From 72d6a5dbc936a73dc3b06da189a668a4e417f732 Mon Sep 17 00:00:00 2001
From: madeve-unipi <56154049+madeve-unipi@users.noreply.github.com>
Date: Mon, 22 Jul 2024 12:55:45 +0200
Subject: [PATCH] Three lines lemma on any strip from the Mathlib version on
 the unit strip (#38)

Not sure this actually needs to be merged, but Hadamard.lean is a
generalization of the Mathlib version of the three lines lemma to any
vertical strip in the complex plane.
---
 BonnAnalysis/Hadamard.lean              | 251 ++++++++++++++++++++++++
 blueprint/src/chapter/interpolation.tex |   5 +-
 2 files changed, 254 insertions(+), 2 deletions(-)
 create mode 100644 BonnAnalysis/Hadamard.lean

diff --git a/BonnAnalysis/Hadamard.lean b/BonnAnalysis/Hadamard.lean
new file mode 100644
index 0000000..fd20e79
--- /dev/null
+++ b/BonnAnalysis/Hadamard.lean
@@ -0,0 +1,251 @@
+import Mathlib.Analysis.Complex.Hadamard
+
+open Set Filter Function Complex Topology HadamardThreeLines
+
+lemma Real.sub_ne_zero_of_lt {a b : ℝ} (hab: a < b) : b - a ≠ 0 := by apply ne_of_gt; simp[hab]
+
+namespace Complex
+
+lemma DiffContOnCl.id {s: Set ℂ} : DiffContOnCl ℂ id s :=
+  DiffContOnCl.mk differentiableOn_id continuousOn_id
+
+namespace HadamardThreeLines
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
+
+/--An auxiliary function to prove the general statement of Hadamard's three lines theorem.-/
+def scale (f: ℂ → E) (l u : ℝ) : ℂ → E := fun z ↦ f (l + z • (u-l))
+
+/--The transformation on ℂ that is used for `scale` maps the strip ``re ⁻¹' (l,u)``
+  to the strip ``re ⁻¹' (0,1)``-/
+lemma scale_mapsto_dom {l u : ℝ} (hul: l<u) {z: ℂ} (hz: z ∈ verticalStrip 0 1) :
+    l + z * (u-l)  ∈ verticalStrip l u := by {
+  simp only [verticalStrip, mem_preimage, mem_Ioo] at hz
+  simp only [verticalStrip, mem_preimage, add_re, ofReal_re, mul_re, sub_re, sub_im, ofReal_im,
+    sub_self, mul_zero, sub_zero, mem_Ioo, lt_add_iff_pos_right]
+  obtain ⟨hz₁, hz₂⟩ := hz
+  simp only [hz₁, mul_pos_iff_of_pos_left, sub_pos, hul, true_and]
+  rw [add_comm, ← sub_lt_sub_iff_right l, add_sub_assoc, sub_self, add_zero]
+  nth_rewrite 2 [← one_mul (u-l)]
+  gcongr
+  simp only [sub_pos]
+  exact hul
+}
+
+/--The transformation on ℂ that is used for `scale` maps the closed strip ``re ⁻¹' [l,u]``
+  to the closed strip ``re ⁻¹' [0,1]``-/
+lemma scale_mapsto_dom_cl {l u : ℝ} (hul: l<u) {z: ℂ} (hz: z ∈ verticalClosedStrip 0 1) :
+    l + z * (u-l)  ∈ verticalClosedStrip l u := by {
+  simp only [verticalClosedStrip, mem_preimage, add_re, ofReal_re, mul_re, sub_re, sub_im,
+    ofReal_im, sub_self, mul_zero, sub_zero, mem_Icc, le_add_iff_nonneg_right]
+  simp only [verticalClosedStrip, mem_preimage, mem_Icc] at hz
+  obtain ⟨hz₁, hz₂⟩ := hz
+  simp only [sub_pos, hul, mul_nonneg_iff_of_pos_right, hz₁, true_and]
+  rw [add_comm, ← sub_le_sub_iff_right l, add_sub_assoc, sub_self, add_zero]
+  nth_rewrite 2 [← one_mul (u-l)]
+  gcongr
+  simp only [sub_nonneg]
+  exact le_of_lt hul
+}
+
+/--If z is on the closed strip `re ⁻¹' [l,u]`, then `(z-l)/(u-l)` is on the closed strip
+  `re ⁻¹' [0,1]`.-/
+lemma scale_mem_strip {z : ℂ} {l u : ℝ} (hul: l < u) (hz: z ∈ verticalClosedStrip l u) :
+    z/(u - l) - l/(u-l) ∈ verticalClosedStrip 0 1 := by{
+  simp only [verticalClosedStrip, Complex.div_re, mem_preimage, sub_re, mem_Icc,
+    sub_nonneg, tsub_le_iff_right, ofReal_re, ofReal_im, sub_im, sub_self, mul_zero, zero_div,
+    add_zero]
+  simp only [verticalClosedStrip] at hz
+  norm_cast
+  simp_rw [Complex.normSq_ofReal, mul_div_assoc, div_mul_eq_div_div_swap,
+    div_self (Real.sub_ne_zero_of_lt hul), ← div_eq_mul_one_div]
+  constructor
+  · gcongr
+    · apply le_of_lt; simp [hul]
+    · exact hz.1
+  · rw [← sub_le_sub_iff_right (l / (u-l)), add_sub_assoc, sub_self, add_zero, div_sub_div_same,
+      div_le_one (by simp[hul]), sub_le_sub_iff_right l]
+    exact hz.2
+}
+
+/-- The function `scale f l u` is `diffContOnCl`. -/
+lemma scale_diffContOnCl {f: ℂ → E} {l u : ℝ} (hul: l < u)
+    (hd : DiffContOnCl ℂ f (verticalStrip l u)) : DiffContOnCl ℂ (scale f l u)
+    (verticalStrip 0 1) := by{
+  unfold scale
+  apply DiffContOnCl.comp (s:= verticalStrip l u) hd
+  · apply DiffContOnCl.const_add
+    apply DiffContOnCl.smul_const
+    apply DiffContOnCl.id
+  · rw [MapsTo]
+    intro z hz
+    exact scale_mapsto_dom hul hz
+}
+
+/-- The norm of the function `scale f l u` is bounded above on the closed strip `re⁻¹' [0, 1]`-/
+lemma scale_bddAbove {f: ℂ → E} {l u : ℝ} (hul: l< u)
+    (hB : BddAbove ((norm ∘ f) '' (verticalClosedStrip l u))) :
+    BddAbove ((norm ∘ (scale f l u)) '' (verticalClosedStrip 0 1)) := by{
+  obtain ⟨R, hR⟩ := bddAbove_def.mp hB
+  rw [bddAbove_def]
+  use R
+  intro r hr
+  obtain ⟨w, hw₁, hw₂, _⟩ := hr
+  simp only [comp_apply, scale, smul_eq_mul]
+  have : ‖f (↑l + w * (↑u - ↑l))‖ ∈ norm ∘ f '' verticalClosedStrip l u := by{
+    simp only [comp_apply, mem_image]
+    use ↑l + w * (↑u - ↑l)
+    simp only [and_true]
+    exact scale_mapsto_dom_cl hul hw₁
+  }
+  exact hR ‖f (↑l + w * (↑u - ↑l))‖ this
+}
+
+/--A bound to the norm of `f` on the line `z.re=l` induces a bound to the norm of
+  `scale f l u z` on the line `z.re=0`. -/
+lemma scale_bound_left {f: ℂ → E} {l u a : ℝ} (ha : ∀ z ∈ re ⁻¹' {l}, ‖f z‖ ≤ a) :
+    ∀ z ∈ re ⁻¹' {0}, ‖scale f l u z‖ ≤ a := by{
+  simp only [mem_preimage, mem_singleton_iff, scale, smul_eq_mul]
+  intro z hz
+  exact ha (↑l + z * (↑u - ↑l)) (by simp[hz])
+}
+
+/--A bound to the norm of `f` on the line `z.re=u` induces a bound to the norm of `scale f l u z`
+  on the line `z.re=1`. -/
+lemma scale_bound_right {f: ℂ → E} {l u b : ℝ} (hb : ∀ z ∈ re ⁻¹' {u}, ‖f z‖ ≤ b) :
+    ∀ z ∈ re ⁻¹' {1}, ‖scale f l u z‖ ≤ b := by{
+  simp only [scale, mem_preimage, mem_singleton_iff, smul_eq_mul]
+  intro z hz
+  exact hb (↑l + z * (↑u - ↑l)) (by simp[hz])
+}
+
+/--A technical lemma needed in the proof-/
+lemma fun_arg_eq {l u: ℝ} (hul: l < u) (z: ℂ) :
+    (↑l + (z / (↑u - ↑l) - ↑l / (↑u - ↑l)) * (↑u - ↑l)) = z := by{
+  rw [sub_mul, div_mul_comm, div_self (by norm_cast; exact Real.sub_ne_zero_of_lt hul ),
+    div_mul_comm, div_self (by norm_cast; exact Real.sub_ne_zero_of_lt hul)]
+  simp
+}
+
+/--Another technical lemma needed in the proof-/
+lemma bound_exp_eq {l u: ℝ} (hul : l < u) (z:ℂ) :
+    (z / (↑u - ↑l)).re - ((l:ℂ) / (↑u - ↑l)).re = (z.re - l) / (u - l) := by{
+  norm_cast
+  rw [Complex.div_re, Complex.normSq_ofReal, Complex.ofReal_re, Complex.ofReal_im, mul_div_assoc,
+    div_mul_eq_div_div_swap, div_self (by norm_cast; exact Real.sub_ne_zero_of_lt hul),
+    ← div_eq_mul_one_div]
+  simp only [mul_zero, zero_div, add_zero]
+  rw [← div_sub_div_same]
+}
+
+/--The correct generalization of `interpStrip` to produce bounds in the general case-/
+noncomputable def interpStrip' (f: ℂ → E) (l u : ℝ) (z : ℂ) : ℂ :=
+  if (sSupNormIm f l) = 0 ∨ (sSupNormIm f u) = 0
+    then 0
+    else (sSupNormIm f l) ^ (1-((z-l)/(u-l))) * (sSupNormIm f u) ^ ((z-l)/(u-l))
+
+
+/--The supremum of the norm of `scale f l u` on the line `z.re = 0` is the same as the supremum
+  of `f` on the line `z.re=l`.-/
+lemma sSupNormIm_scale_left (f: ℂ → E) {l u : ℝ} (hul: l < u) :
+    sSupNormIm (scale f l u) 0 = sSupNormIm f l := by{
+  simp_rw [sSupNormIm, image_comp]
+  have : scale f l u '' (re ⁻¹' {0}) = f '' (re ⁻¹' {l}) := by{
+    ext e
+    simp only [scale, smul_eq_mul, mem_image, mem_preimage, mem_singleton_iff]
+    constructor
+    · intro h
+      obtain ⟨z, hz₁, hz₂⟩ := h
+      use ↑l + z * (↑u - ↑l)
+      simp [hz₁, hz₂]
+    · intro h
+      obtain ⟨z, hz₁, hz₂⟩ := h
+      use ((z-l)/(u-l))
+      constructor
+      · norm_cast
+        rw [Complex.div_re, Complex.normSq_ofReal, Complex.ofReal_re]
+        simp[hz₁]
+      · rw [div_mul_comm, div_self (by norm_cast; exact Real.sub_ne_zero_of_lt hul)]
+        simp [hz₂]
+  }
+  rw [this]
+}
+
+/--The supremum of the norm of `scale f l u` on the line `z.re = 1` is the same as
+  the supremum of `f` on the line `z.re=u`.-/
+lemma sSupNormIm_scale_right (f: ℂ → E) {l u : ℝ} (hul: l < u) :
+    sSupNormIm (scale f l u) 1 = sSupNormIm f u := by{
+  simp_rw [sSupNormIm, image_comp]
+  have : scale f l u '' (re ⁻¹' {1}) = f '' (re ⁻¹' {u}) := by{
+    ext e
+    simp only [scale, smul_eq_mul, mem_image, mem_preimage, mem_singleton_iff]
+    constructor
+    · intro h
+      obtain ⟨z, hz₁, hz₂⟩ := h
+      use ↑l + z * (↑u - ↑l)
+      simp only [add_re, ofReal_re, mul_re, hz₁, sub_re, one_mul, sub_im, ofReal_im, sub_self,
+        mul_zero, sub_zero, add_sub_cancel, hz₂, and_self]
+    · intro h
+      obtain ⟨z, hz₁, hz₂⟩ := h
+      use ((z-l)/(u-l))
+      constructor
+      · norm_cast
+        rw [Complex.div_re, Complex.normSq_ofReal, Complex.ofReal_re]
+        simp only [sub_re, hz₁, ofReal_re, sub_im, ofReal_im, sub_zero, ofReal_sub, sub_self,
+          mul_zero, zero_div, add_zero]
+        rw [div_mul_eq_div_div_swap, mul_div_assoc,
+          div_self (by norm_cast; exact Real.sub_ne_zero_of_lt hul),
+          mul_one, div_self (by norm_cast; exact Real.sub_ne_zero_of_lt hul)]
+      · rw [div_mul_comm, div_self (by norm_cast; exact Real.sub_ne_zero_of_lt hul)]
+        simp only [one_mul, add_sub_cancel, hz₂]
+  }
+  rw [this]
+}
+
+/--A technical lemma relating the bounds given by the three lines lemma on a general strip to
+the bounds for its scaled version on the strip ``re ⁻¹' [0,1]` to the bounds on a general strip.-/
+lemma interpStrip_scale (f: ℂ → E) {l u : ℝ} (hul: l < u) (z : ℂ)  : interpStrip (scale f l u)
+    ((z - ↑l) / (↑u - ↑l)) = interpStrip' f l u z := by{
+  simp only [interpStrip, interpStrip']
+  simp_rw [sSupNormIm_scale_left f hul, sSupNormIm_scale_right f hul]
+}
+
+/--
+**Hadamard three-line theorem**: If `f` is a bounded function, continuous on the
+closed strip `re ⁻¹' [a,b]` and differentiable on open strip `re ⁻¹' (a,b)`, then for
+`M(x) := sup ((norm ∘ f) '' (re ⁻¹' {x}))` we have that for all `z` in the closed strip
+`re ⁻¹' [a,b]` the inequality `‖f(z)‖ ≤ M(0) ^ (1 - ((z.re-a)/(b-a))) * M(1) ^ ((z.re-a)/(b-a))`
+holds. -/
+lemma norm_le_interpStrip_of_mem {l u : ℝ} (hul: l < u) {f : ℂ → E} {z : ℂ}
+    (hz : z ∈ verticalClosedStrip l u) (hd : DiffContOnCl ℂ f (verticalStrip l u))
+    (hB : BddAbove ((norm ∘ f) '' (verticalClosedStrip l u))) :
+    ‖f z‖ ≤ ‖interpStrip' f l u z‖ := by{
+  have hgoal := norm_le_interpStrip_of_mem_verticalClosedStrip (scale f l u)
+    (scale_mem_strip hul hz) (scale_diffContOnCl hul hd) (scale_bddAbove hul hB)
+  simp only [scale, smul_eq_mul, norm_eq_abs] at hgoal
+  rw [fun_arg_eq hul, div_sub_div_same, interpStrip_scale f hul z] at hgoal
+  exact hgoal
+}
+
+
+/-- **Hadamard three-line theorem** (Variant in simpler terms): Let `f` be a
+bounded function, continuous on the closed strip `re ⁻¹' [l,u]` and differentiable on open strip
+`re ⁻¹' (l,u)`. If, for all `z.re = l`, `‖f z‖ ≤ a` for some `a ∈ ℝ` and, similarly, for all
+`z.re = u`, `‖f z‖ ≤ b` for some `b ∈ ℝ` then for all `z` in the closed strip
+`re ⁻¹' [l,u]` the inequality `‖f(z)‖ ≤ a ^ (1 - (z.re-l)/(u-l)) * b ^ ((z.re-l)/(u-l))` holds. -/
+lemma norm_le_interp_of_mem' {f : ℂ → E} {z : ℂ} {a b l u : ℝ} (hul: l < u)
+    (hz : z ∈ verticalClosedStrip l u) (hd : DiffContOnCl ℂ f (verticalStrip l u))
+    (hB : BddAbove ((norm ∘ f) '' (verticalClosedStrip l u)))
+    (ha : ∀ z ∈ re ⁻¹' {l}, ‖f z‖ ≤ a) (hb : ∀ z ∈ re ⁻¹' {u}, ‖f z‖ ≤ b) :
+    ‖f z‖ ≤ a ^ (1 - (z.re-l)/(u-l)) * b ^ ((z.re-l)/(u-l)) := by{
+
+  have hgoal := norm_le_interp_of_mem_verticalClosedStrip' (scale f l u)
+    (scale_mem_strip hul hz) (scale_diffContOnCl hul hd) (scale_bddAbove hul hB)
+    (scale_bound_left ha) (scale_bound_right hb)
+  simp only [scale, smul_eq_mul, sub_re] at hgoal
+  rw [fun_arg_eq hul, bound_exp_eq hul] at hgoal
+  exact hgoal
+}
+
+end HadamardThreeLines
+end Complex
diff --git a/blueprint/src/chapter/interpolation.tex b/blueprint/src/chapter/interpolation.tex
index c3925e6..764e48c 100644
--- a/blueprint/src/chapter/interpolation.tex
+++ b/blueprint/src/chapter/interpolation.tex
@@ -29,7 +29,7 @@ \section{Rietz-Thorin's Interpolation Theorem}
     \label{lem:threelines}
     %\uses{}
     \lean{DiffContOnCl.norm_le_pow_mul_pow}
-    %\leanok
+    \leanok
     Let $S$ be the strip $S:=\{z \in \C \ | \ 0 < \mathrm{Re} \, z < 1 \}$. Let $f: \overline{S} \to \C$
     be a function that is holomorphic on $S$ and continuous and bounded on $\overline{S}$.\\
     Assume $M_0, M_1$ are positive real numbers such that for all values of $y$ in $\R$, we have
@@ -40,7 +40,8 @@ \section{Rietz-Thorin's Interpolation Theorem}
 \end{lemma}
 \begin{proof}
     % \uses{} % Put any results used in the proof but not in the statement here
-    % \leanok % uncomment if the lemma has been proven
+    % Not sure what to put here exactly
+    \leanok % uncomment if the lemma has been proven
     ~\\
     If $|\phi|$ is constant, everything holds trivially by setting $M_0$ and $M_1$ to be the value of $|\phi|$ at a point. Assume $|\phi|$ non-constant.\\
     \begin{itemize}