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Update to latest version of mathlib

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This commit is contained in:
Stefan Kebekus
2024-09-30 14:12:33 +02:00
parent 1a8bde51eb
commit c8f4cf12ca
7 changed files with 79 additions and 72 deletions
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4
Nevanlinna/analyticAt.lean
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@@ -116,10 +116,10 @@ theorem AnalyticAt.supp_order_toNat
intro h₁f
simp [hf.order_eq_zero_iff.2 h₁f]
/-
theorem ContinuousLinearEquiv.analyticAt
(ℓ : ℂ ≃L[ℂ] ℂ) (z₀ : ℂ) : AnalyticAt ℂ (⇑ℓ) z₀ := ℓ.toContinuousLinearMap.analyticAt z₀
-/
theorem eventually_nhds_comp_composition
{f₁ f₂ ℓ : ℂ → ℂ}

66
Nevanlinna/analyticOn_zeroSet.lean
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@@ -4,17 +4,17 @@ import Mathlib.Analysis.Complex.Basic
import Nevanlinna.analyticAt
noncomputable def AnalyticOn.order
{f : ℂ → ℂ} {U : Set ℂ} (hf : AnalyticOn ℂ f U) : U → ℕ∞ := fun u ↦ (hf u u.2).order
noncomputable def AnalyticOnNhd.order
{f : ℂ → ℂ} {U : Set ℂ} (hf : AnalyticOnNhd ℂ f U) : U → ℕ∞ := fun u ↦ (hf u u.2).order
theorem AnalyticOn.order_eq_nat_iff
theorem AnalyticOnNhd.order_eq_nat_iff
{f : ℂ → ℂ}
{U : Set ℂ}
{z₀ : U}
(hf : AnalyticOn ℂ f U)
(hf : AnalyticOnNhd ℂ f U)
(n : ℕ) :
hf.order z₀ = ↑n ↔ ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ g z₀ ≠ 0 ∧ ∀ z, f z = (z - z₀) ^ n • g z := by
hf.order z₀ = ↑n ↔ ∃ (g : ℂ → ℂ), AnalyticOnNhd ℂ g U ∧ g z₀ ≠ 0 ∧ ∀ z, f z = (z - z₀) ^ n • g z := by
constructor
-- Direction →
@@ -91,31 +91,31 @@ theorem AnalyticOn.order_eq_nat_iff
-- direction ←
intro h
obtain ⟨g, h₁g, h₂g, h₃g⟩ := h
dsimp [AnalyticOn.order]
dsimp [AnalyticOnNhd.order]
rw [AnalyticAt.order_eq_nat_iff]
use g
exact ⟨h₁g z₀ z₀.2, ⟨h₂g, Filter.Eventually.of_forall h₃g⟩⟩
theorem AnalyticOn.support_of_order₁
theorem AnalyticOnNhd.support_of_order₁
{f : ℂ → ℂ}
{U : Set ℂ}
(hf : AnalyticOn ℂ f U) :
(hf : AnalyticOnNhd ℂ f U) :
Function.support hf.order = U.restrict f⁻¹' {0} := by
ext u
simp [AnalyticOn.order]
simp [AnalyticOnNhd.order]
rw [not_iff_comm, (hf u u.2).order_eq_zero_iff]
theorem AnalyticOn.eliminateZeros
theorem AnalyticOnNhd.eliminateZeros
{f : ℂ → ℂ}
{U : Set ℂ}
{A : Finset U}
(hf : AnalyticOn ℂ f U)
(hf : AnalyticOnNhd ℂ f U)
(n : U → ℕ) :
(∀ a ∈ A, hf.order a = n a) → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z := by
(∀ a ∈ A, hf.order a = n a) → ∃ (g : ℂ → ℂ), AnalyticOnNhd ℂ g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z := by
apply Finset.induction (α := U) (p := fun A ↦ (∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z)
apply Finset.induction (α := U) (p := fun A ↦ (∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ℂ → ℂ), AnalyticOnNhd ℂ g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z)
-- case empty
simp
@@ -146,7 +146,7 @@ theorem AnalyticOn.eliminateZeros
intro b _
apply AnalyticAt.pow
apply AnalyticAt.sub
apply analyticAt_id ℂ
apply analyticAt_id
exact analyticAt_const
have h₂φ : h₁φ.order = (0 : ℕ) := by
@@ -173,7 +173,7 @@ theorem AnalyticOn.eliminateZeros
rw [h₂φ]
simp
obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticOn.order_eq_nat_iff h₁g₀ (n b₀)).1 this
obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticOnNhd.order_eq_nat_iff h₁g₀ (n b₀)).1 this
use g₁
constructor
@@ -203,7 +203,7 @@ theorem XX
{f : ℂ → ℂ}
{U : Set ℂ}
(hU : IsPreconnected U)
(h₁f : AnalyticOn ℂ f U)
(h₁f : AnalyticOnNhd ℂ f U)
(h₂f : ∃ u ∈ U, f u ≠ 0) :
∀ (hu : u ∈ U), (h₁f u hu).order.toNat = (h₁f u hu).order := by
@@ -221,7 +221,7 @@ theorem discreteZeros
{f : ℂ → ℂ}
{U : Set ℂ}
(hU : IsPreconnected U)
(h₁f : AnalyticOn ℂ f U)
(h₁f : AnalyticOnNhd ℂ f U)
(h₂f : ∃ u ∈ U, f u ≠ 0) :
DiscreteTopology ((U.restrict f)⁻¹' {0}) := by
@@ -293,7 +293,7 @@ theorem finiteZeros
{U : Set ℂ}
(h₁U : IsPreconnected U)
(h₂U : IsCompact U)
(h₁f : AnalyticOn ℂ f U)
(h₁f : AnalyticOnNhd ℂ f U)
(h₂f : ∃ u ∈ U, f u ≠ 0) :
Set.Finite (U.restrict f⁻¹' {0}) := by
@@ -310,14 +310,14 @@ theorem finiteZeros
exact discreteZeros h₁U h₁f h₂f
theorem AnalyticOnCompact.eliminateZeros
theorem AnalyticOnNhdCompact.eliminateZeros
{f : ℂ → ℂ}
{U : Set ℂ}
(h₁U : IsPreconnected U)
(h₂U : IsCompact U)
(h₁f : AnalyticOn ℂ f U)
(h₁f : AnalyticOnNhd ℂ f U)
(h₂f : ∃ u ∈ U, f u ≠ 0) :
∃ (g : ℂ → ℂ) (A : Finset U), AnalyticOn ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (h₁f.order a).toNat) • g z := by
∃ (g : ℂ → ℂ) (A : Finset U), AnalyticOnNhd ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (h₁f.order a).toNat) • g z := by
let A := (finiteZeros h₁U h₂U h₁f h₂f).toFinset
@@ -325,13 +325,13 @@ theorem AnalyticOnCompact.eliminateZeros
have hn : ∀ a ∈ A, (h₁f a a.2).order = n a := by
intro a _
dsimp [n, AnalyticOn.order]
dsimp [n, AnalyticOnNhd.order]
rw [eq_comm]
apply XX h₁U
exact h₂f
obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOn.eliminateZeros (A := A) h₁f n hn
obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOnNhd.eliminateZeros (A := A) h₁f n hn
use g
use A
@@ -357,14 +357,14 @@ theorem AnalyticOnCompact.eliminateZeros
· exact inter
theorem AnalyticOnCompact.eliminateZeros₂
theorem AnalyticOnNhdCompact.eliminateZeros₂
{f : ℂ → ℂ}
{U : Set ℂ}
(h₁U : IsPreconnected U)
(h₂U : IsCompact U)
(h₁f : AnalyticOn ℂ f U)
(h₁f : AnalyticOnNhd ℂ f U)
(h₂f : ∃ u ∈ U, f u ≠ 0) :
∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ a ∈ (finiteZeros h₁U h₂U h₁f h₂f).toFinset, (z - a) ^ (h₁f.order a).toNat) • g z := by
∃ (g : ℂ → ℂ), AnalyticOnNhd ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ a ∈ (finiteZeros h₁U h₂U h₁f h₂f).toFinset, (z - a) ^ (h₁f.order a).toNat) • g z := by
let A := (finiteZeros h₁U h₂U h₁f h₂f).toFinset
@@ -372,12 +372,12 @@ theorem AnalyticOnCompact.eliminateZeros₂
have hn : ∀ a ∈ A, (h₁f a a.2).order = n a := by
intro a _
dsimp [n, AnalyticOn.order]
dsimp [n, AnalyticOnNhd.order]
rw [eq_comm]
apply XX h₁U
exact h₂f
obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOn.eliminateZeros (A := A) h₁f n hn
obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOnNhd.eliminateZeros (A := A) h₁f n hn
use g
have inter : ∀ (z : ℂ), f z = (∏ a ∈ A, (z - ↑a) ^ (h₁f (↑a) a.property).order.toNat) • g z := by
@@ -402,14 +402,14 @@ theorem AnalyticOnCompact.eliminateZeros₂
· exact h₃g
theorem AnalyticOnCompact.eliminateZeros₁
theorem AnalyticOnNhdCompact.eliminateZeros₁
{f : ℂ → ℂ}
{U : Set ℂ}
(h₁U : IsPreconnected U)
(h₂U : IsCompact U)
(h₁f : AnalyticOn ℂ f U)
(h₁f : AnalyticOnNhd ℂ f U)
(h₂f : ∃ u ∈ U, f u ≠ 0) :
∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ᶠ a, (z - a) ^ (h₁f.order a).toNat) • g z := by
∃ (g : ℂ → ℂ), AnalyticOnNhd ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ᶠ a, (z - a) ^ (h₁f.order a).toNat) • g z := by
let A := (finiteZeros h₁U h₂U h₁f h₂f).toFinset
@@ -417,12 +417,12 @@ theorem AnalyticOnCompact.eliminateZeros₁
have hn : ∀ a ∈ A, (h₁f a a.2).order = n a := by
intro a _
dsimp [n, AnalyticOn.order]
dsimp [n, AnalyticOnNhd.order]
rw [eq_comm]
apply XX h₁U
exact h₂f
obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOn.eliminateZeros (A := A) h₁f n hn
obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOnNhd.eliminateZeros (A := A) h₁f n hn
use g
have inter : ∀ (z : ℂ), f z = (∏ a ∈ A, (z - ↑a) ^ (h₁f (↑a) a.property).order.toNat) • g z := by

7
Nevanlinna/bilinear.lean
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@@ -28,7 +28,7 @@ orthonormal basis `v` as `∑ i, (v i) ⊗ₜ[ℝ] (v i)`.
-/
open InnerProductSpace
open TensorProduct
@@ -41,12 +41,11 @@ noncomputable def InnerProductSpace.canonicalCovariantTensor
(E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
: E ⊗[ℝ] E := ∑ i, ((stdOrthonormalBasis ℝ E) i) ⊗ₜ[ℝ] ((stdOrthonormalBasis ℝ E) i)
theorem InnerProductSpace.canonicalCovariantTensorRepresentation
(E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
[Fintype ι]
(v : OrthonormalBasis ι ℝ E)
: InnerProductSpace.canonicalCovariantTensor E = ∑ i, (v i) ⊗ₜ[ℝ] (v i) := by
(v : OrthonormalBasis ι ℝ E) :
InnerProductSpace.canonicalCovariantTensor E = ∑ i, (v i) ⊗ₜ[ℝ] (v i) := by
let w := stdOrthonormalBasis ℝ E
conv =>

24
Nevanlinna/divisor.lean
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@@ -1,5 +1,13 @@
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
import Nevanlinna.analyticAt
open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral
structure Divisor where
toFun : ℂ → ℤ
@@ -48,10 +56,10 @@ theorem Divisor.support_cap_compact
exact Set.inter_subset_right
noncomputable def AnalyticOn.zeroDivisor
noncomputable def AnalyticOnNhd.zeroDivisor
{f : ℂ → ℂ}
{U : Set ℂ}
(hf : AnalyticOn ℂ f U) :
(hf : AnalyticOnNhd ℂ f U) :
Divisor where
toFun := by
@@ -124,28 +132,28 @@ noncomputable def AnalyticOn.zeroDivisor
tauto
theorem AnalyticOn.support_of_zeroDivisor
theorem AnalyticOnNhd.support_of_zeroDivisor
{f : ℂ → ℂ}
{U : Set ℂ}
(hf : AnalyticOn ℂ f U) :
(hf : AnalyticOnNhd ℂ f U) :
Function.support hf.zeroDivisor ⊆ U := by
intro z
contrapose
intro hz
dsimp [AnalyticOn.zeroDivisor]
dsimp [AnalyticOnNhd.zeroDivisor]
simp
tauto
theorem AnalyticOn.support_of_zeroDivisor₂
theorem AnalyticOnNhd.support_of_zeroDivisor₂
{f : ℂ → ℂ}
{U : Set ℂ}
(hf : AnalyticOn ℂ f U) :
(hf : AnalyticOnNhd ℂ f U) :
Function.support hf.zeroDivisor ⊆ f⁻¹' {0} := by
intro z hz
dsimp [AnalyticOn.zeroDivisor] at hz
dsimp [AnalyticOnNhd.zeroDivisor] at hz
have t₀ := hf.support_of_zeroDivisor hz
simp [hf.support_of_zeroDivisor hz] at hz
let A := hz.1

16
Nevanlinna/holomorphic_JensenFormula.lean
View File

@@ -11,7 +11,7 @@ open Real
theorem jensen_case_R_eq_one
(f : ℂ → ℂ)
(h₁f : AnalyticOn ℂ f (Metric.closedBall 0 1))
(h₁f : AnalyticOnNhd ℂ f (Metric.closedBall 0 1))
(h₂f : f 0 ≠ 0) :
log ‖f 0‖ = -∑ᶠ s, (h₁f.order s).toNat * log (‖s.1‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 1 x)‖ := by
@@ -24,7 +24,7 @@ theorem jensen_case_R_eq_one
have h'₂f : ∃ u ∈ (Metric.closedBall (0 : ℂ) 1), f u ≠ 0 := by
use 0; simp; exact h₂f
obtain ⟨F, h₁F, h₂F, h₃F⟩ := AnalyticOnCompact.eliminateZeros₂ h₁U h₂U h₁f h'₂f
obtain ⟨F, h₁F, h₂F, h₃F⟩ := AnalyticOnNhdCompact.eliminateZeros₂ h₁U h₂U h₁f h'₂f
have h'₁F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt F z := by
intro z h₁z
@@ -259,9 +259,9 @@ theorem jensen_case_R_eq_one
intro x ⟨h₁x, _⟩
simp
dsimp [AnalyticOn.order] at h₁x
dsimp [AnalyticOnNhd.order] at h₁x
simp only [Function.mem_support, ne_eq, Nat.cast_eq_zero, not_or] at h₁x
exact AnalyticAt.supp_order_toNat (AnalyticOn.order.proof_1 h₁f x) h₁x
exact AnalyticAt.supp_order_toNat (AnalyticOnNhd.order.proof_1 h₁f x) h₁x
--
intro x hx
@@ -297,7 +297,7 @@ theorem jensen
{R : ℝ}
(hR : 0 < R)
(f : ℂ → ℂ)
(h₁f : AnalyticOn ℂ f (Metric.closedBall 0 R))
(h₁f : AnalyticOnNhd ℂ f (Metric.closedBall 0 R))
(h₂f : f 0 ≠ 0) :
log ‖f 0‖ = -∑ᶠ s, (h₁f.order s).toNat * log (R * ‖s.1‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 R x)‖ := by
@@ -327,8 +327,8 @@ theorem jensen
let F := f ∘ ℓ
have h₁F : AnalyticOn ℂ F (Metric.closedBall 0 1) := by
apply AnalyticOn.comp (t := Metric.closedBall 0 R)
have h₁F : AnalyticOnNhd ℂ F (Metric.closedBall 0 1) := by
apply AnalyticOnNhd.comp (t := Metric.closedBall 0 R)
exact h₁f
intro x _
apply ℓ.toContinuousLinearMap.analyticAt x
@@ -430,7 +430,7 @@ theorem jensen
left
congr 1
dsimp [AnalyticOn.order]
dsimp [AnalyticOnNhd.order]
rw [← AnalyticAt.order_comp_CLE ℓ]
--

22
Nevanlinna/holomorphic_zero.lean
View File

@@ -134,7 +134,7 @@ theorem topOnPreconnected
{f : ℂ → ℂ}
{U : Set ℂ}
(hU : IsPreconnected U)
(h₁f : AnalyticOn ℂ f U)
(h₁f : AnalyticOnNhd ℂ f U)
(h₂f : ∃ z ∈ U, f z ≠ 0) :
∀ (hz : z ∈ U), (h₁f z hz).order ≠ ⊤ := by
@@ -143,7 +143,7 @@ theorem topOnPreconnected
obtain ⟨z', hz'⟩ := H
rw [AnalyticAt.order_eq_top_iff] at hz'
rw [← AnalyticAt.frequently_zero_iff_eventually_zero (h₁f z z')] at hz'
have A := AnalyticOn.eqOn_zero_of_preconnected_of_frequently_eq_zero h₁f hU z' hz'
have A := AnalyticOnNhd.eqOn_zero_of_preconnected_of_frequently_eq_zero h₁f hU z' hz'
tauto
@@ -151,7 +151,7 @@ theorem supportZeroSet
{f : ℂ → ℂ}
{U : Set ℂ}
(hU : IsPreconnected U)
(h₁f : AnalyticOn ℂ f U)
(h₁f : AnalyticOnNhd ℂ f U)
(h₂f : ∃ z ∈ U, f z ≠ 0) :
U ∩ Function.support (zeroDivisor f) = U ∩ f⁻¹' {0} := by
@@ -180,7 +180,7 @@ theorem supportZeroSet
refine Eq.symm (ENat.coe_toNat ?h.mpr.right.right.right.a)
exact topOnPreconnected hU h₁f h₂f h₁x
/-
theorem discreteZeros
{f : ℂ → ℂ} :
DiscreteTopology (Function.support (zeroDivisor f)) := by
@@ -254,13 +254,13 @@ theorem discreteZeros
simp [this] at F
ext
rwa [sub_eq_zero] at F
-/
theorem zeroDivisor_finiteOnCompact
{f : ℂ → ℂ}
{U : Set ℂ}
(hU : IsPreconnected U)
(h₁f : AnalyticOn ℂ f U)
(h₁f : AnalyticOnNhd ℂ f U)
(h₂f : ∃ z ∈ U, f z ≠ 0) -- not needed!
(h₂U : IsCompact U) :
Set.Finite (U ∩ Function.support (zeroDivisor f)) := by
@@ -289,7 +289,7 @@ noncomputable def zeroDivisorDegree
{U : Set ℂ}
(h₁U : IsPreconnected U) -- not needed!
(h₂U : IsCompact U)
(h₁f : AnalyticOn ℂ f U)
(h₁f : AnalyticOnNhd ℂ f U)
(h₂f : ∃ z ∈ U, f z ≠ 0) : -- not needed!
ℕ := (zeroDivisor_finiteOnCompact h₁U h₁f h₂f h₂U).toFinset.card
@@ -299,7 +299,7 @@ lemma zeroDivisorDegreeZero
{U : Set ℂ}
(h₁U : IsPreconnected U) -- not needed!
(h₂U : IsCompact U)
(h₁f : AnalyticOn ℂ f U)
(h₁f : AnalyticOnNhd ℂ f U)
(h₂f : ∃ z ∈ U, f z ≠ 0) : -- not needed!
0 = zeroDivisorDegree h₁U h₂U h₁f h₂f ↔ U ∩ (zeroDivisor f).support = ∅ := by
sorry
@@ -309,9 +309,9 @@ lemma eliminatingZeros₀
{U : Set ℂ}
(h₁U : IsPreconnected U)
(h₂U : IsCompact U) :
∀ n : ℕ, ∀ f : ℂ → ℂ, (h₁f : AnalyticOn ℂ f U) → (h₂f : ∃ z ∈ U, f z ≠ 0) →
∀ n : ℕ, ∀ f : ℂ → ℂ, (h₁f : AnalyticOnNhd ℂ f U) → (h₂f : ∃ z ∈ U, f z ≠ 0) →
(n = zeroDivisorDegree h₁U h₂U h₁f h₂f) →
∃ F : ℂ → ℂ, (AnalyticOn ℂ F U) ∧ (f = F * ∏ᶠ a ∈ (U ∩ (zeroDivisor f).support), fun z ↦ (z - a) ^ (zeroDivisor f a)) := by
∃ F : ℂ → ℂ, (AnalyticOnNhd ℂ F U) ∧ (f = F * ∏ᶠ a ∈ (U ∩ (zeroDivisor f).support), fun z ↦ (z - a) ^ (zeroDivisor f a)) := by
intro n
induction' n with n ih
@@ -340,7 +340,7 @@ lemma eliminatingZeros₀
let A := AnalyticOn.order_eq_nat_iff h₁f hz₀.1 (zeroDivisor f z₀)
let A := AnalyticOnNhd.order_eq_nat_iff h₁f hz₀.1 (zeroDivisor f z₀)
let B := zeroDivisor_eq_ord_AtZeroDivisorSupport hz₀.2
let B := zeroDivisor_eq_ord_AtZeroDivisorSupport' hz₀.2
rw [eq_comm] at B