From eaa424962fe1e0138bf0431d5eb043c526b1b2cc Mon Sep 17 00:00:00 2001
From: Antoine-dSG <adsg96@hku.hk>
Date: Mon, 29 Jul 2024 14:56:33 +0800
Subject: [PATCH] finished continuant proofs

---
 FriezePatterns/Chapter1/section1.lean | 95 ++++++++++++++-------------
 1 file changed, 48 insertions(+), 47 deletions(-)

diff --git a/FriezePatterns/Chapter1/section1.lean b/FriezePatterns/Chapter1/section1.lean
index a02a12c..8959a2c 100644
--- a/FriezePatterns/Chapter1/section1.lean
+++ b/FriezePatterns/Chapter1/section1.lean
@@ -9,73 +9,74 @@ class pattern (F : Type*) [Field F] (f : ℕ × ℕ → F) : Prop where
   topBordOnes : ∀ m, f (0,m) =1
   diamond : ∀ m, ∀ i,  f (i+1,m) * f (i+1,m+1) -1= f (i+2,m)*f (i,m+1)
 
--- In the following class, dom stands for domestic. Roughly, domesticated pattern ⊆ tame pattern ⊆ pattern
-class domPattern (F : Type*) [Field F] (f : ℕ × ℕ → F) extends pattern F f where
+-- In the following class, nz stands for nowhere-zero pattern. we have nowhere-zero patterns ⊆ tame pattern ⊆ pattern
+class nzPattern (F : Type*) [Field F] (f : ℕ × ℕ → F) extends pattern F f where
   non_zero : ∀ i, ∀ m, f (i,m) ≠ 0
 
-class tamePattern (F : Type*) [Field F] (f : ℕ × ℕ → F) extends domPattern F f where
+class tamePattern (F : Type*) [Field F] (f : ℕ × ℕ → F) extends nzPattern F f where
   -- tame condition ?
 
--- lemma scaledContinuant (f : ℕ×ℕ → ℚ) [inftyFrieze f] (i : ℕ) : ∀m, f (i+2,m) * f (i,m+1) = (f (1,m+i+1)*f (i+1,m) - f (i,m))* f (i,m+1) := by
-lemma scaledContinuant (F : Type*) [Field F] (f : ℕ×ℕ → F) [domPattern F f] (i : ℕ) : ∀m, f (i+2,m) * f (i,m+1) = (f (1,m+i+1)*f (i+1,m) - f (i,m))* f (i,m+1) := by
+-- The following proposition should be extended to include tame frieze patterns
+-- Continuant property of nowhere-zero infinite patterns
+lemma inftyContinuant (F : Type*) [Field F] (f : ℕ×ℕ → F) [nzPattern F f] (i : ℕ) : ∀m, f (i+2,m) = f (1,m+i+1)*f (i+1,m) - f (i,m) := by
   induction i with
   | zero =>
   intro m
-  have h₀ : f (0,m+1) = 1 := by exact pattern.topBordOnes (m+1)
-  have h₁ : f (0,m) = 1 := by exact pattern.topBordOnes m
-  rw [h₀,h₁]
   simp
+  have h₀ : f (0,m) = 1 ∧ f (0,m+1) = 1 := by exact ⟨pattern.topBordOnes m, pattern.topBordOnes (m + 1)⟩
+  rw [h₀.1, ← zero_add 2]
   nth_rw 1 [← zero_add 1]
   nth_rw 3 [← zero_add 1]
-  rw [mul_comm, pattern.diamond m 0, h₀]
+  rw[mul_comm, pattern.diamond m 0, h₀.2]
   simp
   | succ n ih =>
   intro m
-  have h₂ : f (n+2,m+1) * f (n,m+1+1) = (f (1,m+1+n+1)*f (n+1,m+1) - f (n,m+1)) * f (n,m+2) := by exact ih (m + 1)
-  -- have h₃ : f (n,m+2) ≠ 0 :=  by exact friezeNonZero f n (m+2)
-  have h₃ : f (n,m+2) ≠ 0 :=  by exact domPattern.non_zero n (m + 2)
-  have h₄ : f (n+2,m+1) = f (n+2,m+1) * f (n,m+2) * (f (n,m+2))⁻¹ := by rw [mul_inv_cancel_right₀ h₃ (f (n+2,m+1))]
-  -- have h₅ : f (n,m+2) * (f (n,m+2))⁻¹ = 1 := by exact Rat.mul_inv_cancel (f (n, m + 2)) h₃
-  have h₅ : f (n,m+2) * (f (n,m+2))⁻¹ = 1 := by exact CommGroupWithZero.mul_inv_cancel (f (n, m + 2)) h₃
-  calc f (n + 1 + 2, m) * f (n + 1, m + 1) = f (n+3,m)* f (n+1, m+1) := by simp
-  _= f (n+2, m) * f (n+2,m+1) - 1 := by exact Eq.symm (pattern.diamond m (n + 1))
-  _= f (n+2, m) * (f (n+2,m+1) * f (n,m+2) * (f (n,m+2))⁻¹) -1 := by exact congrFun (congrArg HSub.hSub (congrArg (HMul.hMul (f (n + 2, m))) h₄)) 1  -- we need to prove that a = f (n,m+1+1) has an inverse
-  _= f (n+2, m) * ((f (n+2,m+1) * f (n,m+1+1)) * (f (n,m+2))⁻¹) -1 := by rw [mul_assoc]
-  _= f (n+2, m) * ((f (1,m+1+n+1)*f (n+1,m+1) - f (n,m+1)) * f (n,m+2) * (f (n,m+2))⁻¹) -1 := by rw [h₂]
-  _= f (n+2, m) * (f (1,m+1+n+1)*f (n+1,m+1) - f (n,m+1)) * (f (n,m+2) * (f (n,m+2))⁻¹) -1 := by ring
-  _= f (n+2, m) * ((f (1,m+1+n+1)*f (n+1,m+1) - f (n,m+1))) * 1 -1 := by rw [h₅]
-  _= f (n+2, m) * f (1,m+(n+1)+1)*f (n+1,m+1) - f (n+2, m) *f (n,m+1) -1 := by ring_nf
-  _= f (n+2, m) * f (1,m+(n+1)+1)*f (n+1,m+1) - ( f (n+1, m)*f (n+1,m+1) -1 ) -1 := by rw [Eq.symm (pattern.diamond m n)]
-  _= (f (n+2, m) * f (1,m+(n+1)+1) - f (n+1, m))*f (n+1,m+1) := by ring
-  _= (f (1,m+(n+1)+1)*f (n+2, m) - f (n+1, m))*f (n+1,m+1) := by rw [mul_comm (f (n+2, m))]
-  _= (f (1,m+(n+1)+1)*f (n+1+1, m) - f (n+1, m))*f (n+1,m+1) := by ring
-
-
--- lemma continuant (f : ℕ×ℕ → ℚ) [inftyFrieze f] (i : ℕ) : ∀m, f (i+2,m) = f (1,m+i+1)*f (i+1,m) - f (i,m) := by
-lemma inftyContinuant (F : Type*) [Field F] (f : ℕ×ℕ → F) [domPattern F f] (i : ℕ) : ∀m, f (i+2,m) = f (1,m+i+1)*f (i+1,m) - f (i,m) := by
-  intro m
-  have h : f (i+2,m) * f (i,m+1) = (f (1,m+i+1)*f (i+1,m) - f (i,m))* f (i,m+1) := by exact scaledContinuant F f i m
-  --have h₁ : f (i,m+1) ≠ 0 := by exact friezeNonZero f  i (m + 1)
-  have h₁ : f (i,m+1) ≠ 0 := by exact domPattern.non_zero i (m + 1)
-  calc f (i+2,m) = f (i+2,m) * f (i,m+1) * (f (i,m+1))⁻¹ := by rw [mul_inv_cancel_right₀ h₁ (f (i+2,m))]
-  _= (f (1,m+i+1)*f (i+1,m) - f (i,m))* f (i,m+1) * (f (i,m+1))⁻¹ := by rw [h]
-  _= (f (1,m+i+1)*f (i+1,m) - f (i,m))* (f (i,m+1) * (f (i,m+1))⁻¹) := by ring
-  --_= (f (1,m+i+1)*f (i+1,m) - f (i,m))*1 := by rw [Rat.mul_inv_cancel (f (i,m+1)) h₁]
-  _= (f (1,m+i+1)*f (i+1,m) - f (i,m))*1 := by rw [CommGroupWithZero.mul_inv_cancel (f (i,m+1)) h₁]
-  _= f (1,m+i+1)*f (i+1,m) - f (i,m) := by simp
-
+  have h : f (n + 1, m+1) ≠ 0 := by exact nzPattern.non_zero (n + 1) (m+1)
+  calc f (n + 1 + 2, m) = f (n + 1 + 2, m) * f (n + 1, m+1) * (f (n + 1, m+1))⁻¹ := by rw [mul_inv_cancel_right₀ h (f (n + 1 + 2, m))]
+  _= (f (n+2,m)*f (n+2,m+1) - 1) * (f (n + 1, m+1))⁻¹ := by rw [pattern.diamond m (n+1)]
+  _= (f (n+2,m)*(f (1, m +1 + n + 1) * f (n + 1, m+1) - f (n, m+1)) - 1) * (f (n + 1, m+1))⁻¹ := by rw [ih (m+1)]
+  _= (f (1, m +1 + n + 1)*f (n+2,m)* f (n + 1, m+1) - f (n+2,m)*f (n, m+1) - 1) * (f (n + 1, m+1))⁻¹ := by ring
+  _= (f (1, m +1 + n + 1)*f (n+2,m)* f (n + 1, m+1) - (f (n+1,m)*f (n+1,m+1) - 1) - 1) * (f (n + 1, m+1))⁻¹ := by rw [pattern.diamond]
+  _= f (1, m +1 + n + 1)*f (n+2,m)*f (n + 1, m+1)* (f (n + 1, m+1))⁻¹ - f (n+1,m)*f (n + 1, m+1)* (f (n + 1, m+1))⁻¹ := by ring
+  _= f (1, m +1 + n + 1)*f (n+2,m) - f (n+1,m) := by rw [mul_inv_cancel_right₀ h (f (1, m +1 + n + 1)*f (n+2,m)), mul_inv_cancel_right₀ h (f (n+1,m))]
+  _= f (1, m + (n + 1) + 1) * f (n + 1 + 1, m) - f (n + 1, m) := by ring
 
+-- It is not clear if the following definition is useful
 -- Bounded field-valued patterns
 class closedPattern (F : Type*) [Field F] (f : ℕ × ℕ → F) extends pattern F f where
   bounded: ∃ (n : ℕ), ∀m, f (n+1,m) = 1
 
 class pattern_n (F : Type*) [Field F] (f : ℕ × ℕ → F) (n : ℕ) : Prop where
   botBordOnes_n : ∀ m, f (n, m) = 1
+  topBordOnes : ∀ m, f (0,m) =1
+  diamond : ∀ m, ∀ i, f (i+1,m) * f (i+1,m+1)-1 = f (i+2,m)*f (i,m+1)
 
-class domPattern_n (F : Type*) [Field F] (f : ℕ × ℕ → F) (n : ℕ) extends pattern_n F f n where
-  non_zero : ∀ i, ∀ m, i ≤ n ∧ f (i,m) ≠ 0
+class nzPattern_n (F : Type*) [Field F] (f : ℕ × ℕ → F) (n : ℕ) extends pattern_n F f n where
+  non_zero : ∀ i, ∀ m, i ≤ n → f (i,m) ≠ 0
 
-lemma continuantFrieze (F : Type*) [Field F] (f : ℕ×ℕ → F) (n : ℕ) [domPattern_n F f n] (i : ℕ) (h: i ≤ n) : ∀m, f (i+2,m) = f (1,m+i+1)*f (i+1,m) - f (i,m) := by
+-- Continuant property for patterns of finite width
+lemma finiteContinuant (F : Type*) [Field F] (f : ℕ×ℕ → F) (n i: ℕ) [nzPattern_n F f n] (hi: i ≤ n) : ∀m, f (i+2,m) = f (1,m+i+1)*f (i+1,m) - f (i,m) := by
  induction i with
- | zero => sorry
- | succ k ih => sorry
+ | zero =>
+  intro m
+  simp
+  have h₀ : f (0,m) = 1 ∧ f (0,m+1) = 1 := by exact ⟨pattern_n.topBordOnes n m, pattern_n.topBordOnes n (m + 1)⟩
+  rw [h₀.1, ← zero_add 2]
+  nth_rw 1 [← zero_add 1]
+  nth_rw 3 [← zero_add 1]
+  rw[mul_comm, pattern_n.diamond n m 0, h₀.2]
+  simp
+ | succ k ih =>
+ intro m
+ have h₁ : f (k+2,m+1) = f (1, m +1 + k + 1) * f (k + 1, m+1) - f (k, m+1) := by
+  rw [ih]
+  linarith
+ have h₂ : f (k + 1, m+1) ≠ 0 := by exact nzPattern_n.non_zero (k + 1) (m + 1) hi
+ calc f (k + 1 + 2, m) = f (k + 1 + 2, m) * f (k + 1, m+1) * (f (k + 1, m+1))⁻¹ := by rw [mul_inv_cancel_right₀ h₂ (f (k + 1 + 2, m))]
+  _= (f (k+2,m)*f (k+2,m+1) - 1) * (f (k + 1, m+1))⁻¹ := by rw [pattern_n.diamond n m (k+1)]
+  _= (f (k+2,m)*(f (1, m +1 + k + 1) * f (k + 1, m+1) - f (k, m+1)) - 1) * (f (k + 1, m+1))⁻¹ := by rw [h₁]
+  _= (f (1, m +1 + k + 1)*f (k+2,m)* f (k + 1, m+1) - f (k+2,m)*f (k, m+1) - 1) * (f (k + 1, m+1))⁻¹ := by ring
+  _= (f (1, m +1 + k + 1)*f (k+2,m)* f (k + 1, m+1) - (f (k+1,m)*f (k+1,m+1) - 1) - 1) * (f (k + 1, m+1))⁻¹ := by rw [pattern_n.diamond n m]
+  _= f (1, m +1 + k + 1)*f (k+2,m)*f (k + 1, m+1)* (f (k + 1, m+1))⁻¹ - f (k+1,m)*f (k + 1, m+1)* (f (k + 1, m+1))⁻¹ := by ring
+  _= f (1, m +1 + k + 1)*f (k+2,m) - f (k+1,m) := by rw [mul_inv_cancel_right₀ h₂ (f (1, m +1 + k + 1)*f (k+2,m)), mul_inv_cancel_right₀ h₂ (f (k+1,m))]
+  _= f (1, m + (k + 1) + 1) * f (k + 1 + 1, m) - f (k + 1, m) := by ring