working…
This commit is contained in:
Stefan Kebekus
parent
9a7c8b82a4
commit
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@ -1,6 +1,13 @@
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import Nevanlinna.harmonicAt_examples
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import Nevanlinna.harmonicAt_meanValue
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lemma int₀
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{a : ℂ}
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(ha : a ∈ Metric.ball 0 1)
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:
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∫ (x : ℝ) in (0)..2 * Real.pi, Real.log ‖circleMap 0 1 x - a‖ = 0 := by
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sorry
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theorem jensen_case_R_eq_one
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(f : ℂ → ℂ)
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@ -64,21 +71,32 @@ theorem jensen_case_R_eq_one
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exact hz A
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have s₁ : ∀ z, f z ≠ 0 → logAbsF z = logAbsf z - ∑ s, Real.log ‖z - a s‖ := by
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sorry
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intro z hz
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rw [s₀ z hz]
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simp
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rw [s₁ 0 h₂f] at t₁
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have {x : ℝ} : f (circleMap 0 1 x) ≠ 0 := by
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sorry
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have h₀ {x : ℝ} : f (circleMap 0 1 x) ≠ 0 := by
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rw [h₃F]
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simp
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constructor
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· exact h₂F (circleMap 0 1 x)
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· by_contra h'
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obtain ⟨s, _, h₂s⟩ := Finset.prod_eq_zero_iff.1 h'
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have : circleMap 0 1 x = a s := by
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rw [← sub_zero (circleMap 0 1 x)]
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nth_rw 2 [← h₂s]
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simp
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let A := ha s
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rw [← this] at A
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simp at A
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simp_rw [s₁ (circleMap 0 1 _) this] at t₁
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simp_rw [s₁ (circleMap 0 1 _) h₀] at t₁
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rw [intervalIntegral.integral_sub] at t₁
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rw [intervalIntegral.integral_finset_sum] at t₁
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have {i : S} : ∫ (x : ℝ) in (0)..2 * Real.pi, Real.log ‖circleMap 0 1 x - a i‖ = 0 := by
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sorry
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simp_rw [this] at t₁
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simp_rw [int₀ (ha _)] at t₁
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simp at t₁
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rw [t₁]
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simp
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@ -89,7 +107,7 @@ theorem jensen_case_R_eq_one
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simp
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rfl
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-- ∀ i ∈ Finset.univ, IntervalIntegrable (fun x => Real.log ‖circleMap 0 1 x - a i‖) MeasureTheory.volume 0 (2 * Real.pi)
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intro i hi
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intro i _
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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@ -115,7 +133,7 @@ theorem jensen_case_R_eq_one
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rw [this]
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apply ContinuousAt.comp
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simp
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sorry
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exact h₀
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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apply ContinuousAt.comp
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@ -125,7 +143,7 @@ theorem jensen_case_R_eq_one
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-- IntervalIntegrable (fun x => ∑ s : { x // x ∈ S }, Real.log ‖circleMap 0 1 x - a s‖) MeasureTheory.volume 0 (2 * Real.pi)
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apply Continuous.intervalIntegrable
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apply continuous_finset_sum
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intro i hi
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intro i _
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apply continuous_iff_continuousAt.2
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intro x
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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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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "d2b1546c5fc05a06426e3f6ee1cb020e71be5592",
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"rev": "0f3e143dffdc3a591662f3401ce1d7a3405227c0",
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"name": "batteries",
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"manifestFile": "lake-manifest.json",
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"inputRev": "main",
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@ -25,7 +25,7 @@
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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "622d52c803db99ff4ea4fb442c1db9e91aed944c",
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"rev": "209712c78b16c795453b6da7f7adbda4589a8f21",
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"name": "aesop",
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"manifestFile": "lake-manifest.json",
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"inputRev": "master",
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@ -35,27 +35,27 @@
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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "d1b33202c3a29a079f292de65ea438648123b635",
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"rev": "c87908619cccadda23f71262e6898b9893bffa36",
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"name": "proofwidgets",
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"manifestFile": "lake-manifest.json",
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"inputRev": "v0.0.39",
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"inputRev": "v0.0.40",
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"inherited": true,
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"configFile": "lakefile.lean"},
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{"url": "https://github.com/leanprover/lean4-cli",
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"type": "git",
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"subDir": null,
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"scope": "",
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"rev": "a11566029bd9ec4f68a65394e8c3ff1af74c1a29",
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"rev": "2cf1030dc2ae6b3632c84a09350b675ef3e347d0",
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"name": "Cli",
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"manifestFile": "lake-manifest.json",
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"inputRev": "main",
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"inherited": true,
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"configFile": "lakefile.lean"},
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"configFile": "lakefile.toml"},
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{"url": "https://github.com/leanprover-community/import-graph",
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"type": "git",
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"subDir": null,
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"scope": "leanprover-community",
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"rev": "68b518c9b352fbee16e6d632adcb7a6d0760e2b7",
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"rev": "e7e90d90a62e6d12cbb27cbbfc31c094ee4ecc58",
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"name": "importGraph",
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"manifestFile": "lake-manifest.json",
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"inputRev": "main",
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@ -65,7 +65,7 @@
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"type": "git",
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"subDir": null,
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"scope": "",
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"rev": "cc495260156b40dcbd55b947c047061e15344000",
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"rev": "c96401c7496392a2200dc78c0b334df0561466dc",
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"name": "mathlib",
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"manifestFile": "lake-manifest.json",
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"inputRev": null,
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@ -1 +1 @@
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leanprover/lean4:v4.10.0-rc2
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leanprover/lean4:v4.10.0
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