack := (∑i, min (pos i) (neg i))
+  let rs := λ i => (pos i - neg i,neg i - pos i)
+  PBIneq rs xs (K - slack)
+
+theorem Reduction
+  (xs : Fin n → Fin 2)
+  (ks : Coeff n) (K : ℕ) (ha : PBIneq ks xs K)
+  : ReductionProp xs ks K
+\end{minted}
+
+Com esses dois teoremas, definimos \textit{Addition} compondo-os:
+
+\begin{minted}[fontsize=\small]{lean4}
+theorem Addition
+  {xs : Fin n → Fin 2}
+  {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
+  {bs : Coeff n} {B : ℕ} (hb : PBIneq bs xs B)
+  : ReductionProp xs (as + bs) (A + B) := by
+  have hk := Addition' xs as A ha bs B hb
+  exact Reduction xs (as + bs) (A + B) hk
+  done
+\end{minted}
 
 % * Implementação do Toy Example
 \subsection{Implementação do Exemplo}
@@ -320,27 +383,25 @@ \section*{Conclusões}
 \vspace{12pt}
 
 \newpage
-
-\section{Apêndice}
+\onecolumn
+\appendix
 
 \subsection{Definição de Pseudo-Boolean}
+\begin{minted}{lean4}
 
-% \begin{minted}{lean4}
-
-% open FinVec BigOperators
+open FinVec BigOperators
 
-% -- TODO: Add support for unicode chars in LaTeX
-% abbrev Coeff (n : .) := Fin n → (. . .)
+-- TODO: Add support for unicode chars in LaTeX
+abbrev Coeff (n : .) := Fin n → (. . .)
 
 
-% def PBSum (cs : Coeff n) (xs : Fin n → Fin 2) :=
-%   . i, let (p,n) := cs i;
-%     if xs i = 1 then p else n
-
-% def PBIneq (cs : Coeff n) (xs : Fin n → Fin 2) (const : .) :=
-%   PBSum cs xs . const
+def PBSum (cs : Coeff n) (xs : Fin n → Fin 2) :=
+. i, let (p,n) := cs i;
+if xs i = 1 then p else n
 
-% \end{minted}
+def PBIneq (cs : Coeff n) (xs : Fin n → Fin 2) (const : .) :=
+PBSum cs xs . const
+\end{minted}
 
 
 \end{document}

From c0d66a8dee1c428fb56747bb0f453f6ccdeb3999 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Fri, 28 Jun 2024 19:06:13 -0300
Subject: [PATCH 78/93] Appendix with proofs

---
 article/cutting-planes-article.pdf | Bin 217469 -> 236210 bytes
 article/cutting-planes-article.tex | 340 ++++++++++++++++++++++++++++-
 2 files changed, 328 insertions(+), 12 deletions(-)

diff --git a/article/cutting-planes-article.pdf b/article/cutting-planes-article.pdf
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 4df33f3..1134135 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -194,8 +194,7 @@ \subsection{Prova da Regra Multiplicação}
   : PBIneq (c • as) xs (c • A)
 \end{minted}
 
-% TODO: Criar apêndice A com código completo
-A prova completa se encontra no Apêndice A. Como a muliplicação por escalares já está definida na \textit{mathlib4} para
+A prova completa se encontra no Apêndice B. Como a muliplicação por escalares já está definida na \textit{mathlib4} para
 os \textit{FinVec}s pelo teorema \texttt{Finset.sum\_nsmul} da biblioteca.
 
 % * Prova da Regra Saturation
@@ -216,8 +215,7 @@ \subsection{Prova da Regra Saturação}
   : PBIneq (map (mapBoth (min A)) as) xs A
 \end{minted}
 
-% TODO: Criar apêndice B com código completo
-Este teorema (Apêndice B) envolveu mais passos, pois não havia o mesmo suporte nativo da \textit{mathlib4}.
+Este teorema (Apêndice C) envolveu mais passos, pois não havia o mesmo suporte nativo da \textit{mathlib4}.
 O lema que provamos, chamado \texttt{le\_sum\_min\_of\_le\_sum} é o caso mais simples, onde trabalhamos com uma
 lista de naturais e desejamos mostrar que a relação menor-ou-igual se mantém ao aplicar o mínimo de $A$:
 \begin{minted}[fontsize=\small]{lean4}
@@ -386,21 +384,339 @@ \section*{Conclusões}
 \onecolumn
 \appendix
 
-\subsection{Definição de Pseudo-Boolean}
+\subsection{Definição de Inequações Pseudo-Booleanas}
 \begin{minted}{lean4}
+import Mathlib.Data.Fin.Tuple.Reflection
 
-open FinVec BigOperators
+namespace PseudoBoolean
 
--- TODO: Add support for unicode chars in LaTeX
-abbrev Coeff (n : .) := Fin n → (. . .)
+open FinVec BigOperators
 
+abbrev Coeff (n : ℕ) := Fin n → (ℕ × ℕ)
 
 def PBSum (cs : Coeff n) (xs : Fin n → Fin 2) :=
-. i, let (p,n) := cs i;
-if xs i = 1 then p else n
+  ∑ i, let (p,n) := cs i;
+    if xs i = 1 then p else n
+
+def PBIneq (cs : Coeff n) (xs : Fin n → Fin 2) (const : ℕ) :=
+  PBSum cs xs ≥ const
+
+example : PBIneq ![(1,0)] ![1] 1 := by
+  reduce                -- Expand the goal to 1 * 1 ≥ 1
+  exact Nat.le_refl 1   -- Prove that 1 * 1 ≥ 1
+  done
+
+example : PBIneq ![(1,0),(2,0)] ![0,1] 2 := by
+  reduce                -- Change goal to 1 * 0 + 2 * 1 ≥ 2
+  exact Nat.le_refl 2   -- Prove 1 * 0 + 2 * 1 ≥ 2
+  done
+
+example : PBIneq ![(3,0),(4,0)] ![0,1] 2 := by
+  reduce
+  simp
+  done
+
+def mapBoth (f : α → β) (t : α × α) : β × β := Prod.map f f t
+
+end PseudoBoolean
+\end{minted}
+\newpage
+
+\subsection{Prova da Regra Multiplicação}
+\begin{minted}{lean4}
+import <<LeanCuttingPlanes>>.Basic
+
+namespace PseudoBoolean
+
+open BigOperators FinVec
+
+theorem Multiplication
+  {xs : Fin n → Fin 2}
+  {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
+  (c : ℕ)
+  : PBIneq (c • as) xs (c • A) := by
+  unfold PBIneq PBSum at *
+  simp only [Fin.isValue, ge_iff_le, nsmul_eq_smul, smul_eq_mul] at *
+  apply nsmul_le_nsmul_right at ha
+  specialize ha c
+  rw [←Finset.sum_nsmul] at ha
+  simp only [smul_eq_mul, Fin.isValue, mul_ite] at ha
+  exact ha
+  done
+
+example
+  (ha : PBIneq ![(1,0)] xs 3)
+  : PBIneq ![(2,0)] xs 6 := by
+  apply Multiplication ha 2
+  done
+
+end PseudoBoolean
+\end{minted}
+\newpage
+
+\subsection{Prova da Regra Saturação}
+\begin{minted}{lean4}
+import <<LeanCuttingPlanes>>.Basic
+
+namespace PseudoBoolean
+open FinVec Matrix BigOperators Finset
+
+-- @collares
+lemma split_summation (n : ℕ) (as : Fin n → ℕ) (k : Fin n) :
+    (∑i with i≠k, as i) + as k = (∑i, as i) := by
+  have : (∑i with i=k, as i) = as k := by rw [Finset.sum_eq_single_of_mem] <;> simp
+  rw [← this, ← Finset.sum_filter_add_sum_filter_not Finset.univ (· ≠ k)]
+  simp only [ne_eq, Decidable.not_not]
+
+lemma le_sum_min_of_le_sum {n A : ℕ} {as : Fin n → ℕ}
+  (h : A ≤ ∑i, as i)
+  : A ≤ ∑i, min A (as i) := by
+  by_cases ha : ∀i, as i ≤ A
+  . -- Assume all elements of as are ≤ A
+    simp_rw [@Nat.min_eq_right A (as _) (ha _)]
+    -- rewrite min A (as i) to (as i)
+    exact h
+
+  . -- Otherwise, ∃k, (as k) > A
+    simp only [not_forall, not_le] at ha
+    obtain ⟨k,hk⟩ := ha
+
+    rw [←split_summation]
+    -- Split goal from  ⊢ A ≤  ∑i, min A (as i)
+    -- to               ⊢ A ≤ (∑i with i ≠ k, min A (as i)) + min A (as k)
+
+    -- min A (as k) = A
+    rw [min_eq_left_of_lt hk]
+
+    -- ⊢ A ≤ (∑i,min A (as i) - A) + A
+    exact Nat.le_add_left A _
+
+theorem Saturation
+  {xs : Fin n → Fin 2}
+  {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
+  : PBIneq (map (mapBoth (min A)) as) xs A := by
+  unfold PBIneq PBSum FinVec.map mapBoth at *
+  simp only [Fin.isValue, ge_iff_le, Prod_map, seq_eq] at *
+  have h := le_sum_min_of_le_sum ha
+  simp_rw [apply_ite (min A) ((xs _ = 1)) ((as _).1) ((as _).2)] at h
+  exact h
+  done
+
+example
+  (ha : PBIneq ![(3,0),(4,0)] xs 3)
+  : PBIneq ![(3,0),(3,0)] xs 3 := by
+  apply Saturation ha
+  done
+
+end PseudoBoolean
+\end{minted}
+\newpage
+
+\subsection{Prova da Regra Divisão}
+\begin{minted}{lean4}
+import <<LeanCuttingPlanes>>.Basic
+import Mathlib.Algebra.Order.Floor.Div
+
+namespace PseudoBoolean
+open Finset FinVec BigOperators
+
+def ceildiv (c : ℕ) (a : ℕ) := a ⌈/⌉ c
+
+lemma ceildiv_le_ceildiv_right {a b : ℕ} (c : ℕ) (hab : a ≤ b)
+  : a ⌈/⌉ c ≤ b ⌈/⌉ c := by
+  repeat rw [Nat.ceilDiv_eq_add_pred_div]
+  apply Nat.div_le_div_right
+  apply Nat.sub_le_sub_right
+  apply Nat.add_le_add_right
+  exact hab
+  done
+
+-- @kbuzzard
+theorem Nat.add_ceildiv_le_add_ceildiv (a b c : ℕ)
+  : (a + b) ⌈/⌉ c ≤ (a ⌈/⌉ c) + (b ⌈/⌉ c) := by
+  -- deal with c=0 separately
+  obtain (rfl | hc) := Nat.eq_zero_or_pos c
+  · simp
+  -- 0 < c case
+  -- First use the "Galois connection"
+  rw [ceilDiv_le_iff_le_smul hc, smul_eq_mul]
+  rw [mul_add]
+  -- now use a standard fact
+  gcongr <;> exact le_smul_ceilDiv hc
+  done
 
-def PBIneq (cs : Coeff n) (xs : Fin n → Fin 2) (const : .) :=
-PBSum cs xs . const
+-- @Ruben-VandeVelde
+theorem Finset.ceildiv_le_ceildiv {α : Type*} (as : α → ℕ) (s : Finset α) (c : ℕ)
+  : (∑i in s, as i) ⌈/⌉ c ≤ ∑i in s,(as i ⌈/⌉ c) := by
+  classical
+  induction s using Finset.cons_induction_on with
+  | h₁ => simp
+  | @h₂ a s ha ih =>
+    rw [sum_cons, sum_cons]
+    have h := Nat.add_ceildiv_le_add_ceildiv (as a) (∑x ∈ s, as x) c
+    exact le_add_of_le_add_left h ih
+    done
+
+lemma ceildiv_ite (P : Prop) [Decidable P] (a b c : ℕ)
+  : (if P then b else c) ⌈/⌉ a = if P then (b ⌈/⌉ a) else (c ⌈/⌉ a) := by
+  split_ifs <;> rfl
+  done
+
+
+
+
+
+
+
+theorem Division
+  {xs : Fin n → Fin 2}
+  {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
+  (c : ℕ)
+  : PBIneq (map (mapBoth (ceildiv c)) as) xs (ceildiv c A) := by
+  unfold PBIneq PBSum ceildiv mapBoth at *
+  simp only [Fin.isValue, ge_iff_le, gt_iff_lt,
+             Prod_map, map_eq, Function.comp_apply] at *
+  apply ceildiv_le_ceildiv_right c at ha
+  apply le_trans ha
+  simp only [←ceildiv_ite]
+  apply Finset.ceildiv_le_ceildiv
+  done
+
+example
+  (ha : PBIneq ![(3,0),(4,0)] xs 3)
+  : PBIneq ![(2,0),(2,0)] xs 2 := by
+  apply Division ha 2
+  done
+
+end PseudoBoolean
+
+\end{minted}
+\newpage
+
+\subsection{Prova da Regra Adição}
+\begin{minted}{lean4}
+import <<LeanCuttingPlanes>>.Basic
+
+namespace PseudoBoolean
+
+open BigOperators FinVec Matrix
+
+theorem Addition'
+  (xs : Fin n → Fin 2)
+  (as : Coeff n) (A : ℕ) (ha : PBIneq as xs A)
+  (bs : Coeff n) (B : ℕ) (hb : PBIneq bs xs B)
+  : PBIneq (as + bs) xs (A + B) := by
+  unfold PBIneq PBSum at *
+  simp only [Fin.isValue, ge_iff_le] at *
+  simp_rw [←ite_add_ite]
+  rw [Finset.sum_add_distrib]
+  exact Nat.add_le_add ha hb
+  done
+
+def ReductionProp
+  (xs : Fin n → Fin 2) (ks : Coeff n) (K : ℕ)
+  : Prop :=
+  let pos := λ i => ks i |>.1
+  let neg := λ i => ks i |>.2
+  let slack := (∑i, min (pos i) (neg i))
+  let rs := λ i => (pos i - neg i,neg i - pos i)
+  PBIneq rs xs (K - slack)
+
+lemma ite_eq_bmul (x y : ℕ) (b : Fin 2)
+  : (if b = 1 then x else y) = (x * b + y * (1 - b)) := by
+  by_cases h : b = 0
+  . rw [h]
+    rw [if_neg]
+    . simp only [Fin.isValue, Fin.val_zero, mul_zero, tsub_zero, mul_one, zero_add]
+    trivial
+  . -- b = 1
+    apply Fin.eq_one_of_neq_zero b at h
+    rw [h]
+    simp only [Fin.isValue, ↓reduceIte, Fin.val_one, mul_one, ge_iff_le, le_refl,
+      tsub_eq_zero_of_le, mul_zero, add_zero]
+
+lemma reduce_terms (p n : ℕ) (x : Fin 2)
+  : p * x + n * (1 - x) = (p - n) * x + (n - p) * (1 - x) + min p n  := by
+  by_cases h : x = 0
+  . rw [h]
+    simp only [Fin.isValue, Fin.val_zero, mul_zero, tsub_zero, mul_one, zero_add]
+    rw [Nat.min_comm]
+    exact Nat.sub_add_min_cancel n p |>.symm
+
+  . -- x = 1
+    apply Fin.eq_one_of_neq_zero x at h
+    rw [h]
+    simp only [Fin.isValue, Fin.val_one, mul_one, ge_iff_le,
+                le_refl, tsub_eq_zero_of_le, mul_zero, add_zero]
+    exact Nat.sub_add_min_cancel p n |>.symm
+
+theorem Reduction
+  (xs : Fin n → Fin 2)
+  (ks : Coeff n) (K : ℕ) (ha : PBIneq ks xs K)
+  : ReductionProp xs ks K := by
+  unfold ReductionProp PBIneq PBSum at *
+  simp only [Fin.isValue, ge_iff_le, tsub_le_iff_right] at *
+  simp_rw [ite_eq_bmul] at *
+  rw [←Finset.sum_add_distrib]
+  simp_rw [←reduce_terms]
+  exact ha
+  done
+
+def AdditionProp
+  (xs : Fin n → Fin 2)
+  (as : Coeff n) (A : ℕ)
+  (bs : Coeff n) (B : ℕ)
+  : Prop :=
+  ReductionProp xs (as + bs) (A + B)
+
+theorem Addition
+  {xs : Fin n → Fin 2}
+  {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
+  {bs : Coeff n} {B : ℕ} (hb : PBIneq bs xs B)
+  : AdditionProp xs as A bs B := by
+  have hk := Addition' xs as A ha bs B hb
+  exact Reduction xs (as + bs) (A + B) hk
+  done
+
+example
+  (ha : PBIneq ![(1,0),(0,0)] xs 1)
+  (hb : PBIneq ![(1,0),(1,0)] xs 2)
+  : PBIneq ![(2,0),(1,0)] xs 3 := by
+  apply Addition ha hb
+  done
+
+-- Reduction happens automatically
+example
+  (ha : PBIneq ![(3,0),(0,0),(1,0)] xs 1)
+  (hb : PBIneq ![(0,0),(2,0),(0,1)] xs 2)
+  : PBIneq ![(3,0),(2,0),(0,0)] xs 2 := by
+  apply Addition ha hb
+  done
+
+end PseudoBoolean
+
+\end{minted}
+\newpage
+
+\subsection{Implementação do Exemplo}
+\begin{minted}{lean4}
+import <<LeanCuttingPlanes>>
+
+open PseudoBoolean
+
+example
+  (xs : Fin 4 → Fin 2)
+  (c1 : PBIneq ![(1,0),(2,0),(1,0),(0,0)] xs 2)
+  (c2 : PBIneq ![(1,0),(2,0),(4,0),(2,0)] xs 5)
+  (c3 : PBIneq ![(0,0),(0,0),(0,0),(0,1)] xs 0)
+  : PBIneq ![(1,0),(2,0),(2,0),(0,0)] xs 3
+  := by
+  let t1 : PBIneq ![(2,0),(4,0),(2,0),(0,0)] xs 4  := by apply Multiplication c1 2
+  let t2 : PBIneq ![(3,0),(6,0),(6,0),(2,0)] xs 9  := by apply Addition t1 c2
+  let t3 : PBIneq ![(0,0),(0,0),(0,0),(0,2)] xs 0  := by apply Multiplication c3 2
+  let t4 : PBIneq ![(3,0),(6,0),(6,0),(0,0)] xs 7  := by apply Addition t2 t3
+  exact Division t4 3
+  done
 \end{minted}
 
 

From cf6e70627f8b7fc60979973bd6aa87f1da460807 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Sat, 29 Jun 2024 15:45:03 -0300
Subject: [PATCH 79/93] Finished first version of article

---
 article/cutting-planes-article.pdf | Bin 236210 -> 236065 bytes
 article/cutting-planes-article.tex |  35 ++++++++++++++++++++++++-----
 2 files changed, 29 insertions(+), 6 deletions(-)

diff --git a/article/cutting-planes-article.pdf b/article/cutting-planes-article.pdf
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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 1134135..d77d2e2 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -51,7 +51,13 @@ \section{Introdução}
 %   corretude. Alem disso, mostrar exemplos de uso dessa biblioteca com o "Toy Example" presente na apresentacao de Jakob.
 %   Criacao de uma documentacao em formato mdbook, que estara disponivel para pesquisadores e matematicos que desejem usar
 %   a biblioteca.
-Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+O cálculo formal de Planos de Corte é uma técnica matemática utilizada na área de otimização,
+particularmente em Programação Linear.
+Nesse trabalho, procuramos trazer essa lógica para os resultados formalizados em \texttt{Lean 4},
+seguindo a tendência crescente no mundo da matemática e atendendo a necessidade de confiança
+nos resultados de solucionadores. Além da formalização, procuramos aqui disponibilizar estes
+resultados para a comunidade em geral, para que mais integrações entre projetos possam ser possíveis
+no futuro.
 
 \section{Referencial Teórico}
 % Capítulo referencial: identificação de trabalhos correlatos, referencial teórico.
@@ -277,7 +283,8 @@ \subsection{Prova da Regra Divisão}
 \begin{minted}[fontsize=\small]{lean4}
 theorem Finset.ceildiv_le_ceildiv {α : Type*}
     (as : α → ℕ) (s : Finset α) (c : ℕ)
-  : (∑i in s, as i) ⌈/⌉ c ≤ ∑i in s,(as i ⌈/⌉ c)
+  : (∑i in s, as i) ⌈/⌉ c
+  ≤ ∑i in s,(as i ⌈/⌉ c)
 \end{minted}
 
 O último detalhe que tivemos que provar é que podemos distribuir o \textit{ceildiv} sobre a expressão \texttt{if-then-else}:
@@ -359,11 +366,29 @@ \subsection{Prova da Regra Adição}
 
 % * Implementação do Toy Example
 \subsection{Implementação do Exemplo}
-Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+Com todas as regras necessárias, prosseguimos para um exemplo retirado da apresentação
+(Apêncide F). Aqui podemos ver como usar as regras, com auxílio da tática \texttt{apply}.
+O leitor fica convidado a visitar o repositório público em \texttt{github.com/bernborgess/lean-cutting-planes}
+e tentar por conta própria.
+
 
 \section*{Conclusões}
 % Capítulo de fechamento: conclusões e relação de trabalhos futuros.
-Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris luctus lacus eu varius fermentum. Nullam lacus urna, semper a euismod sed, hendrerit eu nisi. Donec nec risus turpis. Vestibulum placerat justo sit amet aliquam mattis. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Sed vulputate vulputate maximus. Sed euismod turpis vitae libero consectetur gravida. Pellentesque eu nibh ipsum. Ut porta tortor vel pharetra euismod. Nulla quis nunc bibendum, condimentum sem a, tempus lectus.
+Nós demonstramos a lógica de planos de corte como método correto para trabalhar com pseudo-booleanos.
+Com nossa biblioteca \texttt{lean-cutting-planes} agora podemos utilizar o sistema de tipos de Lean
+para validar com confiança passos dessa lógica. Com a documentação convidamos novos pesquisadores e
+matemáticos a usar os resultados em verificadores.
+
+
+% 6. Conclusion
+% We have demonstrated Euclidean geometry as an attractive
+% target for autoformalization. With our SMT-based symbolic
+% engine, it is feasible to (1) automatically evaluate autofor-
+% malized theorems statements and (2) have the model auto-
+% formalize only explicit proof steps, leaving diagrammatic
+% reasoning implicit. We have constructed the LeanEuclid
+% benchmark to facilitate future research on autoformalizing
+% Euclidean geometry.
 
 \begin{thebibliography}{00}
     \bibitem{SatLive}       D. Le Berre, ``Sat Live! keep up to date with research on the satisfiability problem'', Acesso em http://www.satlive.org/.
@@ -378,8 +403,6 @@ \section*{Conclusões}
     \bibitem{mathlib4}      The mathlib Community. 2020. ``The lean mathematical library''. In CPP 2020. 367-381. https://doi.org/10.1145/3372885.3373824
 \end{thebibliography}
 
-\vspace{12pt}
-
 \newpage
 \onecolumn
 \appendix

From e75f852714f37a50c15d123335307486761109d5 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 2 Jul 2024 11:08:29 -0300
Subject: [PATCH 80/93] Article Feedback Changes 1

---
 article/cutting-planes-article.pdf | Bin 236065 -> 266910 bytes
 article/cutting-planes-article.tex |  92 ++++++++++++++++++-----------
 2 files changed, 56 insertions(+), 36 deletions(-)

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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index d77d2e2..8c29cfb 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -30,8 +30,7 @@
 
 \begin{abstract}
     Esta pesquisa envolve a definição e verificação da lógica pseudo-booleana de
-    Planos de Corte utilizando o provador de teoremas Lean 4
-
+    Planos de Corte utilizando o provador de teoremas Lean 4.
 \end{abstract}
 
 \begin{IEEEkeywords}
@@ -40,20 +39,9 @@
 
 \section{Introdução}
 
-% TODO
-% O cálculo formal de Planos de Corte é uma técnica matemática utilizada na área de otimização,
-% particularmente em Programação Linear.  
-% Capítulo introdutório: caracterização do problema, motivação, objetivos, breve descrição da estrutura do trabalho.
-% * problema de utilizar pseudo booleanos para representacao eficiente de problemas SAT
-% * motivacao: Ter certeza que solucoes que utilizam os pseudo-booleans estao corretas
-% * objetivos: Usar metodos formais para validar a logica pseudo-booleana e provas que a utilizem
-% * estrutura do trabalho: biblioteca de codigo em Lean 4 que implementa as regras da logica pseudo-booleana e prova sua
-%   corretude. Alem disso, mostrar exemplos de uso dessa biblioteca com o "Toy Example" presente na apresentacao de Jakob.
-%   Criacao de uma documentacao em formato mdbook, que estara disponivel para pesquisadores e matematicos que desejem usar
-%   a biblioteca.
-O cálculo formal de Planos de Corte é uma técnica matemática utilizada na área de otimização,
+O cálculo formal de Planos de Corte\cite{CutPlane} é uma técnica matemática utilizada na área de otimização,
 particularmente em Programação Linear.
-Nesse trabalho, procuramos trazer essa lógica para os resultados formalizados em \texttt{Lean 4},
+Neste trabalho, procuramos trazer essa lógica para os resultados formalizados em \texttt{Lean 4},
 seguindo a tendência crescente no mundo da matemática e atendendo a necessidade de confiança
 nos resultados de solucionadores. Além da formalização, procuramos aqui disponibilizar estes
 resultados para a comunidade em geral, para que mais integrações entre projetos possam ser possíveis
@@ -64,23 +52,27 @@ \section{Referencial Teórico}
 
 \subsection{Satisfiabilidade}
 O problema da satisfabilidade booleana (SAT) é muito importante para a Ciência da Computação, sendo o
-primeiro demonstrado ser da class NP-Completo. Ele é o problema de decidir para dada expressão booleana,
+primeiro demonstrado ser NP-Completo. Ele é o problema de decidir para dada expressão booleana
 se é possível escolher valores para as variáveis de forma que a expressão como um todo seja verdadeira.
 Este problema é extensivamente pesquisado na área de métodos formais\cite{SatLive} e existem inclusive competições para
 definir qual é o melhor solucionador do mundo \cite{SatComp}.
 
-\subsection{Pseudo-Booleans}
+\subsection{Pseudo-Booleanos}
 Um formato comumente utilizado para representar expressões booleanas é a Forma Normal Conjuntiva (CNF), que
 consiste da conjunção de cláusulas, em que cada cláusula é a disjunção de variáveis ou negação de variáveis\cite{CNF}.
-Neste trabalho, introduzimos uma outra representação para expressões, chamados Pseudo-Booleanos, funções estudadas
+Neste trabalho, usamos uma outra representação para expressões, chamados Pseudo-Booleanos, funções estudadas
 desde os anos 1960 na área de pesquisa operacional, em programação inteira. Este formato consiste de um somatório
 do produto de um coeficiente por um literal, que é maior ou igual a uma constante natural.
 Este formato é exponencialmente mais compacto que o CNF, o que motiva seu uso\cite{PBSolve}.
 
-\subsection{Pseudo-Boolean Reasoning}
-A lógica formal de Cutting Planes introduz 2 axiomas e 4 regras de inferência, nomeadamente
+\subsection{Raciocínio Pseudo-Booleano}
+A lógica formal de Planos de Corte introduz 2 axiomas e 4 regras de inferência, nomeadamente
 \textbf{Adição}, \textbf{Multiplicação}, \textbf{Divisão} e \textbf{Saturação},
-que permitem derivar novas inequações a partir de outras\cite{CutPlane}.
+que permitem derivar novas inequações a partir de outras\cite{CutPlane},
+o que será detalhado na próxima seção. Com essa regras, podemos manipular as equações
+de forma algébrica e construir uma árvore de derivação que demonstra novos resultados,
+unindo somatórios diferentes pela adição, e variando seus coeficientes pela multiplicação,
+divisão ou saturação.
 
 \subsection{Lean Theorem Prover}
 Lean é uma linguagem de programação e provador de teoremas criado por Leonardo de Moura em 2013\cite{LeanProver}, com
@@ -95,9 +87,11 @@ \subsection{Lean Theorem Prover}
 \section{Contribuições}
 % Capítulo de contribuição: descrição organizada das atividades conduzidas pelo aluno.
 A contribuição deste trabalho está na criação de código em Lean 4 para lidar com a lógica de cutting planes.
-Primeiramente, as inequações foram formalmente definidas em Lean, utilizando teoremas e definições de sua biblioteca
-matemática mathlib4\cite{mathlib4}. Assim as quatro regras foram definidas e formalizadas, e em seguida um exemplo de
-raciocínio, apresentado por Jakob Nordström foi formalizado nessa biblioteca.
+Com essa implementação conseguimos demonstrar formalmente que as regras são corretas, ao verificá-las pelo sistema de tipos de Lean.
+Primeiramente, as inequações foram formalmente definidas utilizando teoremas e definições da biblioteca
+matemática \textit{mathlib4}\cite{mathlib4}.
+Assim, as quatro regras foram definidas e formalizadas, e, em seguida,
+um exemplo de raciocínio apresentado por Jakob Nordström foi formalizado nessa biblioteca.
 
 % * Definição da inequação pseudo-booleana
 \subsection{Definição de Inequações Pseudo-Booleanas}
@@ -105,7 +99,7 @@ \subsection{Definição de Inequações Pseudo-Booleanas}
 \begin{equation}
     C_1 \land C_2 \land \dots \land C_n
 \end{equation}
-onde,
+em que,
 \begin{equation}
     C_i = a_1^i \lor a_2^i \lor \dots \lor a_k^i
 \end{equation}
@@ -118,13 +112,13 @@ \subsection{Definição de Inequações Pseudo-Booleanas}
 \begin{equation}
     (x_1 \lor x_2) \land (x_1 \lor \neg x_3) \land (x_2 \lor \neg x_3)
 \end{equation}
-Onde $x_1$, $x_2$ e $x_3$ são nossas variáveis booleanas e a atribuição $x_1=T$, $x_2=T$, $x_3=T$ satisfaz essa fórmula.
+Em que $x_1$, $x_2$ e $x_3$ são nossas variáveis booleanas e a atribuição $x_1=T$, $x_2=T$, $x_3=T$ satisfaz essa fórmula.
 
 O formato pseudo-booleano consiste de:
 \begin{equation}
     \sum_i{a_i l_i} \ge A
 \end{equation}
-onde,
+em que,
 \begin{equation}
     \begin{gathered}
         A, a_i \in \mathbb{N} \\
@@ -137,11 +131,14 @@ \subsection{Definição de Inequações Pseudo-Booleanas}
     x_1 + x_2 + \overline x_3 \ge 2
 \end{equation}
 
-Na implementação em Lean 4 utilizamos o tipo \texttt{Fin 2}, que só permite os valores $0$ ou $1$
-para os pseudo-booleanos.
-Para os coeficientes \texttt{Coeff}, usamos a estrutura \texttt{FinVec},
-que permite termos uma lista com $n$ elementos, onde cada elemento da lista será um par
-de dois números naturais, o tipo $\mathbb{N}$ em Lean:
+Lean é uma linguagem de tipos dependentes, ou seja, é possível definir tipos que tem relações com valores concretos.
+Um caso útil é o tipo \texttt{Fin n}, que contém os números naturais até \texttt{n}.
+Para modelar os Pseudo-Booleanos utilizamos o tipo \texttt{Fin 2}, que contém os valores $0$ e $1$.
+
+Para os coeficientes \texttt{Coeff}, utilisamos a estrutura \texttt{FinVec},
+que permite definir uma lista com exatamente \texttt{n} elementos,
+expressa por uma função que toma um \texttt{Fin n} e retorna o elemento,
+em que cada um será um par de dois números naturais, o tipo $\mathbb{N}$ em Lean:
 \begin{minted}[fontsize=\small]{lean4}
 abbrev Coeff (n : ℕ) := Fin n → (ℕ × ℕ)
 \end{minted}
@@ -167,6 +164,7 @@ \subsection{Definição de Inequações Pseudo-Booleanas}
 
 Para criar uma expressão desse tipo basta fornecer as listas de coeficientes, pseudo-booleanos e a constante,
 e em seguida provar que a propriedade vale:
+\\
 \begin{minted}[fontsize=\small]{lean4}
 example : PBIneq ![(1,0),(2,0)] ![0,1] 2 := by
   -- Change goal to 1 * 0 + 2 * 1 ≥ 2
@@ -200,8 +198,25 @@ \subsection{Prova da Regra Multiplicação}
   : PBIneq (c • as) xs (c • A)
 \end{minted}
 
-A prova completa se encontra no Apêndice B. Como a muliplicação por escalares já está definida na \textit{mathlib4} para
-os \textit{FinVec}s pelo teorema \texttt{Finset.sum\_nsmul} da biblioteca.
+Podemos usar o fato \texttt{nsmul\_le\_nsmul\_right} que, se $a \le b$, então $c \cdot a \le c \cdot b$, aplicamos isso em nossa premissa para obter:
+\begin{equation}
+    c \cdot \sum_i{a_i l_i} \ge c \cdot A
+\end{equation}
+
+Como a muliplicação por escalares já está definida na \textit{mathlib4} para
+os \texttt{FinVec}s pelo teorema \texttt{Finset.sum\_nsmul} da biblioteca,
+que diz:
+\begin{equation}
+    \sum_{x \in s}{n \cdot f(x)} = n \cdot \sum_{x \in s}{f(x)}
+\end{equation}
+
+Podemos aplicá-la da direita para esquerda para obter:
+\begin{equation}
+    \sum_i{c \cdot a_i l_i} \ge c \cdot A
+\end{equation}
+
+A prova completa se encontra no Apêndice B.
+
 
 % * Prova da Regra Saturation
 \subsection{Prova da Regra Saturação}
@@ -237,7 +252,7 @@ \subsection{Prova da Regra Saturação}
           Nesse caso $\min(A,as_i) = as_i$, para todo $i$, logo temos a mesma lista.
           % os elementos,
           Então a afirmação vale pela hipótese.
-    \item Caso contrário, existe ao menos um índice $k$ da lista $as$, onde $as_k > A$.
+    \item Caso contrário, existe ao menos um índice $k$ da lista $as$, em que $as_k > A$.
           Podemos dividir o somatório em $\sum_{i\neq k}{\min(A,as_i) + \min(A,as_k)}$, separando esse índice em específico.
           Como $as_k > A$, substituímos $\min(A,as_k)$ por $A$.
           Como queremos mostrar que $A \le \sum_{i\neq k}{\min(A,as_i) + A}$,
@@ -333,7 +348,7 @@ \subsection{Prova da Regra Adição}
 \end{minted}
 
 Em seguida, implementamos a regra \textit{Reduction}, que toma uma inequação e aniquila coeficientes
-onde ambos elementos do par são maiores que $0$.
+em que ambos elementos do par são maiores que $0$.
 Quando isso acontece, essa diferença "\textit{slack}" deve ser subtraída da constante:
 
 \begin{minted}[fontsize=\small]{lean4}
@@ -365,9 +380,14 @@ \subsection{Prova da Regra Adição}
 \end{minted}
 
 % * Implementação do Toy Example
+
+\newpage
 \subsection{Implementação do Exemplo}
 Com todas as regras necessárias, prosseguimos para um exemplo retirado da apresentação
 (Apêncide F). Aqui podemos ver como usar as regras, com auxílio da tática \texttt{apply}.
+
+\includegraphics[scale=0.17]{_minted-cutting-planes-article/toy_example.png}
+
 O leitor fica convidado a visitar o repositório público em \texttt{github.com/bernborgess/lean-cutting-planes}
 e tentar por conta própria.
 

From 8ca76669567dfb66df795976d74c53545919e4e6 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 2 Jul 2024 20:44:57 -0300
Subject: [PATCH 81/93] Fix some typos

---
 article/cutting-planes-article.pdf | Bin 266910 -> 267078 bytes
 article/cutting-planes-article.tex |  22 ++++++----------------
 2 files changed, 6 insertions(+), 16 deletions(-)

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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 8c29cfb..1607e17 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -16,7 +16,7 @@
 \begin{document}
 
 \title{Modelando e verificando o cálculo formal de Planos de Corte com Lean 4\\
-{\footnotesize Projeto Orientadado em Computação I - Pesquisa Científica}
+{\footnotesize Projeto Orientado em Computação I - Pesquisa Científica}
 }
 
 \author{\IEEEauthorblockN{Bernardo Borges}
@@ -50,8 +50,8 @@ \section{Introdução}
 \section{Referencial Teórico}
 % Capítulo referencial: identificação de trabalhos correlatos, referencial teórico.
 
-\subsection{Satisfiabilidade}
-O problema da satisfabilidade booleana (SAT) é muito importante para a Ciência da Computação, sendo o
+\subsection{Satisfatibilidade}
+O problema da satisfatibilidade booliana (SAT) é muito importante para a Ciência da Computação, sendo o
 primeiro demonstrado ser NP-Completo. Ele é o problema de decidir para dada expressão booleana
 se é possível escolher valores para as variáveis de forma que a expressão como um todo seja verdadeira.
 Este problema é extensivamente pesquisado na área de métodos formais\cite{SatLive} e existem inclusive competições para
@@ -135,7 +135,7 @@ \subsection{Definição de Inequações Pseudo-Booleanas}
 Um caso útil é o tipo \texttt{Fin n}, que contém os números naturais até \texttt{n}.
 Para modelar os Pseudo-Booleanos utilizamos o tipo \texttt{Fin 2}, que contém os valores $0$ e $1$.
 
-Para os coeficientes \texttt{Coeff}, utilisamos a estrutura \texttt{FinVec},
+Para os coeficientes \texttt{Coeff}, utilizamos a estrutura \texttt{FinVec},
 que permite definir uma lista com exatamente \texttt{n} elementos,
 expressa por uma função que toma um \texttt{Fin n} e retorna o elemento,
 em que cada um será um par de dois números naturais, o tipo $\mathbb{N}$ em Lean:
@@ -203,7 +203,7 @@ \subsection{Prova da Regra Multiplicação}
     c \cdot \sum_i{a_i l_i} \ge c \cdot A
 \end{equation}
 
-Como a muliplicação por escalares já está definida na \textit{mathlib4} para
+Como a multiplicação por escalares já está definida na \textit{mathlib4} para
 os \texttt{FinVec}s pelo teorema \texttt{Finset.sum\_nsmul} da biblioteca,
 que diz:
 \begin{equation}
@@ -384,7 +384,7 @@ \subsection{Prova da Regra Adição}
 \newpage
 \subsection{Implementação do Exemplo}
 Com todas as regras necessárias, prosseguimos para um exemplo retirado da apresentação
-(Apêncide F). Aqui podemos ver como usar as regras, com auxílio da tática \texttt{apply}.
+(Apêndice F). Aqui podemos ver como usar as regras, com auxílio da tática \texttt{apply}.
 
 \includegraphics[scale=0.17]{_minted-cutting-planes-article/toy_example.png}
 
@@ -400,16 +400,6 @@ \section*{Conclusões}
 matemáticos a usar os resultados em verificadores.
 
 
-% 6. Conclusion
-% We have demonstrated Euclidean geometry as an attractive
-% target for autoformalization. With our SMT-based symbolic
-% engine, it is feasible to (1) automatically evaluate autofor-
-% malized theorems statements and (2) have the model auto-
-% formalize only explicit proof steps, leaving diagrammatic
-% reasoning implicit. We have constructed the LeanEuclid
-% benchmark to facilitate future research on autoformalizing
-% Euclidean geometry.
-
 \begin{thebibliography}{00}
     \bibitem{SatLive}       D. Le Berre, ``Sat Live! keep up to date with research on the satisfiability problem'', Acesso em http://www.satlive.org/.
     \bibitem{SatComp}       M. Heule, M. Jävisalo, M. Suda, ``The International SAT Competition Web Page'', Acesso em https://satcompetition.github.io/.

From 104de0cdc70dcf854c7d425938301ea427adadc6 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 2 Jul 2024 21:13:04 -0300
Subject: [PATCH 82/93] Reorder references

---
 article/cutting-planes-article.pdf | Bin 267078 -> 267080 bytes
 article/cutting-planes-article.tex |   2 +-
 2 files changed, 1 insertion(+), 1 deletion(-)

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diff --git a/article/cutting-planes-article.tex b/article/cutting-planes-article.tex
index 1607e17..47304c9 100644
--- a/article/cutting-planes-article.tex
+++ b/article/cutting-planes-article.tex
@@ -401,11 +401,11 @@ \section*{Conclusões}
 
 
 \begin{thebibliography}{00}
+    \bibitem{CutPlane}      J. Nordström, ``A Unified Proof System for Discrete Combinatorial Problems'', Novembro de 2023.
     \bibitem{SatLive}       D. Le Berre, ``Sat Live! keep up to date with research on the satisfiability problem'', Acesso em http://www.satlive.org/.
     \bibitem{SatComp}       M. Heule, M. Jävisalo, M. Suda, ``The International SAT Competition Web Page'', Acesso em https://satcompetition.github.io/.
     \bibitem{CNF}           ``Conjunctive normal form'', Encyclopedia of Mathematics, EMS Press, 2001.
     \bibitem{PBSolve}       J. Nordström, ``Pseudo-Boolean Solving and Optimization'', Fevereiro de 2021.
-    \bibitem{CutPlane}      J. Nordström, ``A Unified Proof System for Discrete Combinatorial Problems'', Novembro de 2023.
     \bibitem{LeanProver}    L. de Moura, S. Kong, J. Avigad, F. van Doorn, J. von Raumer ``The Lean Theorem Prover'', 25th International Conference on Automated Deduction (CADE-25), Berlin, Germany, 2015. Acesso em https://lean-lang.org/papers/system.pdf.
     \bibitem{Lean4}         L. de Moura, S. Ullrich ``The Lean 4 Theorem Prover and Programming Language'', 28th International Conference on Automated Deduction (CADE-28), Pittsburgh, USA, 2021. Acesso em https://lean-lang.org/papers/lean4.pdf.
     \bibitem{LTE}           J. Commelin, P. Scholze ``Liquid Tensor Experiment''. Acesso em https://math.commelin.net/files/LTE.pdf

From 08f7ab27e5d8124bad9d5d94513980b513a5f9a3 Mon Sep 17 00:00:00 2001
From: bernborgess <62393097+bernborgess@users.noreply.github.com>
Date: Mon, 29 Jul 2024 13:11:31 -0300
Subject: [PATCH 83/93] Update .gitattributes

---
 .gitattributes | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/.gitattributes b/.gitattributes
index 5f387b9..bca3c19 100644
--- a/.gitattributes
+++ b/.gitattributes
@@ -1 +1,2 @@
-theme/** linguist-detectable=false
\ No newline at end of file
+theme/** linguist-detectable=false
+article/** linguist-detectable=false

From 80ed92087e8c88206a8d002d638b8d413248166e Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Mon, 7 Oct 2024 10:22:47 -0300
Subject: [PATCH 84/93] Started alethe cutting planes tex file

---
 alethe/cutting-planes-alethe.pdf | Bin 0 -> 210930 bytes
 alethe/cutting-planes-alethe.tex | 103 +++++++++++++++++++++++++++++++
 2 files changed, 103 insertions(+)
 create mode 100644 alethe/cutting-planes-alethe.pdf
 create mode 100644 alethe/cutting-planes-alethe.tex

diff --git a/alethe/cutting-planes-alethe.pdf b/alethe/cutting-planes-alethe.pdf
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diff --git a/alethe/cutting-planes-alethe.tex b/alethe/cutting-planes-alethe.tex
new file mode 100644
index 0000000..45d3a70
--- /dev/null
+++ b/alethe/cutting-planes-alethe.tex
@@ -0,0 +1,103 @@
+\documentclass[12pt]{article}
+
+% https://canizo.org/templates -> article
+
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage[a4paper, margin=2.7cm]{geometry}
+\usepackage{amsmath, amsthm, amsfonts, amssymb}
+\usepackage{mathrsfs}           % \mathscr font.
+\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue,breaklinks]{hyperref}
+\usepackage{bbm} 
+\usepackage{breakurl}
+\usepackage{natbib}
+\usepackage{url}
+\bibliographystyle{plainnat-linked}
+
+\usepackage{xcolor}
+\definecolor{light-gray}{gray}{0.95}
+\newcommand{\code}[1]{\colorbox{light-gray}{\texttt{#1}}}
+
+% Paper title and author
+\title{\textbf{The Alethe Proof Format}}
+\author{Bernardo Borges \and Haniel Barbosa}
+
+\begin{document}
+
+\maketitle
+
+\begin{abstract}
+    Specifying more rules for alethe in the Cutting Planes reasoning
+\end{abstract}
+
+\tableofcontents
+
+\section{Introduction}
+In this article we describe how the cutting planes reasoning can be expressed in the alethe
+proof format, in order to make use of checkers (carcara).
+
+\section{Pseudo Boolean Inequality in Alethe}
+The Pseudo Boolean format consists of:
+\begin{equation}
+    \sum_i{a_i l_i} \ge A
+\end{equation}
+where,
+\begin{equation}
+    \begin{gathered}
+        A, a_i \in \mathbb{N} \\
+        l_i \in \{ x_i, \overline x_i \}, \qquad x_i + \overline x_i = 1
+    \end{gathered}
+\end{equation}
+
+In order to express it in alethe, we define an expression:
+
+\centerline{\code{(>= (+ <TERMS> 0) <A>)}}
+
+where \code{<TERMS>} is a list of either:
+\begin{enumerate}
+    \item \code{(* <a\_i> <l\_i>)} in a plain literal $a_i x_i$.
+    \item \code{(* (- 1 <a\_i>) <l\_i>)} in a negated literal $a_i \overline x_i$
+\end{enumerate}
+
+and \code{<A>} is the natural constant $A$.
+
+\section{Cutting Planes Rules in Alethe}
+
+\subsection{Rule 1: cp\_addition}
+We can apply the rules, using the `step' syntax:
+
+\begin{verbatim}
+(assume c1 (>= (+ (* 2 x1) 0) 1))
+(assume c2 (>= (+ (* 1 (- 1 x1)) 0) 1))
+(step t1 (cl (>= (+ (* 1 x1) 0) 1))
+    :rule cp\_addition :premises (c1 c2)
+)
+\end{verbatim}
+
+\subsection{Rule 2: cp\_multiplication}
+\begin{verbatim}
+(assume c1 (>= (+ (* 1 x1) 0) 1))
+(step t1 (cl (>= (+ (* 2 x1) 0) 2))
+    :rule cp_multiplication :premises (c1) :args (2)
+)
+\end{verbatim}
+
+\subsection{Rule 3: cp\_division}
+\begin{verbatim}
+(assume c1 (>= (+ (* 2 x1) 0) 2))
+(step t1 (cl (>= (+ (* 1 x1) 0) 1))
+    :rule cp_division :premises (c1) :args (2)
+)
+\end{verbatim}
+
+\subsection{Rule 4: cp\_saturation}
+\begin{verbatim}
+(assume c1 (>= (+ (* 2 x1) 0) 1))
+(step t1 (cl (>= (+ (* 1 x1) 0) 1))
+    :rule cp_saturation :premises (c1)
+)
+\end{verbatim}
+
+\bibliography{bibliography}
+
+\end{document}

From 3023da0f0d2870e2c3b55b4b9ec5b4f2db653100 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 8 Oct 2024 10:01:14 -0300
Subject: [PATCH 85/93] Inserted alethe docs format

---
 alethe/.gitignore                |  14 +++++
 alethe/cutting-planes-alethe.pdf | Bin 210930 -> 0 bytes
 alethe/cutting-planes-alethe.tex | 103 -------------------------------
 alethe/cutting_planes.tex        |  56 +++++++++++++++++
 alethe/doc.pdf                   | Bin 0 -> 360983 bytes
 5 files changed, 70 insertions(+), 103 deletions(-)
 create mode 100644 alethe/.gitignore
 delete mode 100644 alethe/cutting-planes-alethe.pdf
 delete mode 100644 alethe/cutting-planes-alethe.tex
 create mode 100644 alethe/cutting_planes.tex
 create mode 100644 alethe/doc.pdf

diff --git a/alethe/.gitignore b/alethe/.gitignore
new file mode 100644
index 0000000..7bd2381
--- /dev/null
+++ b/alethe/.gitignore
@@ -0,0 +1,14 @@
+*.tar.gz
+bib.bib
+changelog.tex
+doc.aux
+doc.tex
+highlight.py
+Makefile
+publications.bib
+rule_index_style.ist
+rule_list.tex
+shell.nix
+Title.jpg
+version.sh
+.vscode
diff --git a/alethe/cutting-planes-alethe.pdf b/alethe/cutting-planes-alethe.pdf
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index 15e1e25fb51e48cdfd6b4f532fe1466a8c0f8468..0000000000000000000000000000000000000000
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diff --git a/alethe/cutting-planes-alethe.tex b/alethe/cutting-planes-alethe.tex
deleted file mode 100644
index 45d3a70..0000000
--- a/alethe/cutting-planes-alethe.tex
+++ /dev/null
@@ -1,103 +0,0 @@
-\documentclass[12pt]{article}
-
-% https://canizo.org/templates -> article
-
-\usepackage[utf8]{inputenc}
-\usepackage[T1]{fontenc}
-\usepackage[a4paper, margin=2.7cm]{geometry}
-\usepackage{amsmath, amsthm, amsfonts, amssymb}
-\usepackage{mathrsfs}           % \mathscr font.
-\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue,breaklinks]{hyperref}
-\usepackage{bbm} 
-\usepackage{breakurl}
-\usepackage{natbib}
-\usepackage{url}
-\bibliographystyle{plainnat-linked}
-
-\usepackage{xcolor}
-\definecolor{light-gray}{gray}{0.95}
-\newcommand{\code}[1]{\colorbox{light-gray}{\texttt{#1}}}
-
-% Paper title and author
-\title{\textbf{The Alethe Proof Format}}
-\author{Bernardo Borges \and Haniel Barbosa}
-
-\begin{document}
-
-\maketitle
-
-\begin{abstract}
-    Specifying more rules for alethe in the Cutting Planes reasoning
-\end{abstract}
-
-\tableofcontents
-
-\section{Introduction}
-In this article we describe how the cutting planes reasoning can be expressed in the alethe
-proof format, in order to make use of checkers (carcara).
-
-\section{Pseudo Boolean Inequality in Alethe}
-The Pseudo Boolean format consists of:
-\begin{equation}
-    \sum_i{a_i l_i} \ge A
-\end{equation}
-where,
-\begin{equation}
-    \begin{gathered}
-        A, a_i \in \mathbb{N} \\
-        l_i \in \{ x_i, \overline x_i \}, \qquad x_i + \overline x_i = 1
-    \end{gathered}
-\end{equation}
-
-In order to express it in alethe, we define an expression:
-
-\centerline{\code{(>= (+ <TERMS> 0) <A>)}}
-
-where \code{<TERMS>} is a list of either:
-\begin{enumerate}
-    \item \code{(* <a\_i> <l\_i>)} in a plain literal $a_i x_i$.
-    \item \code{(* (- 1 <a\_i>) <l\_i>)} in a negated literal $a_i \overline x_i$
-\end{enumerate}
-
-and \code{<A>} is the natural constant $A$.
-
-\section{Cutting Planes Rules in Alethe}
-
-\subsection{Rule 1: cp\_addition}
-We can apply the rules, using the `step' syntax:
-
-\begin{verbatim}
-(assume c1 (>= (+ (* 2 x1) 0) 1))
-(assume c2 (>= (+ (* 1 (- 1 x1)) 0) 1))
-(step t1 (cl (>= (+ (* 1 x1) 0) 1))
-    :rule cp\_addition :premises (c1 c2)
-)
-\end{verbatim}
-
-\subsection{Rule 2: cp\_multiplication}
-\begin{verbatim}
-(assume c1 (>= (+ (* 1 x1) 0) 1))
-(step t1 (cl (>= (+ (* 2 x1) 0) 2))
-    :rule cp_multiplication :premises (c1) :args (2)
-)
-\end{verbatim}
-
-\subsection{Rule 3: cp\_division}
-\begin{verbatim}
-(assume c1 (>= (+ (* 2 x1) 0) 2))
-(step t1 (cl (>= (+ (* 1 x1) 0) 1))
-    :rule cp_division :premises (c1) :args (2)
-)
-\end{verbatim}
-
-\subsection{Rule 4: cp\_saturation}
-\begin{verbatim}
-(assume c1 (>= (+ (* 2 x1) 0) 1))
-(step t1 (cl (>= (+ (* 1 x1) 0) 1))
-    :rule cp_saturation :premises (c1)
-)
-\end{verbatim}
-
-\bibliography{bibliography}
-
-\end{document}
diff --git a/alethe/cutting_planes.tex b/alethe/cutting_planes.tex
new file mode 100644
index 0000000..5488f54
--- /dev/null
+++ b/alethe/cutting_planes.tex
@@ -0,0 +1,56 @@
+\begin{RuleDescription}{cp_addition}
+    \begin{AletheX}
+        $i$. & \ctxsep & $\varphi$ & \currule \\
+    \end{AletheX}
+    TODO where $\varphi$ corresponds up to the orientation of equalities
+    to a formula asserted in the input problem, or $\varphi$ is a local assumption
+    in a subproof.
+    \ruleparagraph{Remark.}
+    This rule can not be used by the
+    \inlineAlethe{(step }\dots\inlineAlethe{)} command. Instead it corresponds to the dedicated
+    \inlineAlethe{(assume }\dots\inlineAlethe{)} command.
+\end{RuleDescription}
+
+
+\begin{RuleDescription}{cp_multiplication}
+    \begin{AletheX}
+        $i$. & \ctxsep & $\varphi$ & \currule \\
+    \end{AletheX}
+    TODO where $\varphi$ corresponds up to the orientation of equalities
+    to a formula asserted in the input problem, or $\varphi$ is a local assumption
+    in a subproof.
+    \ruleparagraph{Remark.}
+    This rule can not be used by the
+    \inlineAlethe{(step }\dots\inlineAlethe{)} command. Instead it corresponds to the dedicated
+    \inlineAlethe{(assume }\dots\inlineAlethe{)} command.
+\end{RuleDescription}
+
+
+\begin{RuleDescription}{cp_divison}
+    \begin{AletheX}
+        $i$. & \ctxsep & $\varphi$ & \currule \\
+    \end{AletheX}
+    TODO where $\varphi$ corresponds up to the orientation of equalities
+    to a formula asserted in the input problem, or $\varphi$ is a local assumption
+    in a subproof.
+    \ruleparagraph{Remark.}
+    This rule can not be used by the
+    \inlineAlethe{(step }\dots\inlineAlethe{)} command. Instead it corresponds to the dedicated
+    \inlineAlethe{(assume }\dots\inlineAlethe{)} command.
+\end{RuleDescription}
+
+
+\begin{RuleDescription}{cp_saturation}
+    \begin{AletheX}
+        $i$. & \ctxsep & $\varphi$ & \currule \\
+    \end{AletheX}
+    TODO where $\varphi$ corresponds up to the orientation of equalities
+    to a formula asserted in the input problem, or $\varphi$ is a local assumption
+    in a subproof.
+    \ruleparagraph{Remark.}
+    This rule can not be used by the
+    \inlineAlethe{(step }\dots\inlineAlethe{)} command. Instead it corresponds to the dedicated
+    \inlineAlethe{(assume }\dots\inlineAlethe{)} command.
+\end{RuleDescription}
+
+
diff --git a/alethe/doc.pdf b/alethe/doc.pdf
new file mode 100644
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From d499ceb5ed1c2cc6ca450d7f9ca53b0ba6d8288d Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 8 Oct 2024 10:53:18 -0300
Subject: [PATCH 86/93] Added PBIneq definition to alethe docs

---
 alethe/cutting_planes.tex |  74 ++++++++++++++++++++++----------------
 alethe/doc.pdf            | Bin 360983 -> 364395 bytes
 2 files changed, 43 insertions(+), 31 deletions(-)

diff --git a/alethe/cutting_planes.tex b/alethe/cutting_planes.tex
index 5488f54..66ebb67 100644
--- a/alethe/cutting_planes.tex
+++ b/alethe/cutting_planes.tex
@@ -1,14 +1,44 @@
 \begin{RuleDescription}{cp_addition}
-    \begin{AletheX}
-        $i$. & \ctxsep & $\varphi$ & \currule \\
-    \end{AletheX}
-    TODO where $\varphi$ corresponds up to the orientation of equalities
-    to a formula asserted in the input problem, or $\varphi$ is a local assumption
-    in a subproof.
-    \ruleparagraph{Remark.}
-    This rule can not be used by the
-    \inlineAlethe{(step }\dots\inlineAlethe{)} command. Instead it corresponds to the dedicated
-    \inlineAlethe{(assume }\dots\inlineAlethe{)} command.
+    A step of the \currule{} rule represents a step in the cutting planes reasoning, that works on
+    Pseudo-Boolean Constraints, that are 0-1 integer linear inequalities.
+
+    A pseudo boolean inequality is a term of the form
+    \[
+        \sum_i{a_i l_i} \ge A
+    \]
+    where we have:
+    \begin{itemize}
+        \item constant $A \in \mathbb{N}$
+        \item coefficients $ a_i \in \mathbb{N}$
+        \item literals $ l_i: x_i\ \text{or}\ \overline{x_i}\ (\text{where } x_i + \overline{x_i} = 1)$
+        \item variables $ x_i $ take values $ 0 = false $ or $ 1 = true$
+    \end{itemize}
+
+    In alethe such formula will be defined as:
+
+    \centerline{\inlineAlethe{(>= (+ <TERMS> 0) <A>)}}
+    \hfill
+
+    where \inlineAlethe{<TERMS>} is a list of either:
+    \begin{enumerate}
+        \item \inlineAlethe{(* <a_i> <x_i>)} in a plain literal $a_i x_i$.
+        \item \inlineAlethe{(* (- 1 <a_i>) <x_i>)} in a negated literal $a_i \overline x_i$
+    \end{enumerate}
+
+    and \inlineAlethe{<A>} is the natural constant $A$. \\
+
+    Using this format, two constraints can be added together, adding the coefficients and the
+    constants. Another behaviour is that $x_i$ and $\overline x_i$ that may appear together
+    cancel each other:
+
+    \begin{AletheXS}
+        $j$. & ${\sum_i{a_i l_i} \ge A}$ & ${\sum_i{b_i l_i} \ge B}$ \\
+        \spsep
+        $k$. & $\Gamma$ & \ctxsep  & ${\sum_i{(a_i + b_i) l_i} \ge (A+B)}$ & \currule{} \\
+    \end{AletheXS}
+
+    \textit{TODO: make the above line fit the "rule" format appropriately}
+
 \end{RuleDescription}
 
 
@@ -16,13 +46,7 @@
     \begin{AletheX}
         $i$. & \ctxsep & $\varphi$ & \currule \\
     \end{AletheX}
-    TODO where $\varphi$ corresponds up to the orientation of equalities
-    to a formula asserted in the input problem, or $\varphi$ is a local assumption
-    in a subproof.
-    \ruleparagraph{Remark.}
-    This rule can not be used by the
-    \inlineAlethe{(step }\dots\inlineAlethe{)} command. Instead it corresponds to the dedicated
-    \inlineAlethe{(assume }\dots\inlineAlethe{)} command.
+    TODO
 \end{RuleDescription}
 
 
@@ -30,13 +54,7 @@
     \begin{AletheX}
         $i$. & \ctxsep & $\varphi$ & \currule \\
     \end{AletheX}
-    TODO where $\varphi$ corresponds up to the orientation of equalities
-    to a formula asserted in the input problem, or $\varphi$ is a local assumption
-    in a subproof.
-    \ruleparagraph{Remark.}
-    This rule can not be used by the
-    \inlineAlethe{(step }\dots\inlineAlethe{)} command. Instead it corresponds to the dedicated
-    \inlineAlethe{(assume }\dots\inlineAlethe{)} command.
+    TODO
 \end{RuleDescription}
 
 
@@ -44,13 +62,7 @@
     \begin{AletheX}
         $i$. & \ctxsep & $\varphi$ & \currule \\
     \end{AletheX}
-    TODO where $\varphi$ corresponds up to the orientation of equalities
-    to a formula asserted in the input problem, or $\varphi$ is a local assumption
-    in a subproof.
-    \ruleparagraph{Remark.}
-    This rule can not be used by the
-    \inlineAlethe{(step }\dots\inlineAlethe{)} command. Instead it corresponds to the dedicated
-    \inlineAlethe{(assume }\dots\inlineAlethe{)} command.
+    TODO
 \end{RuleDescription}
 
 
diff --git a/alethe/doc.pdf b/alethe/doc.pdf
index 6c05464f625dc745e3d5ea4c2624d5fb28ee0ebb..8ffb886af45ef208efbadff8af7f1499a4db5a50 100644
GIT binary patch
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From bb1ec3c6ece82883e861cbb70ceb1f00efb8680e Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Mon, 14 Oct 2024 11:05:30 -0300
Subject: [PATCH 87/93] Specify the 4 cp rules in alethe docs

---
 alethe/cutting_planes.tex |  48 ++++++++++++++++++++++----------------
 alethe/doc.pdf            | Bin 364395 -> 365831 bytes
 2 files changed, 28 insertions(+), 20 deletions(-)

diff --git a/alethe/cutting_planes.tex b/alethe/cutting_planes.tex
index 66ebb67..5fd9f7e 100644
--- a/alethe/cutting_planes.tex
+++ b/alethe/cutting_planes.tex
@@ -25,44 +25,52 @@
         \item \inlineAlethe{(* (- 1 <a_i>) <x_i>)} in a negated literal $a_i \overline x_i$
     \end{enumerate}
 
-    and \inlineAlethe{<A>} is the natural constant $A$. \\
+    and \inlineAlethe{<A>} is the natural constant $A$.
 
     Using this format, two constraints can be added together, adding the coefficients and the
     constants. Another behaviour is that $x_i$ and $\overline x_i$ that may appear together
     cancel each other:
 
-    \begin{AletheXS}
-        $j$. & ${\sum_i{a_i l_i} \ge A}$ & ${\sum_i{b_i l_i} \ge B}$ \\
-        \spsep
-        $k$. & $\Gamma$ & \ctxsep  & ${\sum_i{(a_i + b_i) l_i} \ge (A+B)}$ & \currule{} \\
-    \end{AletheXS}
-
-    \textit{TODO: make the above line fit the "rule" format appropriately}
+    \begin{AletheS}
+        $i_1$. & \ctxsep & \, & ${\sum_i{a_i l_i} \ge A}$  \\
+        $i_2$. & \ctxsep  & \, & ${\sum_i{b_i l_i} \ge B}$ \\
+        $j$. & \ctxsep  & \, & ${\sum_i{(a_i + b_i) l_i} \ge (A+B)}$  & (\currule\; $i_1$, $i_2$)
+    \end{AletheS}
 
 \end{RuleDescription}
 
 
 \begin{RuleDescription}{cp_multiplication}
-    \begin{AletheX}
-        $i$. & \ctxsep & $\varphi$ & \currule \\
-    \end{AletheX}
-    TODO
+    A constraint can be multiplied by any $c \in \mathbb{N}^+$:
+
+    \begin{AletheS}
+        $i$. & \ctxsep & \, & ${\sum_i{a_i l_i} \ge A}$  \\
+        $j$. & \ctxsep  & \, & ${\sum_i{c a_i l_i} \ge c A}$  & (\currule\; $i$)\,[$c$]
+    \end{AletheS}
+
 \end{RuleDescription}
 
 
 \begin{RuleDescription}{cp_divison}
-    \begin{AletheX}
-        $i$. & \ctxsep & $\varphi$ & \currule \\
-    \end{AletheX}
-    TODO
+    A constraint can be divided by any $c \in \mathbb{N}^+$,
+    and the the ceiling of this division in applied:
+
+    \begin{AletheS}
+        $i$. & \ctxsep & \, & ${\sum_i{a_i l_i} \ge A}$  \\
+        $j$. & \ctxsep  & \, & ${\sum_i{ \lceil \frac{a_i}{c} \rceil l_i} \ge \lceil \frac{A}{c} \rceil}$  & (\currule\; $i$)\,[$c$]
+    \end{AletheS}
+
 \end{RuleDescription}
 
 
 \begin{RuleDescription}{cp_saturation}
-    \begin{AletheX}
-        $i$. & \ctxsep & $\varphi$ & \currule \\
-    \end{AletheX}
-    TODO
+    A constraint can replace its coefficients by the minimum between them and the constant:
+
+    \begin{AletheS}
+        $i$. & \ctxsep & \, & ${\sum_i{a_i l_i} \ge A}$  \\
+        $j$. & \ctxsep  & \, & ${\sum_i{ \min(a_i,A)\cdot l_i} \ge A}$  & (\currule\; $i$)
+    \end{AletheS}
+
 \end{RuleDescription}
 
 
diff --git a/alethe/doc.pdf b/alethe/doc.pdf
index 8ffb886af45ef208efbadff8af7f1499a4db5a50..49c45a149021692d1aed879ee3e6d51aa9ccb4db 100644
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From 64bf3bb41a213631ef0cf5022f1d78b314999da2 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 15 Oct 2024 09:46:44 -0300
Subject: [PATCH 88/93] Expand cp_addtion in la_generic style

---
 alethe/cutting_planes.tex |  85 ++++++++++++++++++++++++++++++++++++++
 alethe/doc.pdf            | Bin 365831 -> 370187 bytes
 2 files changed, 85 insertions(+)

diff --git a/alethe/cutting_planes.tex b/alethe/cutting_planes.tex
index 5fd9f7e..24855a1 100644
--- a/alethe/cutting_planes.tex
+++ b/alethe/cutting_planes.tex
@@ -39,6 +39,91 @@
 
 \end{RuleDescription}
 
+\newpage
+
+\begin{RuleDescription}{cp_addition2}
+    A step of the \currule{} rule represents the addition of two pseudo-Boolean
+    constraints using cutting planes reasoning. Pseudo-Boolean constraints are
+    0-1 integer linear inequalities of the form:
+    \[
+        \sum_i a_i \times l_i \geq A
+    \]
+    where $A \in \mathbb{N}$ is a constant, $a_i \in \mathbb{N}$ are coefficients,
+    and $l_i$ are literals of the form $x_i$ or $\overline{x_i}$ (where $x_i + \overline{x_i} = 1$),
+    with $x_i$ taking pseudo-boolean values $0$ (false) or $1$ (true).
+
+    The \proofRule{cp_addition2} rule allows two pseudo-Boolean constraints to be
+    added together, combining their coefficients and constants. Literals $x_i$ and
+    $\overline{x_i}$ that appear together cancel each other.
+
+    Formally, given two constraints:
+    \[
+        \sum_i a_i \times l_i \geq A \quad \text{and} \quad \sum_i b_i \times l_i \geq B,
+    \]
+    the result of applying the \currule{} rule is:
+    \[
+        \sum_i (a_i + b_i) \times l_i \geq (A + B).
+    \]
+
+    A {\currule} step in the proof has the form:
+
+    \begin{AletheS}
+        $i_1$. & \ctxsep & \, & ${\sum_i{a_i l_i} \ge A}$  \\
+        $i_2$. & \ctxsep  & \, & ${\sum_i{b_i l_i} \ge B}$ \\
+        $j$. & \ctxsep  & \, & ${\sum_i{(a_i + b_i) l_i} \ge (A+B)}$  & (\currule\; $i_1$, $i_2$)
+    \end{AletheS}
+
+    To verify a \proofRule{cp_addition2} step, one must check that the two given
+    pseudo-Boolean constraints are valid and that their combination satisfies
+    the addition rule.
+
+\end{RuleDescription}
+
+\begin{RuleExample}
+    A simple \proofRule{cp_addition2} step might look like this:
+
+    \begin{AletheVerb}
+        (assume c1 (>= (+ (* 1 x1) 0) 1))
+        (assume c2 (>= (+ (* 1 x2) 0) 1))
+        (step t1 (cl (>= (+ (* 1 x1) (* 1 x2) 0) 2))
+        :rule cp_addition :premises (c1 c2))
+    \end{AletheVerb}
+
+    In this example, we are adding two constraints.
+    \[
+        x_1 \geq 1 \quad \text{and} \quad x_2 \geq 1.
+    \]
+    After applying the \proofRule{cp_addition2} rule,
+    the combined constraint is:
+    \[
+        x_1 + x_2 \geq 1
+    \]
+    After simplification, this results in a contradiction.
+\end{RuleExample}
+
+\begin{RuleExample}
+    This \proofRule{cp_addition2} example has negated literals that cancel each other:
+
+    \begin{AletheVerb}
+        (assume c1 (>= (+ (* 2 x1) (* 3 x2) 0) 2))
+        (assume c2 (>= (+ (* 1 (- 1 x1)) (* 3 (- 1 x2)) 0) 4))
+        (step t1 (cl (>= (+ (* 1 x1) 0) 2))
+        :rule cp_addition :premises (c1 c2))
+    \end{AletheVerb}
+
+    In this example, we are adding two constraints.
+    \[
+        2 x_1 + 3 x_2 \geq 2 \quad \text{and} \quad \overline{x_1} + 3 \overline{x_2} \geq 4.
+    \]
+    After applying the \proofRule{cp_addition2} rule,
+    the combined constraint is:
+    \[
+        x_1 \geq 2
+    \]
+    After simplification, this results in a contradiction.
+\end{RuleExample}
+
+
 
 \begin{RuleDescription}{cp_multiplication}
     A constraint can be multiplied by any $c \in \mathbb{N}^+$:
diff --git a/alethe/doc.pdf b/alethe/doc.pdf
index 49c45a149021692d1aed879ee3e6d51aa9ccb4db..36f03fc6e029ad3f06c594b5cf527b491c849978 100644
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From 5f99409d63aa58647a01dbcda5a6431fb697f3bf Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Tue, 15 Oct 2024 10:33:57 -0300
Subject: [PATCH 89/93] Added bit blastic rules to docs

---
 alethe/.gitignore         |   1 +
 alethe/bit_blasting.tex   | 130 ++++++++++++++++++++++++++++++++++++++
 alethe/cutting_planes.tex |   3 +-
 alethe/doc.pdf            | Bin 370187 -> 377534 bytes
 4 files changed, 132 insertions(+), 2 deletions(-)
 create mode 100644 alethe/bit_blasting.tex

diff --git a/alethe/.gitignore b/alethe/.gitignore
index 7bd2381..522003f 100644
--- a/alethe/.gitignore
+++ b/alethe/.gitignore
@@ -12,3 +12,4 @@ shell.nix
 Title.jpg
 version.sh
 .vscode
+BitBlasting_Atoms_and_Terms.pdf
diff --git a/alethe/bit_blasting.tex b/alethe/bit_blasting.tex
new file mode 100644
index 0000000..9b41337
--- /dev/null
+++ b/alethe/bit_blasting.tex
@@ -0,0 +1,130 @@
+\begin{RuleDescription}{bv_equal}
+    Consider bitvectors $x$ of length $k$ with respective PB-blasted vectors $x$
+    - i.e. $x$ is a vector of Pseudo-Boolean variables.
+
+    A step of the \currule{} rule means bitwise equality, expressed by:
+    \[
+        \sum_{i=0}^{k-1} 2^i\mathbf{x}_{k-i-1} - \sum_{i=0}^{k-1} 2^i\mathbf{y}_{k-i-1} = 0.
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_not_equal}
+    The \currule{} rule, meaning bitwise inequality, can be concisely expressed by:
+    \[
+        \mathbf{z} = \text{XOR}(x, y); \qquad \sum_{i=0}^{k-1} \mathbf{z}_i \geq 0
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_ult}
+    The \currule{} rule for unsigned less than, $x < y$, is defined as:
+    \[
+        \sum_{i=0}^{k-1} 2^i\mathbf{y}_{k-i-1} - \sum_{i=0}^{k-1} 2^i\mathbf{x}_{k-i-1} \geq 1.
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_not_ult}
+    The negation of the \currule{} rule, meaning $x \geq y$ for unsigned integers, can be expressed as:
+    \[
+        \neg ULT(x, y) \equiv UGE(x, y)
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_uge}
+    The \currule{} rule for unsigned greater than or equal, $x \geq y$, is defined as:
+    \[
+        \sum_{i=0}^{k-1} 2^i\mathbf{x}_{k-i-1} - \sum_{i=0}^{k-1} 2^i\mathbf{y}_{k-i-1} \geq 0.
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_not_uge}
+    The negation of the \currule{} rule, meaning $x < y$ for unsigned integers, is expressed by:
+    \[
+        \neg UGE(x, y) \equiv ULT(x, y)
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_ule}
+    The \currule{} rule for unsigned less than or equal, $x \leq y$, is equivalent to:
+    \[
+        ULE(x, y) \equiv UGE(y, x)
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_not_ule}
+    The negation of the \currule{} rule, meaning $x > y$, is expressed by:
+    \[
+        \neg ULE(x, y) \equiv UGT(x, y)
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_ugt}
+    The \currule{} rule for unsigned greater than, $x > y$, is equivalent to:
+    \[
+        UGT(x, y) \equiv ULT(y, x)
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_not_ugt}
+    The negation of the \currule{} rule, meaning $x \leq y$, is expressed by:
+    \[
+        \neg UGT(x, y) \equiv ULE(x, y)
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_slt}
+    The \currule{} rule for signed less than, $x < y$, is defined as:
+    \[
+        -(2^{k-1})\mathbf{y}_0 + \sum_{i=0}^{k-2} 2^i\mathbf{y}_{k-i-1} + 2^{k-1} \mathbf{x}_{0} - \sum_{i=0}^{k-2} 2^i\mathbf{x}_{k-i-1} \geq 1.
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_not_slt}
+    The negation of the \currule{} rule, meaning $x \geq y$ for signed integers, is expressed by:
+    \[
+        \neg SLT(x, y) \equiv SGE(x, y)
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_sge}
+    The \currule{} rule for signed greater than or equal, $x \geq y$, is defined as:
+    \[
+        -(2^{k-1})\mathbf{x}_0 + \sum_{i=0}^{k-2} 2^i\mathbf{x}_{k-i-1} + 2^{k-1}\mathbf{y}_0 - \sum_{i=0}^{k-2} 2^i\mathbf{y}_{k-i-1} \geq 0.
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_not_sge}
+    The negation of the \currule{} rule, meaning $x < y$ for signed integers, is expressed by:
+    \[
+        \neg SGE(x, y) \equiv SLT(x, y)
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_sle}
+    The \currule{} rule for signed less than or equal, $x \leq y$, is equivalent to:
+    \[
+        SLE(x, y) \equiv SGE(y, x)
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_not_sle}
+    The negation of the \currule{} rule, meaning $x > y$, is expressed by:
+    \[
+        \neg SLE(x, y) \equiv SGT(x, y)
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_sgt}
+    The \currule{} rule for signed greater than, $x > y$, is equivalent to:
+    \[
+        SGT(x, y) \equiv SLT(y, x)
+    \]
+\end{RuleDescription}
+
+\begin{RuleDescription}{bv_not_sgt}
+    The negation of the \currule{} rule, meaning $x \leq y$, is expressed by:
+    \[
+        \neg SGT(x, y) \equiv SLE(x, y)
+    \]
+\end{RuleDescription}
+
+\newpage
\ No newline at end of file
diff --git a/alethe/cutting_planes.tex b/alethe/cutting_planes.tex
index 24855a1..43d0573 100644
--- a/alethe/cutting_planes.tex
+++ b/alethe/cutting_planes.tex
@@ -98,7 +98,6 @@
     \[
         x_1 + x_2 \geq 1
     \]
-    After simplification, this results in a contradiction.
 \end{RuleExample}
 
 \begin{RuleExample}
@@ -158,4 +157,4 @@
 
 \end{RuleDescription}
 
-
+\newpage
\ No newline at end of file
diff --git a/alethe/doc.pdf b/alethe/doc.pdf
index 36f03fc6e029ad3f06c594b5cf527b491c849978..ae498221a791dde2425d5fab8c1ca501f9e6eb46 100644
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From 7376edaf1ad680667f479c5129d18eb29a750000 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Mon, 21 Oct 2024 11:53:12 -0300
Subject: [PATCH 90/93] [alethe] Reformat the negated literals

---
 alethe/cutting_planes.tex |  76 +++++++++++---------------------------
 alethe/doc.pdf            | Bin 377534 -> 375886 bytes
 2 files changed, 22 insertions(+), 54 deletions(-)

diff --git a/alethe/cutting_planes.tex b/alethe/cutting_planes.tex
index 43d0573..aa1b7a7 100644
--- a/alethe/cutting_planes.tex
+++ b/alethe/cutting_planes.tex
@@ -1,60 +1,28 @@
 \begin{RuleDescription}{cp_addition}
-    A step of the \currule{} rule represents a step in the cutting planes reasoning, that works on
-    Pseudo-Boolean Constraints, that are 0-1 integer linear inequalities.
-
-    A pseudo boolean inequality is a term of the form
-    \[
-        \sum_i{a_i l_i} \ge A
-    \]
-    where we have:
-    \begin{itemize}
-        \item constant $A \in \mathbb{N}$
-        \item coefficients $ a_i \in \mathbb{N}$
-        \item literals $ l_i: x_i\ \text{or}\ \overline{x_i}\ (\text{where } x_i + \overline{x_i} = 1)$
-        \item variables $ x_i $ take values $ 0 = false $ or $ 1 = true$
-    \end{itemize}
-
-    In alethe such formula will be defined as:
-
-    \centerline{\inlineAlethe{(>= (+ <TERMS> 0) <A>)}}
-    \hfill
-
-    where \inlineAlethe{<TERMS>} is a list of either:
-    \begin{enumerate}
-        \item \inlineAlethe{(* <a_i> <x_i>)} in a plain literal $a_i x_i$.
-        \item \inlineAlethe{(* (- 1 <a_i>) <x_i>)} in a negated literal $a_i \overline x_i$
-    \end{enumerate}
-
-    and \inlineAlethe{<A>} is the natural constant $A$.
-
-    Using this format, two constraints can be added together, adding the coefficients and the
-    constants. Another behaviour is that $x_i$ and $\overline x_i$ that may appear together
-    cancel each other:
-
-    \begin{AletheS}
-        $i_1$. & \ctxsep & \, & ${\sum_i{a_i l_i} \ge A}$  \\
-        $i_2$. & \ctxsep  & \, & ${\sum_i{b_i l_i} \ge B}$ \\
-        $j$. & \ctxsep  & \, & ${\sum_i{(a_i + b_i) l_i} \ge (A+B)}$  & (\currule\; $i_1$, $i_2$)
-    \end{AletheS}
-
-\end{RuleDescription}
-
-\newpage
-
-\begin{RuleDescription}{cp_addition2}
     A step of the \currule{} rule represents the addition of two pseudo-Boolean
     constraints using cutting planes reasoning. Pseudo-Boolean constraints are
     0-1 integer linear inequalities of the form:
     \[
         \sum_i a_i \times l_i \geq A
     \]
-    where $A \in \mathbb{N}$ is a constant, $a_i \in \mathbb{N}$ are coefficients,
-    and $l_i$ are literals of the form $x_i$ or $\overline{x_i}$ (where $x_i + \overline{x_i} = 1$),
-    with $x_i$ taking pseudo-boolean values $0$ (false) or $1$ (true).
+    where $A$ is called \textbf{constant}, $a_i$ are \textbf{coefficients},
+    and $l_i$ are \textbf{literals}, that are either:
+    \begin{itemize}
+        \item \textbf{plain} literal, a term \inlineAlethe{x};
+        \item \textbf{negated} literal, a term of the form \inlineAlethe{(- 1 x)}
+    \end{itemize}
+
+    where the \inlineAlethe{x} value is a pseudo-boolean variable,
+    i.e. it will resolve to values \inlineAlethe{0} (false) or \inlineAlethe{1} (true).
+    All these values are of sort \inlineAlethe{Int}.
+
+    To form a summation we use a list of added terms of form,
+    \inlineAlethe{(+ <T1> <T2> ... 0)} and each term is
+    \inlineAlethe{(* a_i <L1>)}, with a coefficient and a literal.\\
 
-    The \proofRule{cp_addition2} rule allows two pseudo-Boolean constraints to be
-    added together, combining their coefficients and constants. Literals $x_i$ and
-    $\overline{x_i}$ that appear together cancel each other.
+    The \currule{} rule allows two pseudo-Boolean constraints to be
+    added together, combining their coefficients and constants. Negated and plain literals
+    over the same variable cancel each other.
 
     Formally, given two constraints:
     \[
@@ -73,14 +41,14 @@
         $j$. & \ctxsep  & \, & ${\sum_i{(a_i + b_i) l_i} \ge (A+B)}$  & (\currule\; $i_1$, $i_2$)
     \end{AletheS}
 
-    To verify a \proofRule{cp_addition2} step, one must check that the two given
+    To verify a \currule{} step, one must check that the two given
     pseudo-Boolean constraints are valid and that their combination satisfies
     the addition rule.
 
 \end{RuleDescription}
 
 \begin{RuleExample}
-    A simple \proofRule{cp_addition2} step might look like this:
+    A simple \proofRule{cp_addition} step might look like this:
 
     \begin{AletheVerb}
         (assume c1 (>= (+ (* 1 x1) 0) 1))
@@ -93,7 +61,7 @@
     \[
         x_1 \geq 1 \quad \text{and} \quad x_2 \geq 1.
     \]
-    After applying the \proofRule{cp_addition2} rule,
+    After applying the \proofRule{cp_addition} rule,
     the combined constraint is:
     \[
         x_1 + x_2 \geq 1
@@ -101,7 +69,7 @@
 \end{RuleExample}
 
 \begin{RuleExample}
-    This \proofRule{cp_addition2} example has negated literals that cancel each other:
+    This \proofRule{cp_addition} example has negated literals that cancel each other:
 
     \begin{AletheVerb}
         (assume c1 (>= (+ (* 2 x1) (* 3 x2) 0) 2))
@@ -114,7 +82,7 @@
     \[
         2 x_1 + 3 x_2 \geq 2 \quad \text{and} \quad \overline{x_1} + 3 \overline{x_2} \geq 4.
     \]
-    After applying the \proofRule{cp_addition2} rule,
+    After applying the \proofRule{cp_addition} rule,
     the combined constraint is:
     \[
         x_1 \geq 2
diff --git a/alethe/doc.pdf b/alethe/doc.pdf
index ae498221a791dde2425d5fab8c1ca501f9e6eb46..08b26b1a9a7a4dcea547096fb16b28be716b44ca 100644
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Mqteq$DNCdNACgHC1ONa4


From 4a08adcd31739b4e2af1ab6b4824fb4e13a56bc4 Mon Sep 17 00:00:00 2001
From: bernborgess <62393097+bernborgess@users.noreply.github.com>
Date: Thu, 19 Dec 2024 11:04:38 -0300
Subject: [PATCH 91/93] Create lean_action_ci.yml

---
 .github/workflows/lean_action_ci.yml | 14 ++++++++++++++
 1 file changed, 14 insertions(+)
 create mode 100644 .github/workflows/lean_action_ci.yml

diff --git a/.github/workflows/lean_action_ci.yml b/.github/workflows/lean_action_ci.yml
new file mode 100644
index 0000000..09cd4ca
--- /dev/null
+++ b/.github/workflows/lean_action_ci.yml
@@ -0,0 +1,14 @@
+name: Lean Action CI
+
+on:
+  push:
+  pull_request:
+  workflow_dispatch:
+
+jobs:
+  build:
+    runs-on: ubuntu-latest
+
+    steps:
+      - uses: actions/checkout@v4
+      - uses: leanprover/lean-action@v1

From ff4b0f781637bb76b2a7f8db07b8c057906132de Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 19 Dec 2024 11:06:54 -0300
Subject: [PATCH 92/93] :boom: Updating lean to v4.15.0

---
 .gitignore                                  |  2 +-
 LeanCuttingPlanes.lean                      | 10 +--
 LeanCuttingPlanes/Rules/Addition.lean       | 14 ++--
 LeanCuttingPlanes/Rules/Division.lean       |  4 +-
 LeanCuttingPlanes/Rules/Multiplication.lean |  5 +-
 LeanCuttingPlanes/Rules/Saturation.lean     |  4 +-
 Main.lean                                   |  2 +-
 lake-manifest.json                          | 87 ++++++++++++++-------
 lakefile.lean                               | 13 ---
 lakefile.toml                               | 14 ++++
 lean-toolchain                              |  2 +-
 11 files changed, 93 insertions(+), 64 deletions(-)
 delete mode 100644 lakefile.lean
 create mode 100644 lakefile.toml

diff --git a/.gitignore b/.gitignore
index 7d3fd48..cd3fdc5 100644
--- a/.gitignore
+++ b/.gitignore
@@ -315,4 +315,4 @@ TSWLatexianTemp*
 #*Notes.bib
 
 # Dowloaded assets
-*.zip
\ No newline at end of file
+*.zip/.lake
diff --git a/LeanCuttingPlanes.lean b/LeanCuttingPlanes.lean
index 2808d6c..92a9fd7 100644
--- a/LeanCuttingPlanes.lean
+++ b/LeanCuttingPlanes.lean
@@ -1,7 +1,7 @@
 -- This module serves as the root of the `LeanCuttingPlanes` library.
 -- Import modules here that should be built as part of the library.
-import «LeanCuttingPlanes».Basic
-import «LeanCuttingPlanes».Rules.Addition
-import «LeanCuttingPlanes».Rules.Multiplication
-import «LeanCuttingPlanes».Rules.Division
-import «LeanCuttingPlanes».Rules.Saturation
+import LeanCuttingPlanes.Basic
+import LeanCuttingPlanes.Rules.Addition
+import LeanCuttingPlanes.Rules.Multiplication
+import LeanCuttingPlanes.Rules.Division
+import LeanCuttingPlanes.Rules.Saturation
diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index 98a8ea9..098287c 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -1,4 +1,4 @@
-import «LeanCuttingPlanes».Basic
+import LeanCuttingPlanes.Basic
 
 namespace PseudoBoolean
 
@@ -30,13 +30,12 @@ lemma ite_eq_bmul (x y : ℕ) (b : Fin 2)
   by_cases h : b = 0
   . rw [h]
     rw [if_neg]
-    . simp only [Fin.isValue, Fin.val_zero, mul_zero, tsub_zero, mul_one, zero_add]
+    . simp only [Fin.isValue, Fin.val_zero, mul_zero, tsub_zero, mul_one, zero_add,Nat.sub_zero, mul_one]
     trivial
   . -- b = 1
     apply Fin.eq_one_of_neq_zero b at h
     rw [h]
-    simp only [Fin.isValue, ↓reduceIte, Fin.val_one, mul_one, ge_iff_le, le_refl,
-      tsub_eq_zero_of_le, mul_zero, add_zero]
+    simp only [Fin.isValue, ↓reduceIte, Fin.val_one, mul_one, Nat.sub_self, mul_zero, add_zero]
 
 lemma reduce_terms (p n : ℕ) (x : Fin 2)
   : p * x + n * (1 - x) = (p - n) * x + (n - p) * (1 - x) + min p n  := by
@@ -44,13 +43,13 @@ lemma reduce_terms (p n : ℕ) (x : Fin 2)
   . rw [h]
     simp only [Fin.isValue, Fin.val_zero, mul_zero, tsub_zero, mul_one, zero_add]
     rw [Nat.min_comm]
-    exact Nat.sub_add_min_cancel n p |>.symm
+    simp only [Nat.sub_zero, mul_one, Nat.sub_add_min_cancel]
 
   . -- x = 1
     apply Fin.eq_one_of_neq_zero x at h
     rw [h]
-    simp only [Fin.isValue, Fin.val_one, mul_one, ge_iff_le, le_refl, tsub_eq_zero_of_le, mul_zero, add_zero]
-    exact Nat.sub_add_min_cancel p n |>.symm
+    simp only [Fin.isValue, Fin.val_one, mul_one, Nat.sub_self, mul_zero, add_zero,
+      Nat.sub_add_min_cancel]
 
 theorem Reduction
   (xs : Fin n → Fin 2)
@@ -59,6 +58,7 @@ theorem Reduction
   unfold ReductionProp PBIneq PBSum at *
   simp only [Fin.isValue, ge_iff_le, tsub_le_iff_right] at *
   simp_rw [ite_eq_bmul] at *
+  simp only [Nat.sub_le_iff_le_add]
   rw [←Finset.sum_add_distrib]
   simp_rw [←reduce_terms]
   exact ha
diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index 2a4697a..86044bf 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -1,4 +1,4 @@
-import «LeanCuttingPlanes».Basic
+import LeanCuttingPlanes.Basic
 import Mathlib.Algebra.Order.Floor.Div
 
 namespace PseudoBoolean
@@ -61,7 +61,7 @@ theorem Division
   simp only [Fin.isValue, ge_iff_le, gt_iff_lt, Prod_map, map_eq, Function.comp_apply] at *
   apply ceildiv_le_ceildiv_right c at ha
   apply le_trans ha
-  simp only [←ceildiv_ite]
+  simp only [←ceildiv_ite, Fin.isValue, Prod.map_fst, Prod.map_snd]
   apply Finset.ceildiv_le_ceildiv
   done
 
diff --git a/LeanCuttingPlanes/Rules/Multiplication.lean b/LeanCuttingPlanes/Rules/Multiplication.lean
index fff10c8..a293121 100644
--- a/LeanCuttingPlanes/Rules/Multiplication.lean
+++ b/LeanCuttingPlanes/Rules/Multiplication.lean
@@ -1,4 +1,5 @@
-import «LeanCuttingPlanes».Basic
+import LeanCuttingPlanes.Basic
+import Mathlib.Algebra.Order.Module.Defs
 
 namespace PseudoBoolean
 
@@ -16,7 +17,7 @@ theorem Multiplication
   (c : ℕ)
   : PBIneq (c • as) xs (c • A) := by
   unfold PBIneq PBSum at *
-  simp only [Fin.isValue, ge_iff_le, nsmul_eq_smul, smul_eq_mul] at *
+  simp only [Fin.isValue, nsmul_eq_smul, smul_eq_mul, ge_iff_le] at *
   apply nsmul_le_nsmul_right at ha
   specialize ha c
   rw [←Finset.sum_nsmul] at ha
diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index e949e74..deed655 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -1,4 +1,4 @@
-import «LeanCuttingPlanes».Basic
+import LeanCuttingPlanes.Basic
 
 namespace PseudoBoolean
 open FinVec Matrix BigOperators Finset
@@ -42,7 +42,7 @@ theorem Saturation
   {as : Coeff n} {A : ℕ} (ha : PBIneq as xs A)
   : PBIneq (map (mapBoth (min A)) as) xs A := by
   unfold PBIneq PBSum FinVec.map mapBoth at *
-  simp only [Fin.isValue, ge_iff_le, Prod_map, seq_eq] at *
+  simp only [Fin.isValue, ge_iff_le, Prod.map_apply, seq_eq] at *
   have h := le_sum_min_of_le_sum ha
   simp_rw [apply_ite (min A) ((xs _ = 1)) ((as _).1) ((as _).2)] at h
   exact h
diff --git a/Main.lean b/Main.lean
index e0ef49b..01e1231 100644
--- a/Main.lean
+++ b/Main.lean
@@ -1,4 +1,4 @@
-import «LeanCuttingPlanes»
+import LeanCuttingPlanes
 
 open PseudoBoolean
 
diff --git a/lake-manifest.json b/lake-manifest.json
index 61f6ac9..ff5ce5a 100644
--- a/lake-manifest.json
+++ b/lake-manifest.json
@@ -1,68 +1,95 @@
-{"version": 7,
+{"version": "1.1.0",
  "packagesDir": ".lake/packages",
  "packages":
- [{"url": "https://github.com/leanprover-community/batteries",
+ [{"url": "https://github.com/leanprover-community/mathlib4",
    "type": "git",
    "subDir": null,
-   "rev": "7b3c48b58fa0ae1c8f27889bdb086ea5e4b27b06",
-   "name": "batteries",
+   "scope": "leanprover-community",
+   "rev": "32ba402fc2048299c9c0e153b72b4d4ea8aade19",
+   "name": "mathlib",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "main",
-   "inherited": true,
+   "inputRev": "master",
+   "inherited": false,
    "configFile": "lakefile.lean"},
-  {"url": "https://github.com/leanprover-community/quote4",
+  {"url": "https://github.com/leanprover-community/plausible",
    "type": "git",
    "subDir": null,
-   "rev": "53156671405fbbd5402ed17a79bd129b961bd8d6",
-   "name": "Qq",
+   "scope": "leanprover-community",
+   "rev": "8e5cb8d424df462f84997dd68af6f40e347c3e35",
+   "name": "plausible",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "master",
+   "inputRev": "v4.15.0-rc1",
    "inherited": true,
-   "configFile": "lakefile.lean"},
-  {"url": "https://github.com/leanprover-community/aesop",
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/LeanSearchClient",
    "type": "git",
    "subDir": null,
-   "rev": "e8c8a42642ceb5af33708b79ae8a3148b681c236",
-   "name": "aesop",
+   "scope": "leanprover-community",
+   "rev": "d7caecce0d0f003fd5e9cce9a61f1dd6ba83142b",
+   "name": "LeanSearchClient",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "master",
+   "inputRev": "main",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/import-graph",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "ed3b856bd8893ade75cafe13e8544d4c2660f377",
+   "name": "importGraph",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "v4.15.0-rc1",
    "inherited": true,
    "configFile": "lakefile.toml"},
   {"url": "https://github.com/leanprover-community/ProofWidgets4",
    "type": "git",
    "subDir": null,
-   "rev": "e6b6247c61280c77ade6bbf0bc3c66a44fe2e0c5",
+   "scope": "leanprover-community",
+   "rev": "2b000e02d50394af68cfb4770a291113d94801b5",
    "name": "proofwidgets",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "v0.0.36",
+   "inputRev": "v0.0.48",
    "inherited": true,
    "configFile": "lakefile.lean"},
-  {"url": "https://github.com/leanprover/lean4-cli",
+  {"url": "https://github.com/leanprover-community/aesop",
    "type": "git",
    "subDir": null,
-   "rev": "a11566029bd9ec4f68a65394e8c3ff1af74c1a29",
-   "name": "Cli",
+   "scope": "leanprover-community",
+   "rev": "a4a08d92be3de00def5298059bf707c72dfd3c66",
+   "name": "aesop",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "main",
+   "inputRev": "master",
+   "inherited": true,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/quote4",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "ad942fdf0b15c38bface6acbb01d63855a2519ac",
+   "name": "Qq",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "v4.14.0",
    "inherited": true,
    "configFile": "lakefile.lean"},
-  {"url": "https://github.com/leanprover-community/import-graph.git",
+  {"url": "https://github.com/leanprover-community/batteries",
    "type": "git",
    "subDir": null,
-   "rev": "35e38eb320982cfd2fcc864e0e0467ca223c8cdb",
-   "name": "importGraph",
+   "scope": "leanprover-community",
+   "rev": "9e583efcea920afa13ee2a53069821a2297a94c0",
+   "name": "batteries",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
    "inherited": true,
    "configFile": "lakefile.toml"},
-  {"url": "https://github.com/leanprover-community/mathlib4",
+  {"url": "https://github.com/leanprover/lean4-cli",
    "type": "git",
    "subDir": null,
-   "rev": "e76a76c36a1c2792438b62656bb94234bc4e5317",
-   "name": "mathlib",
+   "scope": "leanprover",
+   "rev": "0c8ea32a15a4f74143e4e1e107ba2c412adb90fd",
+   "name": "Cli",
    "manifestFile": "lake-manifest.json",
-   "inputRev": null,
-   "inherited": false,
-   "configFile": "lakefile.lean"}],
+   "inputRev": "main",
+   "inherited": true,
+   "configFile": "lakefile.toml"}],
  "name": "«lean-cutting-planes»",
  "lakeDir": ".lake"}
diff --git a/lakefile.lean b/lakefile.lean
deleted file mode 100644
index 0aa9938..0000000
--- a/lakefile.lean
+++ /dev/null
@@ -1,13 +0,0 @@
-import Lake
-open Lake DSL
-
-package «lean-cutting-planes» where
-
-require mathlib from git
-  "https://github.com/leanprover-community/mathlib4"
-
-lean_lib «LeanCuttingPlanes» where
-
-@[default_target]
-lean_exe «lean-cutting-planes» where
-  root := `Main
diff --git a/lakefile.toml b/lakefile.toml
new file mode 100644
index 0000000..b7035cd
--- /dev/null
+++ b/lakefile.toml
@@ -0,0 +1,14 @@
+name = "lean-cutting-planes"
+version = "0.1.0"
+defaultTargets = ["lean-cutting-planes"]
+
+[[require]]
+name = "mathlib"
+scope = "leanprover-community"
+
+[[lean_lib]]
+name = "LeanCuttingPlanes"
+
+[[lean_exe]]
+name = "lean-cutting-planes"
+root = "Main"
diff --git a/lean-toolchain b/lean-toolchain
index 78f39e2..cf25a98 100644
--- a/lean-toolchain
+++ b/lean-toolchain
@@ -1 +1 @@
-leanprover/lean4:v4.8.0-rc2
+leanprover/lean4:v4.15.0-rc1

From 4ef4d983407c613b0c13ee28c461b77426e49f07 Mon Sep 17 00:00:00 2001
From: bernborgess <bernborgess@outlook.com>
Date: Thu, 19 Dec 2024 11:14:30 -0300
Subject: [PATCH 93/93] lint: remove `done`s

---
 LeanCuttingPlanes/Basic.lean                | 3 ---
 LeanCuttingPlanes/Rules/Addition.lean       | 5 -----
 LeanCuttingPlanes/Rules/Division.lean       | 8 +-------
 LeanCuttingPlanes/Rules/Multiplication.lean | 2 --
 LeanCuttingPlanes/Rules/Saturation.lean     | 2 --
 Main.lean                                   | 3 ---
 6 files changed, 1 insertion(+), 22 deletions(-)

diff --git a/LeanCuttingPlanes/Basic.lean b/LeanCuttingPlanes/Basic.lean
index 18b3993..34626a5 100644
--- a/LeanCuttingPlanes/Basic.lean
+++ b/LeanCuttingPlanes/Basic.lean
@@ -58,17 +58,14 @@ def PBIneq (cs : Coeff n) (xs : Fin n → Fin 2) (const : ℕ) :=
 example : PBIneq ![(1,0)] ![1] 1 := by
   reduce                -- Expand the goal to 1 * 1 ≥ 1
   exact Nat.le_refl 1   -- Prove that 1 * 1 ≥ 1
-  done
 
 example : PBIneq ![(1,0),(2,0)] ![0,1] 2 := by
   reduce                -- Change goal to 1 * 0 + 2 * 1 ≥ 2
   exact Nat.le_refl 2   -- Prove 1 * 0 + 2 * 1 ≥ 2
-  done
 
 example : PBIneq ![(3,0),(4,0)] ![0,1] 2 := by
   reduce
   simp
-  done
 
 def mapBoth (f : α → β) (t : α × α) : β × β := Prod.map f f t
 
diff --git a/LeanCuttingPlanes/Rules/Addition.lean b/LeanCuttingPlanes/Rules/Addition.lean
index 098287c..e918350 100644
--- a/LeanCuttingPlanes/Rules/Addition.lean
+++ b/LeanCuttingPlanes/Rules/Addition.lean
@@ -14,7 +14,6 @@ theorem Addition'
   simp_rw [←ite_add_ite]
   rw [Finset.sum_add_distrib]
   exact Nat.add_le_add ha hb
-  done
 
 def ReductionProp
   (xs : Fin n → Fin 2) (ks : Coeff n) (K : ℕ)
@@ -62,7 +61,6 @@ theorem Reduction
   rw [←Finset.sum_add_distrib]
   simp_rw [←reduce_terms]
   exact ha
-  done
 
 def AdditionProp
   (xs : Fin n → Fin 2)
@@ -83,14 +81,12 @@ theorem Addition
   : AdditionProp xs as A bs B := by
   have hk := Addition' xs as A ha bs B hb
   exact Reduction xs (as + bs) (A + B) hk
-  done
 
 example
   (ha : PBIneq ![(1,0),(0,0)] xs 1)
   (hb : PBIneq ![(1,0),(1,0)] xs 2)
   : PBIneq ![(2,0),(1,0)] xs 3 := by
   apply Addition ha hb
-  done
 
 -- Reduction happens automatically
 example
@@ -98,7 +94,6 @@ example
   (hb : PBIneq ![(0,0),(2,0),(0,1)] xs 2)
   : PBIneq ![(3,0),(2,0),(0,0)] xs 2 := by
   apply Addition ha hb
-  done
 
 
 end PseudoBoolean
diff --git a/LeanCuttingPlanes/Rules/Division.lean b/LeanCuttingPlanes/Rules/Division.lean
index 86044bf..32a8aa4 100644
--- a/LeanCuttingPlanes/Rules/Division.lean
+++ b/LeanCuttingPlanes/Rules/Division.lean
@@ -13,7 +13,6 @@ lemma ceildiv_le_ceildiv_right {a b : ℕ} (c : ℕ) (hab : a ≤ b)
   apply Nat.sub_le_sub_right
   apply Nat.add_le_add_right
   exact hab
-  done
 
 -- @kbuzzard
 theorem Nat.add_ceildiv_le_add_ceildiv (a b c : ℕ)
@@ -27,7 +26,6 @@ theorem Nat.add_ceildiv_le_add_ceildiv (a b c : ℕ)
   rw [mul_add]
   -- now use a standard fact
   gcongr <;> exact le_smul_ceilDiv hc
-  done
 
 -- @Ruben-VandeVelde
 theorem Finset.ceildiv_le_ceildiv {α : Type*} (as : α → ℕ) (s : Finset α) (c : ℕ)
@@ -39,12 +37,10 @@ theorem Finset.ceildiv_le_ceildiv {α : Type*} (as : α → ℕ) (s : Finset α)
     rw [sum_cons, sum_cons]
     have h := Nat.add_ceildiv_le_add_ceildiv (as a) (∑x ∈ s, as x) c
     exact le_add_of_le_add_left h ih
-    done
 
 lemma ceildiv_ite (P : Prop) [Decidable P] (a b c : ℕ)
   : (if P then b else c) ⌈/⌉ a = if P then (b ⌈/⌉ a) else (c ⌈/⌉ a) := by
   split_ifs <;> rfl
-  done
 
 -- Division
 -- ∑i (a i * l i) ≥ A
@@ -58,17 +54,15 @@ theorem Division
   (c : ℕ)
   : PBIneq (map (mapBoth (ceildiv c)) as) xs (ceildiv c A) := by
   unfold PBIneq PBSum ceildiv mapBoth at *
-  simp only [Fin.isValue, ge_iff_le, gt_iff_lt, Prod_map, map_eq, Function.comp_apply] at *
+  simp only [Fin.isValue, ge_iff_le, gt_iff_lt, Prod.map_apply, map_eq, Function.comp_apply] at *
   apply ceildiv_le_ceildiv_right c at ha
   apply le_trans ha
   simp only [←ceildiv_ite, Fin.isValue, Prod.map_fst, Prod.map_snd]
   apply Finset.ceildiv_le_ceildiv
-  done
 
 example
   (ha : PBIneq ![(3,0),(4,0)] xs 3)
   : PBIneq ![(2,0),(2,0)] xs 2 := by
   apply Division ha 2
-  done
 
 end PseudoBoolean
diff --git a/LeanCuttingPlanes/Rules/Multiplication.lean b/LeanCuttingPlanes/Rules/Multiplication.lean
index a293121..6f4bc02 100644
--- a/LeanCuttingPlanes/Rules/Multiplication.lean
+++ b/LeanCuttingPlanes/Rules/Multiplication.lean
@@ -23,12 +23,10 @@ theorem Multiplication
   rw [←Finset.sum_nsmul] at ha
   simp only [smul_eq_mul, Fin.isValue, mul_ite] at ha
   exact ha
-  done
 
 example
   (ha : PBIneq ![(1,0)] xs 3)
   : PBIneq ![(2,0)] xs 6 := by
   apply Multiplication ha 2
-  done
 
 end PseudoBoolean
diff --git a/LeanCuttingPlanes/Rules/Saturation.lean b/LeanCuttingPlanes/Rules/Saturation.lean
index deed655..ca9ad90 100644
--- a/LeanCuttingPlanes/Rules/Saturation.lean
+++ b/LeanCuttingPlanes/Rules/Saturation.lean
@@ -46,12 +46,10 @@ theorem Saturation
   have h := le_sum_min_of_le_sum ha
   simp_rw [apply_ite (min A) ((xs _ = 1)) ((as _).1) ((as _).2)] at h
   exact h
-  done
 
 example
   (ha : PBIneq ![(3,0),(4,0)] xs 3)
   : PBIneq ![(3,0),(3,0)] xs 3 := by
   apply Saturation ha
-  done
 
 end PseudoBoolean
diff --git a/Main.lean b/Main.lean
index 01e1231..29210c5 100644
--- a/Main.lean
+++ b/Main.lean
@@ -6,7 +6,6 @@ def pb : PBIneq ![(1,0),(2,0)] ![0,1] 1 := by
   reduce
   rw [Nat.le_eq]
   apply Nat.one_le_ofNat
-  done
 
 def derived
   (xs : Fin 4 → Fin 2)
@@ -20,7 +19,6 @@ def derived
   let t3 : PBIneq ![(0,0),(0,0),(0,0),(0,2)] xs 0  := by apply Multiplication c3 2
   let t4 : PBIneq ![(3,0),(6,0),(6,0),(0,0)] xs 7  := by apply Addition t2 t3
   exact Division t4 3
-  done
 
 def resolved : ∃xs, PBIneq ![(1,0),(2,0),(2,0),(0,0)] xs 3 :=
   let xs := ![1,0,1,0]
@@ -30,7 +28,6 @@ def resolved : ∃xs, PBIneq ![(1,0),(2,0),(2,0),(0,0)] xs 3 :=
   by
   use xs
   exact derived xs c1 c2 c3
-  done
 
 def main : IO Unit := do
   IO.println "This is lean-cutting-planes"