diff --git a/ConNF/BaseType/Params.lean b/ConNF/BaseType/Params.lean
index 6017f1009c..964c22982a 100644
--- a/ConNF/BaseType/Params.lean
+++ b/ConNF/BaseType/Params.lean
@@ -325,6 +325,12 @@ theorem κ_le_zero_iff (i : κ) : i ≤ 0 ↔ i = 0 :=
   by rw [← not_lt, ← Ordinal.typein_le_typein (· < ·), κ_typein_zero, Ordinal.le_zero,
     ← κ_typein_zero, Ordinal.typein_inj]
 
+theorem κ_not_lt_zero (i : κ) : ¬i < 0 := by
+  obtain (h | h | h) := lt_trichotomy i 0
+  · cases h.ne ((κ_le_zero_iff i).mp h.le)
+  · exact h.not_lt
+  · exact h.not_lt
+
 @[simp]
 theorem κ_add_eq_zero_iff (i j : κ) : i + j = 0 ↔ i = 0 ∧ j = 0 :=
   by rw [← Ordinal.typein_inj (α := κ) (· < ·), ← Ordinal.typein_inj (α := κ) (· < ·),
diff --git a/ConNF/Counting.lean b/ConNF/Counting.lean
index d98e05db97..5346bfe24b 100644
--- a/ConNF/Counting.lean
+++ b/ConNF/Counting.lean
@@ -1 +1,2 @@
 import ConNF.Counting.CodingFunction
+import ConNF.Counting.Spec
diff --git a/ConNF/Counting/Spec.lean b/ConNF/Counting/Spec.lean
new file mode 100644
index 0000000000..7595346ecb
--- /dev/null
+++ b/ConNF/Counting/Spec.lean
@@ -0,0 +1,111 @@
+import ConNF.FOA.Basic.Flexible
+import ConNF.Counting.CodingFunction
+
+/-!
+# Specifications
+-/
+
+open Quiver Set Sum WithBot
+
+open scoped Classical Cardinal
+
+universe u
+
+namespace ConNF
+
+variable [Params.{u}] [Level] [FOAAssumptions] {β : Λ}
+
+mutual
+  inductive SpecCondition : Λ → Type u
+    | atom {β : Λ} (A : ExtendedIndex β) (others : Set κ) (nearLitters: Set κ) :
+        SpecCondition β
+    | flexible {β : Λ} (A : ExtendedIndex β) :
+        SpecCondition β
+    | inflexibleCoe {β : Λ} (A : ExtendedIndex β) (h : InflexibleCoePath A)
+        (χ : CodingFunction h.δ) (σ : Spec h.δ) : SpecCondition β
+    | inflexibleBot {β : Λ} (A : ExtendedIndex β) (h : InflexibleBotPath A) (atoms : Set κ) :
+        SpecCondition β
+
+  inductive Spec : Λ → Type u
+    | mk {β : Λ} (max : κ) : ((i : κ) → i < max → SpecCondition β) → Spec β
+end
+
+namespace Spec
+
+def max : Spec β → κ
+  | mk max _ => max
+
+def f : (σ : Spec β) → (i : κ) → i < σ.max → SpecCondition β
+  | mk _ f => f
+
+@[simp]
+theorem mk_max (max : κ) (f : (i : κ) → i < max → SpecCondition β) :
+    (mk max f).max = max :=
+  rfl
+
+@[simp]
+theorem mk_f (max : κ) (f : (i : κ) → i < max → SpecCondition β) :
+    (mk max f).f = f :=
+  rfl
+
+@[ext]
+theorem ext {σ τ : Spec β} (h₁ : σ.max = τ.max)
+    (h₂ : ∀ (i : κ) (hσ : i < σ.max) (hτ : i < τ.max), σ.f i hσ = τ.f i hτ) : σ = τ := by
+  obtain ⟨max, f⟩ := σ
+  obtain ⟨max', f'⟩ := τ
+  subst h₁
+  simp only [mk_max, mk.injEq, heq_eq_eq, true_and]
+  ext i hi
+  exact h₂ i hi hi
+
+end Spec
+
+open scoped Classical in
+noncomputable def SupportCondition.comp (c : SupportCondition β) {γ : TypeIndex}
+    (A : Path (β : TypeIndex) γ) (otherwise : SupportCondition γ) : SupportCondition γ :=
+  if h : ∃ B : ExtendedIndex γ, c.path = A.comp B then
+    ⟨h.choose, c.value⟩
+  else
+    otherwise
+
+noncomputable def Support.comp (S : Support β) {γ : TypeIndex} (A : Path (β : TypeIndex) γ) :
+    Support γ :=
+  if h : ∃ B : ExtendedIndex γ, ∃ x : Atom ⊕ NearLitter, ⟨A.comp B, x⟩ ∈ S then
+    ⟨S.max, fun i hi => (S.f i hi).comp A ⟨h.choose, h.choose_spec.choose⟩⟩
+  else
+    ⟨0, fun _ hi => (κ_not_lt_zero _ hi).elim⟩
+
+instance : WellFoundedRelation Λ where
+  rel := (· < ·)
+  wf := IsWellFounded.wf
+
+instance : WellFoundedRelation Bool where
+  rel := (· < ·)
+  wf := IsWellFounded.wf
+
+open scoped Classical in
+mutual
+  noncomputable def specFor {β : Λ} (S : Support β) : SupportCondition β → SpecCondition β
+    | ⟨A, inl a⟩ => SpecCondition.atom A
+        {i | ∃ hi : i < S.max, S.f i hi = ⟨A, inl a⟩}
+        {i | ∃ N : NearLitter, a ∈ N ∧ ∃ hi : i < S.max, S.f i hi = ⟨A, inr N⟩}
+    | ⟨A, inr N⟩ =>
+        if h : Nonempty (InflexibleCoe A N.1) then
+          SpecCondition.inflexibleCoe A h.some.path
+            (CodingFunction.code h.some.t.support h.some.t (support_supports _))
+            (have : h.some.path.δ < β := coe_lt_coe.mp (h.some.δ_lt_β);
+              spec (S.comp (h.some.path.B.cons h.some.path.hδ) + h.some.t.support))
+        else if h : Nonempty (InflexibleBot A N.1) then
+          SpecCondition.inflexibleBot A h.some.path
+            {i | ∃ hi : i < S.max, S.f i hi = ⟨h.some.path.B.cons (bot_lt_coe _), inl h.some.a⟩}
+        else
+          SpecCondition.flexible A
+
+  noncomputable def spec {β : Λ} (S : Support β) : Spec β :=
+    Spec.mk S.max (fun i hi => specFor S (S.f i hi))
+end
+termination_by
+  specFor β S c => (β, false)
+  spec β S => (β, true)
+
+end ConNF
diff --git a/ConNF/FOA/Complete/LitterCompletion.lean b/ConNF/FOA/Complete/LitterCompletion.lean
index cfee365d53..873dd6ee91 100644
--- a/ConNF/FOA/Complete/LitterCompletion.lean
+++ b/ConNF/FOA/Complete/LitterCompletion.lean
@@ -97,11 +97,10 @@ noncomputable def litterCompletion (π : StructApprox β) (A : ExtendedIndex β)
     if hs : _ ∧ _ then
       fuzz h.some.path.hδε ((ihAction H).hypothesisedAllowable h.some.path hs.1 hs.2 • h.some.t)
     else L
-  else
-    if h : Nonempty (InflexibleBot A L) then
-      fuzz (show (⊥ : TypeIndex) ≠ (h.some.path.ε : Λ) from bot_ne_coe)
-        (H.atomImage (h.some.path.B.cons (bot_lt_coe _)) h.some.a h.some.constrains)
-    else NearLitterApprox.flexibleCompletion (π A) A • L
+  else if h : Nonempty (InflexibleBot A L) then
+    fuzz (show (⊥ : TypeIndex) ≠ (h.some.path.ε : Λ) from bot_ne_coe)
+      (H.atomImage (h.some.path.B.cons (bot_lt_coe _)) h.some.a h.some.constrains)
+  else NearLitterApprox.flexibleCompletion (π A) A • L
 
 theorem litterCompletion_of_flexible (π : StructApprox β) (A : ExtendedIndex β) (L : Litter)
     (H : HypAction ⟨A, inr L.toNearLitter⟩) (hflex : Flexible A L) :