Update holomorphic_zero.lean

This commit is contained in:
Stefan Kebekus 2024-08-19 15:59:14 +02:00
parent a910bd6988
commit 5e7dd06d4c
1 changed files with 130 additions and 16 deletions

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@ -288,18 +288,21 @@ theorem AnalyticOn.order_eq_nat_iff
(hf : AnalyticOn f U)
(hz₀ : z₀ ∈ U)
(n : ) :
(hf z₀ hz₀).order = ↑n ∃ (g : ), AnalyticOn g U ∧ g z₀ ≠ 0 ∧ ∀ z ∈ U, f z = (z - z₀) ^ n • g z := by
(hf z₀ hz₀).order = ↑n ∃ (g : ), AnalyticOn g U ∧ g z₀ ≠ 0 ∧ ∀ z, f z = (z - z₀) ^ n • g z := by
constructor
-- Direction →
intro hn
obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ hz₀) n).1 hn
let g : := fun z ↦ if hz : z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n
-- Define a candidate function; this is (f z) / (z - z₀) ^ n with the
-- removable singularity removed
let g : := fun z ↦ if z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n
have t₀ : ∀ᶠ (z : ) in nhds z₀, g z = gloc z := by
-- Describe g near z₀
have g_near_z₀ : ∀ᶠ (z : ) in nhds z₀, g z = gloc z := by
rw [eventually_nhds_iff]
rw [eventually_nhds_iff] at h₃gloc
obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₃gloc
obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 h₃gloc
use t
constructor
· intro y h₁y
@ -315,19 +318,18 @@ theorem AnalyticOn.order_eq_nat_iff
· assumption
· assumption
have t₁ {z₁ : } : z₁ ≠ z₀ → ∀ᶠ (z : ) in nhds z₁, g z = f z / (z - z₀) ^ n := by
-- Describe g near points z₁ that are different from z₀
have g_near_z₁ {z₁ : } : z₁ ≠ z₀ → ∀ᶠ (z : ) in nhds z₁, g z = f z / (z - z₀) ^ n := by
intro hz₁
rw [eventually_nhds_iff]
use {z₀}ᶜ
constructor
· intro y hy
simp at hy
dsimp [g]
simp [hy]
· constructor
· exact isOpen_compl_singleton
· tauto
simp [g, hy]
· exact ⟨isOpen_compl_singleton, hz₁⟩
-- Use g and show that it has all required properties
use g
constructor
· -- AnalyticOn g U
@ -335,18 +337,130 @@ theorem AnalyticOn.order_eq_nat_iff
by_cases h₂z : z = z₀
· rw [h₂z]
apply AnalyticAt.congr h₁gloc
exact Filter.EventuallyEq.symm t
· simp_rw [eq_comm] at t
apply AnalyticAt.congr _ (t₁ h₂z)
exact Filter.EventuallyEq.symm g_near_z
· simp_rw [eq_comm] at g_near_z
apply AnalyticAt.congr _ (g_near_z₁ h₂z)
apply AnalyticAt.div
exact hf z h₁z
apply AnalyticAt.pow
apply AnalyticAt.sub
apply analyticAt_id
exact analyticAt_const
apply analyticAt_const
simp
rw [sub_eq_zero]
tauto
· constructor
· simp [g]; tauto
· intro z
by_cases h₂z : z = z₀
· rw [h₂z, g_near_z₀.self_of_nhds]
exact h₃gloc.self_of_nhds
· rw [(g_near_z₁ h₂z).self_of_nhds]
simp [h₂z]
rw [div_eq_mul_inv, mul_comm, mul_assoc, inv_mul_cancel]
simp; norm_num
rw [sub_eq_zero]
tauto
-- direction ←
intro h
obtain ⟨g, h₁g, h₂g, h₃g⟩ := h
rw [AnalyticAt.order_eq_nat_iff]
use g
exact ⟨h₁g z₀ hz₀, ⟨h₂g, Filter.eventually_of_forall h₃g⟩⟩
theorem AnalyticOn.order_eq_nat_iff'
{f : }
{U : Set }
{A : Finset U}
(hf : AnalyticOn f U)
(n : A → ) :
∀ a : A, (hf a (Subtype.coe_prop a.val)).order = n a → ∃ (g : ), AnalyticOn g U ∧ (∀ a, g a ≠ 0) ∧ ∀ z, f z = (∏ a, (z - a) ^ (n a)) • g z := by
apply Finset.induction
let a : A := by sorry
let b : := by sorry
let u : U := by sorry
let X := n a
have : a = (3 : ) := by sorry
have : b ∈ ↑A := by sorry
have : ↑a ∈ U := by exact Subtype.coe_prop a.val
let Y := ∀ a : A, (hf a (Subtype.coe_prop a.val)).order = n a
--∀ a : A, (hf (ha a)).order = ↑(n a) →
intro hn
obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ hz₀) n).1 hn
-- Define a candidate function
let g : := fun z ↦ if z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n
-- Describe g near z₀
have g_near_z₀ : ∀ᶠ (z : ) in nhds z₀, g z = gloc z := by
rw [eventually_nhds_iff]
obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 h₃gloc
use t
constructor
· intro y h₁y
by_cases h₂y : y = z₀
· dsimp [g]; simp [h₂y]
· dsimp [g]; simp [h₂y]
rw [div_eq_iff_mul_eq, eq_comm, mul_comm]
exact h₁t y h₁y
norm_num
rw [sub_eq_zero]
tauto
· constructor
· assumption
· assumption
-- Describe g near points z₁ different from z₀
have g_near_z₁ {z₁ : } : z₁ ≠ z₀ → ∀ᶠ (z : ) in nhds z₁, g z = f z / (z - z₀) ^ n := by
intro hz₁
rw [eventually_nhds_iff]
use {z₀}ᶜ
constructor
· intro y hy
simp at hy
simp [g, hy]
· exact ⟨isOpen_compl_singleton, hz₁⟩
-- Use g and show that it has all required properties
use g
constructor
· -- AnalyticOn g U
intro z h₁z
by_cases h₂z : z = z₀
· rw [h₂z]
apply AnalyticAt.congr h₁gloc
exact Filter.EventuallyEq.symm g_near_z₀
· simp_rw [eq_comm] at g_near_z₁
apply AnalyticAt.congr _ (g_near_z₁ h₂z)
apply AnalyticAt.div
exact hf z h₁z
apply AnalyticAt.pow
apply AnalyticAt.sub
apply analyticAt_id
apply analyticAt_const
simp
rw [sub_eq_zero]
tauto
· constructor
· simp [g]; tauto
· intro z
by_cases h₂z : z = z₀
· rw [h₂z, g_near_z₀.self_of_nhds]
exact h₃gloc.self_of_nhds
· rw [(g_near_z₁ h₂z).self_of_nhds]
simp [h₂z]
rw [div_eq_mul_inv, mul_comm, mul_assoc, inv_mul_cancel]
simp; norm_num
rw [sub_eq_zero]
tauto
noncomputable def zeroDivisorDegree