working…

This commit is contained in:
Stefan Kebekus 2024-08-01 10:02:36 +02:00
parent 9a7c8b82a4
commit f65c84a14b
3 changed files with 38 additions and 20 deletions

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@ -1,6 +1,13 @@
import Nevanlinna.harmonicAt_examples import Nevanlinna.harmonicAt_examples
import Nevanlinna.harmonicAt_meanValue import Nevanlinna.harmonicAt_meanValue
lemma int₀
{a : }
(ha : a ∈ Metric.ball 0 1)
:
∫ (x : ) in (0)..2 * Real.pi, Real.log ‖circleMap 0 1 x - a‖ = 0 := by
sorry
theorem jensen_case_R_eq_one theorem jensen_case_R_eq_one
(f : ) (f : )
@ -64,21 +71,32 @@ theorem jensen_case_R_eq_one
exact hz A exact hz A
have s₁ : ∀ z, f z ≠ 0 → logAbsF z = logAbsf z - ∑ s, Real.log ‖z - a s‖ := by have s₁ : ∀ z, f z ≠ 0 → logAbsF z = logAbsf z - ∑ s, Real.log ‖z - a s‖ := by
sorry intro z hz
rw [s₀ z hz]
simp
rw [s₁ 0 h₂f] at t₁ rw [s₁ 0 h₂f] at t₁
have {x : } : f (circleMap 0 1 x) ≠ 0 := by have h₀ {x : } : f (circleMap 0 1 x) ≠ 0 := by
sorry rw [h₃F]
simp
constructor
· exact h₂F (circleMap 0 1 x)
· by_contra h'
obtain ⟨s, _, h₂s⟩ := Finset.prod_eq_zero_iff.1 h'
have : circleMap 0 1 x = a s := by
rw [← sub_zero (circleMap 0 1 x)]
nth_rw 2 [← h₂s]
simp
let A := ha s
rw [← this] at A
simp at A
simp_rw [s₁ (circleMap 0 1 _) this] at t₁ simp_rw [s₁ (circleMap 0 1 _) h₀] at t₁
rw [intervalIntegral.integral_sub] at t₁ rw [intervalIntegral.integral_sub] at t₁
rw [intervalIntegral.integral_finset_sum] at t₁ rw [intervalIntegral.integral_finset_sum] at t₁
have {i : S} : ∫ (x : ) in (0)..2 * Real.pi, Real.log ‖circleMap 0 1 x - a i‖ = 0 := by simp_rw [int₀ (ha _)] at t₁
sorry
simp_rw [this] at t₁
simp at t₁ simp at t₁
rw [t₁] rw [t₁]
simp simp
@ -89,7 +107,7 @@ theorem jensen_case_R_eq_one
simp simp
rfl rfl
-- ∀ i ∈ Finset.univ, IntervalIntegrable (fun x => Real.log ‖circleMap 0 1 x - a i‖) MeasureTheory.volume 0 (2 * Real.pi) -- ∀ i ∈ Finset.univ, IntervalIntegrable (fun x => Real.log ‖circleMap 0 1 x - a i‖) MeasureTheory.volume 0 (2 * Real.pi)
intro i hi intro i _
apply Continuous.intervalIntegrable apply Continuous.intervalIntegrable
apply continuous_iff_continuousAt.2 apply continuous_iff_continuousAt.2
intro x intro x
@ -115,7 +133,7 @@ theorem jensen_case_R_eq_one
rw [this] rw [this]
apply ContinuousAt.comp apply ContinuousAt.comp
simp simp
sorry exact h₀
apply ContinuousAt.comp apply ContinuousAt.comp
apply Complex.continuous_abs.continuousAt apply Complex.continuous_abs.continuousAt
apply ContinuousAt.comp apply ContinuousAt.comp
@ -125,7 +143,7 @@ theorem jensen_case_R_eq_one
-- IntervalIntegrable (fun x => ∑ s : { x // x ∈ S }, Real.log ‖circleMap 0 1 x - a s‖) MeasureTheory.volume 0 (2 * Real.pi) -- IntervalIntegrable (fun x => ∑ s : { x // x ∈ S }, Real.log ‖circleMap 0 1 x - a s‖) MeasureTheory.volume 0 (2 * Real.pi)
apply Continuous.intervalIntegrable apply Continuous.intervalIntegrable
apply continuous_finset_sum apply continuous_finset_sum
intro i hi intro i _
apply continuous_iff_continuousAt.2 apply continuous_iff_continuousAt.2
intro x intro x
have : (fun x => Real.log ‖circleMap 0 1 x - a i‖) = Real.log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - a i) := have : (fun x => Real.log ‖circleMap 0 1 x - a i‖) = Real.log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - a i) :=

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@ -5,7 +5,7 @@
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"manifestFile": "lake-manifest.json", "manifestFile": "lake-manifest.json",
"inputRev": "main", "inputRev": "main",
@ -25,7 +25,7 @@
"type": "git", "type": "git",
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"scope": "leanprover-community", "scope": "leanprover-community",
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"manifestFile": "lake-manifest.json", "manifestFile": "lake-manifest.json",
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@ -35,27 +35,27 @@
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"scope": "leanprover-community", "scope": "leanprover-community",
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"name": "proofwidgets", "name": "proofwidgets",
"manifestFile": "lake-manifest.json", "manifestFile": "lake-manifest.json",
"inputRev": "v0.0.39", "inputRev": "v0.0.40",
"inherited": true, "inherited": true,
"configFile": "lakefile.lean"}, "configFile": "lakefile.lean"},
{"url": "https://github.com/leanprover/lean4-cli", {"url": "https://github.com/leanprover/lean4-cli",
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"scope": "", "scope": "",
"rev": "a11566029bd9ec4f68a65394e8c3ff1af74c1a29", "rev": "2cf1030dc2ae6b3632c84a09350b675ef3e347d0",
"name": "Cli", "name": "Cli",
"manifestFile": "lake-manifest.json", "manifestFile": "lake-manifest.json",
"inputRev": "main", "inputRev": "main",
"inherited": true, "inherited": true,
"configFile": "lakefile.lean"}, "configFile": "lakefile.toml"},
{"url": "https://github.com/leanprover-community/import-graph", {"url": "https://github.com/leanprover-community/import-graph",
"type": "git", "type": "git",
"subDir": null, "subDir": null,
"scope": "leanprover-community", "scope": "leanprover-community",
"rev": "68b518c9b352fbee16e6d632adcb7a6d0760e2b7", "rev": "e7e90d90a62e6d12cbb27cbbfc31c094ee4ecc58",
"name": "importGraph", "name": "importGraph",
"manifestFile": "lake-manifest.json", "manifestFile": "lake-manifest.json",
"inputRev": "main", "inputRev": "main",
@ -65,7 +65,7 @@
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"scope": "", "scope": "",
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"name": "mathlib", "name": "mathlib",
"manifestFile": "lake-manifest.json", "manifestFile": "lake-manifest.json",
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@ -1 +1 @@
leanprover/lean4:v4.10.0-rc2 leanprover/lean4:v4.10.0