kebekus/nevanlinna
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Make things compile again

Browse Source
This commit is contained in:
Stefan Kebekus
2024-07-25 15:52:16 +02:00
parent 34dfad798c
commit c785a85f26
11 changed files with 40 additions and 265 deletions
Download Patch File Download Diff File Expand all files Collapse all files

17
Nevanlinna/bilinear.lean
View File

@@ -32,24 +32,13 @@ orthonormal basis `v` as `∑ i, (v i) ⊗ₜ[ℝ] (v i)`.
open TensorProduct
lemma OrthonormalBasis.sum_repr'
{𝕜 : Type*} [RCLike 𝕜]
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
[Fintype ι]
(b : OrthonormalBasis ι 𝕜 E)
(v : E) :
v = ∑ i, ⟪b i, v⟫_𝕜 • (b i) := by
nth_rw 1 [← (b.sum_repr v)]
simp_rw [b.repr_apply_apply v]
noncomputable def InnerProductSpace.canonicalContravariantTensor
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
: E ⊗[ℝ] E →ₗ[ℝ] ℝ := TensorProduct.lift bilinFormOfRealInner
noncomputable def InnerProductSpace.canonicalCovariantTensor
(E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
(E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
: E ⊗[ℝ] E := ∑ i, ((stdOrthonormalBasis ℝ E) i) ⊗ₜ[ℝ] ((stdOrthonormalBasis ℝ E) i)
@@ -64,9 +53,9 @@ theorem InnerProductSpace.canonicalCovariantTensorRepresentation
right
arg 2
intro i
rw [w.sum_repr' (v i)]
rw [← w.sum_repr' (v i)]
simp_rw [TensorProduct.sum_tmul, TensorProduct.tmul_sum, TensorProduct.smul_tmul_smul]
conv =>
right
rw [Finset.sum_comm]

10
Nevanlinna/cauchyRiemann.lean
View File

@@ -45,8 +45,8 @@ theorem CauchyRiemann₃ : (DifferentiableAt ℂ f z)
theorem CauchyRiemann₄
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
{f : ℂ → F} :
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
{f : ℂ → F} :
(Differentiable ℂ f) → partialDeriv ℝ Complex.I f = Complex.I • partialDeriv ℝ 1 f := by
intro h
unfold partialDeriv
@@ -58,13 +58,13 @@ theorem CauchyRiemann₄
simp
rw [← mul_one Complex.I]
rw [← smul_eq_mul]
rw [ContinuousLinearMap.map_smul_of_tower (fderiv ℂ f w) Complex.I 1]
conv =>
right
right
intro w
rw [DifferentiableAt.fderiv_restrictScalars ℝ (h w)]
funext w
simp
theorem CauchyRiemann₅ {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : ℂ → F} {z : ℂ} : (DifferentiableAt ℂ f z)
→ partialDeriv ℝ Complex.I f z = Complex.I • partialDeriv ℝ 1 f z := by
@@ -77,8 +77,8 @@ theorem CauchyRiemann₅ {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
simp
rw [← mul_one Complex.I]
rw [← smul_eq_mul]
rw [ContinuousLinearMap.map_smul_of_tower (fderiv ℂ f z) Complex.I 1]
conv =>
right
right
rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
simp

2
Nevanlinna/complexHarmonic.lean
View File

@@ -33,7 +33,7 @@ theorem HarmonicAt_iff
· constructor
· exact Set.mem_inter h₂s₁ h₃s₂
· constructor
· exact h₃s₁.mono (Set.inter_subset_left s₁ s₂)
· exact h₃s₁.mono Set.inter_subset_left
· intro z hz
exact h₂t₂ (h₁s₂ hz.2)
· intro hyp

3
Nevanlinna/holomorphic.lean
View File

@@ -137,12 +137,11 @@ theorem CauchyRiemann'₅
simp
rw [← mul_one Complex.I]
rw [← smul_eq_mul]
rw [ContinuousLinearMap.map_smul_of_tower (fderiv ℂ f z) Complex.I 1]
conv =>
right
right
rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
simp
theorem CauchyRiemann'₆
{f : ℂ → F}

23
Nevanlinna/laplace2.lean
View File

@@ -10,15 +10,6 @@ import Mathlib.LinearAlgebra.Contraction
open BigOperators
open Finset
lemma OrthonormalBasis.sum_repr'
{𝕜 : Type*} [RCLike 𝕜]
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
[Fintype ι]
(b : OrthonormalBasis ι 𝕜 E)
(v : E) :
v = ∑ i, ⟪b i, v⟫_𝕜 • (b i) := by
nth_rw 1 [← (b.sum_repr v)]
simp_rw [b.repr_apply_apply v]
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
@@ -27,20 +18,20 @@ noncomputable def realInnerAsElementOfDualTensorprod
: TensorProduct ℝ E E →ₗ[ℝ] ℝ := TensorProduct.lift bilinFormOfRealInner
instance
instance
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [FiniteDimensional ℝ E]
: NormedAddCommGroup (TensorProduct ℝ E E) where
norm := by
: NormedAddCommGroup (TensorProduct ℝ E E) where
norm := by
sorry
dist_self := by
dist_self := by
sorry
sorry
/-
instance
instance
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [FiniteDimensional ℝ E]
: InnerProductSpace ℝ (TensorProduct ℝ E E) where
smul := _

6
Nevanlinna/partialDeriv.lean
View File

@@ -44,6 +44,8 @@ theorem partialDeriv_smul₁ {f : E → F} {a : 𝕜} {v : E} : partialDeriv
left
intro w
rw [map_smul]
funext w
simp
theorem partialDeriv_add₁ {f : E → F} {v₁ v₂ : E} : partialDeriv 𝕜 (v₁ + v₂) f = (partialDeriv 𝕜 v₁ f) + (partialDeriv 𝕜 v₂ f) := by
@@ -52,6 +54,8 @@ theorem partialDeriv_add₁ {f : E → F} {v₁ v₂ : E} : partialDeriv 𝕜 (v
left
intro w
rw [map_add]
funext w
simp
theorem partialDeriv_smul₂ {f : E → F} {a : 𝕜} {v : E} : partialDeriv 𝕜 v (a • f) = a • partialDeriv 𝕜 v f := by
@@ -90,6 +94,8 @@ theorem partialDeriv_add₂ {f₁ f₂ : E → F} (h₁ : Differentiable 𝕜 f
intro w
left
rw [fderiv_add (h₁ w) (h₂ w)]
funext w
simp
theorem partialDeriv_add₂_differentiableAt

40
Nevanlinna/smulImprovement.lean
View File

@@ -1,40 +0,0 @@
import Mathlib.Analysis.Complex.CauchyIntegral
--import Mathlib.Analysis.Complex.Module
example simplificationTest₁
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [IsScalarTower ℝ ℂ E]
{v : E}
{z : ℂ} :
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 ℝ]
{𝕜₂ : Type*} [NontriviallyNormedField ℂ] [NormedAlgebra ℝ ℂ]
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] [IsScalarTower ℝ ℂ E]
{v : E}
{r s : ℂ} :
(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 : ℂ} :
z • v = z.re • v + Complex.I • z.im • v := by
sorry

47
Nevanlinna/tensor.lean
View File

@@ -1,47 +0,0 @@
import Mathlib.Analysis.InnerProductSpace.PiL2
open RCLike Real Filter
open Topology ComplexConjugate
open LinearMap (BilinForm)
open TensorProduct
open InnerProductSpace
open Inner
example
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
{F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
: 1 = 0 := by
let e : E →ₗ[ℝ] E →ₗ[ℝ] ℝ := innerₗ E
let f : F →ₗ[ℝ] F →ₗ[ℝ] ℝ := innerₗ F
let e₂ := TensorProduct.map₂ e f
let l := TensorProduct.lid ℝ ℝ
have : ∀ e1 e2 : E, ∀ f1 f2 : F, e₂ (e1 ⊗ f1) (e2 ⊗ f2) =
let X : InnerProductSpace.Core ℝ (E ⊗[ℝ] F) := {
inner := by
intro a b
exact TensorProduct.lid ℝ ℝ ((TensorProduct.map₂ (innerₗ E) (innerₗ F)) a b)
conj_symm := by
simp
intro x y
unfold innerₗ
simp
sorry
nonneg_re := _
definite := _
add_left := _
smul_left := _
}
let inner : E ⊗[𝕜] E → E ⊗[𝕜] E → 𝕜 :=
--let x := TensorProduct.lift
--E.inner
--TensorProduct.map₂
sorry
sorry

130
Nevanlinna/test.lean
View File

@@ -1,130 +0,0 @@
import Mathlib.Analysis.Complex.CauchyIntegral
--import Mathlib.Analysis.Complex.Module
-- test
open ComplexConjugate
#check DiffContOnCl.circleIntegral_sub_inv_smul
open Real
theorem CauchyIntegralFormula :
∀
{R : ℝ} -- Radius of the ball
{w : ℂ} -- Point in the interior of the ball
{f : ℂ → ℂ}, -- Holomorphic function
DiffContOnCl ℂ f (Metric.ball 0 R)
→ w ∈ Metric.ball 0 R
→ (∮ (z : ℂ) in C(0, R), (z - w)⁻¹ • f z) = (2 * π * Complex.I) • f w := by
exact DiffContOnCl.circleIntegral_sub_inv_smul
#check CauchyIntegralFormula
#check HasDerivAt.continuousAt
#check Real.log
#check ComplexConjugate.conj
#check Complex.exp
theorem SimpleCauchyFormula :
∀
{R : ℝ} -- Radius of the ball
{w : ℂ} -- Point in the interior of the ball
{f : ℂ → ℂ}, -- Holomorphic function
Differentiable ℂ f
→ w ∈ Metric.ball 0 R
→ (∮ (z : ℂ) in C(0, R), (z - w)⁻¹ • f z) = (2 * Real.pi * Complex.I) • f w := by
intro R w f fHyp
apply DiffContOnCl.circleIntegral_sub_inv_smul
constructor
· exact Differentiable.differentiableOn fHyp
· suffices Continuous f from by
exact Continuous.continuousOn this
rw [continuous_iff_continuousAt]
intro x
exact DifferentiableAt.continuousAt (fHyp x)
theorem JensenFormula₂ :
∀
{R : ℝ} -- Radius of the ball
{f : ℂ → ℂ}, -- Holomorphic function
Differentiable ℂ f
→ (∀ z ∈ Metric.ball 0 R, f z ≠ 0)
→ (∫ (θ : ℝ) in Set.Icc 0 (2 * π), Complex.log ‖f (circleMap 0 R θ)‖ ) = 2 * π * Complex.log ‖f 0‖ := by
intro r f fHyp₁ fHyp₂
/- We treat the trivial case r = 0 separately. -/
by_cases rHyp : r = 0
rw [rHyp]
simp
left
unfold ENNReal.ofReal
simp
rw [max_eq_left (mul_nonneg zero_le_two pi_nonneg)]
simp
/- From hereon, we assume that r ≠ 0. -/
/- Replace the integral over 0 … 2π by a circle integral -/
suffices (∮ (z : ℂ) in C(0, r), -(Complex.I * z⁻¹ * Complex.log ↑(Complex.abs (f z)))) = 2 * ↑π * Complex.log ↑‖f 0‖ from by
have : ∫ (θ : ℝ) in Set.Icc 0 (2 * π), Complex.log ↑‖f (circleMap 0 r θ)‖ = (∮ (z : ℂ) in C(0, r), -(Complex.I * z⁻¹ * Complex.log ↑(Complex.abs (f z)))) := by
have : (fun θ ↦ Complex.log ‖f (circleMap 0 r θ)‖) = (fun θ ↦ ((deriv (circleMap 0 r) θ)) • ((deriv (circleMap 0 r) θ)⁻¹ • Complex.log ↑‖f (circleMap 0 r θ)‖)) := by
funext θ
rw [← smul_assoc, smul_eq_mul, smul_eq_mul, mul_inv_cancel, one_mul]
simp
exact rHyp
rw [this]
simp
let tmp := circleIntegral_def_Icc (fun z ↦ -(Complex.I * z⁻¹ * (Complex.log ↑‖f z‖))) 0 r
simp at tmp
rw [← tmp]
rw [this]
have : ∀ z : ℂ, log (Complex.abs z) = 1/2 * Complex.log z + 1/2 * Complex.log (conj z) := by
intro z
by_cases argHyp : Complex.arg z = π
· rw [Complex.log, argHyp, Complex.log]
let ZZ := Complex.arg_conj z
rw [argHyp] at ZZ
simp at ZZ
have : conj z = z := by
apply?
sorry
let ZZ := Complex.log_conj z
nth_rewrite 1 [Complex.log]
simp
let φ := Complex.arg z
let r := Complex.abs z
have : z = r * Complex.exp (φ * Complex.I) := by
exact (Complex.abs_mul_exp_arg_mul_I z).symm
have : Complex.log z = Complex.log r + r*Complex.I := by
apply?
sorry
simp at XX
sorry
sorry