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Update holomorphic.primitive.lean

This commit is contained in:
Stefan Kebekus 2024-06-17 17:22:17 +02:00
parent 8d2339a769
commit 46ededdde7
1 changed files with 38 additions and 16 deletions
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54
Nevanlinna/holomorphic.primitive.lean
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@ -219,52 +219,74 @@ theorem primitive_fderivAtBasepoint
rw [← intervalIntegral.integral_sub t₂ t₃]
rw [Filter.eventually_iff_exists_mem]
let s := f⁻¹' Metric.ball (f 0) c
let s := f⁻¹' Metric.ball (f 0) (c / (4 : ℝ))
have h₁s : IsOpen s := IsOpen.preimage hf Metric.isOpen_ball
have h₂s : 0 ∈ s := by
apply Set.mem_preimage.mpr
exact Metric.mem_ball_self hc
apply Metric.mem_ball_self
linarith
obtain ⟨ε, h₁ε, h₂ε⟩ := Metric.isOpen_iff.1 h₁s 0 h₂s
have h₃ε : ∀ y ∈ Metric.ball 0 ε, ‖(f y) - (f 0)‖ < c := by
have h₃ε : ∀ y ∈ Metric.ball 0 ε, ‖(f y) - (f 0)‖ < (c / (4 : ℝ)) := by
intro y hy
exact mem_ball_iff_norm.mp (h₂ε hy)
apply mem_ball_iff_norm.mp (h₂ε hy)
use Metric.ball 0 ε
constructor
· exact Metric.ball_mem_nhds 0 h₁ε
· intro y hy
have h₁y : |y.re| < ε := by
calc |y.re|
_ ≤ Complex.abs y := by apply Complex.abs_re_le_abs
_ < ε := by
let A := mem_ball_iff_norm.1 hy
simp at A
assumption
sorry
have : ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ ≤ c * |y.re - 0| := by
have t₁ : ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ ≤ (c / (4 : ℝ)) * |y.re - 0| := by
apply intervalIntegral.norm_integral_le_of_norm_le_const
intro x hx
have h₁x : |x| < ε := by sorry
have h₁x : |x| < ε := by
sorry
apply le_of_lt
apply h₃ε { re := x, im := 0 }
rw [mem_ball_iff_norm]
simp
have : { re := x, im := 0 } = (x : ℂ) := by rfl
rw [this]
rw [Complex.abs_ofReal]
exact h₁x
sorry
/-
have t₂ : ‖∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ ≤ (c / (4 : ℝ)) * |y.im - 0| := by
apply intervalIntegral.norm_integral_le_of_norm_le_const
intro x hx
have h₁x : |x| < ε := by
sorry
apply le_of_lt
apply h₃ε { re := y.re, im := x }
simp
have : { re := y.re, im := x } = (x : ℂ) := by
rfl
rw [this]
rw [Complex.abs_ofReal]
exact h₁x
calc ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0) + Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖
_ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by apply norm_add_le
_ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
apply norm_add_le
_ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
simp
rw [norm_smul]
simp
_ ≤ |(∫ (x : ℝ) in (0)..(y.re), ‖f { re := x, im := 0 } - f 0‖)| + |∫ (x : ℝ) in (0)..(y.im), ‖f { re := y.re, im := x } - f 0‖| := by
_ ≤ (c / (4 : ℝ)) * |y.re - 0| + ‖∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
apply add_le_add
apply intervalIntegral.norm_integral_le_abs_integral_norm
apply intervalIntegral.norm_integral_le_abs_integral_norm
_ ≤
-/
exact t₁
rfl
_ ≤ c * ‖y‖ := by sorry
sorry