working…

This commit is contained in:
Stefan Kebekus 2024-06-17 10:11:39 +02:00
parent 321ceba46b
commit 8d2339a769
2 changed files with 78 additions and 34 deletions

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@ -1,5 +1,6 @@
import Mathlib.Analysis.Complex.TaylorSeries
import Mathlib.MeasureTheory.Integral.DivergenceTheorem
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Nevanlinna.cauchyRiemann
import Nevanlinna.partialDeriv
@ -166,8 +167,8 @@ theorem primitive_lem1
rw [this, ← add_smul]
simp
rw [this]
have hc : HasDerivAt (fun (w : ) ↦ w) 1 0 := by
have hc : HasDerivAt (fun (w : ) ↦ w) 1 0 := by
apply hasDerivAt_id'
nth_rewrite 2 [← (one_smul v)]
exact HasDerivAt.smul_const hc v
@ -176,7 +177,7 @@ theorem primitive_lem1
theorem primitive_fderivAtBasepoint
{E : Type u} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E]
(f : → E)
(f : → E)
(hf : Continuous f) :
HasDerivAt (primitive 0 f) (f 0) 0 := by
unfold primitive
@ -186,7 +187,7 @@ theorem primitive_fderivAtBasepoint
rw [Asymptotics.isLittleO_iff]
intro c hc
have {z : } {e : E} : z • e = (∫ (x : ) in (0)..(z.re), e) + Complex.I • ∫ (x : ) in (0)..(z.im), e:= by
have {z : } {e : E} : z • e = (∫ (x : ) in (0)..(z.re), e) + Complex.I • ∫ (x : ) in (0)..(z.im), e:= by
simp
rw [smul_comm]
rw [← smul_assoc]
@ -201,7 +202,7 @@ theorem primitive_fderivAtBasepoint
arg 1
arg 2
rw [this]
have {A B C D :E} : (A + B) - (C + D) = (A - C) + (B - D) := by
have {A B C D :E} : (A + B) - (C + D) = (A - C) + (B - D) := by
abel
have t₀ {r : } : IntervalIntegrable (fun x => f { re := x, im := 0 }) MeasureTheory.volume 0 r := by sorry
have t₁ {r : } :IntervalIntegrable (fun x => f 0) MeasureTheory.volume 0 r := by sorry
@ -219,48 +220,51 @@ theorem primitive_fderivAtBasepoint
rw [Filter.eventually_iff_exists_mem]
let s := f⁻¹' Metric.ball (f 0) c
have : IsOpen s := by
apply IsOpen.mem_nhds
apply IsOpen.preimage hf
exact Metric.isOpen_ball
--sorry
--apply isOpen_iff_ball_subset
use s
constructor
· apply IsOpen.mem_nhds
apply IsOpen.preimage hf
exact Metric.isOpen_ball
apply Set.mem_preimage.mpr
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
obtain ⟨ε, h₁ε, h₂ε⟩ := Metric.isOpen_iff.1 h₁s 0 h₂s
have h₃ε : ∀ y ∈ Metric.ball 0 ε, ‖(f y) - (f 0)‖ < c := by
intro y hy
exact mem_ball_iff_norm.mp (h₂ε hy)
use Metric.ball 0 ε
constructor
· exact Metric.ball_mem_nhds 0 h₁ε
· intro y hy
have : ‖(∫ (x : ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ ≤ c * |y.re| := by
let A := intervalIntegral.norm_integral_le_of_norm_le_const_ae
have h₁y : |y.re| < ε := by
sorry
have : ‖(∫ (x : ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ ≤ c * |y.re - 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 := x, im := 0 }
simp
have : { re := x, im := 0 } = (x : ) := by rfl
rw [this]
rw [Complex.abs_ofReal]
exact h₁x
sorry
/-
calc ‖(∫ (x : ) in (0)..(y.re), f { re := x, im := 0 } - f 0) + Complex.I • ∫ (x : ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖
_ ≤ ‖(∫ (x : ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖Complex.I • ∫ (x : ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by apply norm_add_le
_ ≤ ‖(∫ (x : ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖∫ (x : ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
_ ≤ ‖(∫ (x : ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖∫ (x : ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
simp
rw [norm_smul]
simp
_ ≤ |(∫ (x : ) in (0)..(y.re), ‖f { re := x, im := 0 } - f 0‖)| + |∫ (x : ) in (0)..(y.im), ‖f { re := y.re, im := x } - f 0‖| := by
_ ≤ |(∫ (x : ) in (0)..(y.re), ‖f { re := x, im := 0 } - f 0‖)| + |∫ (x : ) 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
_ ≤
_ ≤
-/
sorry

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@ -0,0 +1,40 @@
import Mathlib.Analysis.Complex.CauchyIntegral
--import Mathlib.Analysis.Complex.Module
example simplificationTest₁
{E : Type*} [NormedAddCommGroup E] [NormedSpace E] [IsScalarTower E]
{v : E}
{z : } :
z • v = z.re • v + Complex.I • z.im • v := by
/-
An attempt to write "rw [add_smul]" will fail with "did not find instance of
the pattern in the target -- expression (?r + ?s) • ?x".
-/
sorry
theorem add_smul'
(𝕜₁ : Type*) [NontriviallyNormedField ]
{𝕜₂ : Type*} [NontriviallyNormedField ] [NormedAlgebra ]
{E : Type*} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] [IsScalarTower E]
{v : E}
{r s : } :
(r + s) • v = r • v + s • v :=
Module.add_smul r s v
theorem smul_add' (a : M) (b₁ b₂ : A) : a • (b₁ + b₂) = a • b₁ + a • b₂ :=
DistribSMul.smul_add _ _ _
#align smul_add smul_add
example simplificationTest₂
{v : E}
{z : } :
z • v = z.re • v + Complex.I • z.im • v := by
sorry