Make things compile again
This commit is contained in:
Stefan Kebekus
parent
34dfad798c
commit
c785a85f26
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@ -32,17 +32,6 @@ orthonormal basis `v` as `∑ i, (v i) ⊗ₜ[ℝ] (v i)`.
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open TensorProduct
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lemma OrthonormalBasis.sum_repr'
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{𝕜 : Type*} [RCLike 𝕜]
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{E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
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[Fintype ι]
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(b : OrthonormalBasis ι 𝕜 E)
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(v : E) :
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v = ∑ i, ⟪b i, v⟫_𝕜 • (b i) := by
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nth_rw 1 [← (b.sum_repr v)]
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simp_rw [b.repr_apply_apply v]
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noncomputable def InnerProductSpace.canonicalContravariantTensor
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{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
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: E ⊗[ℝ] E →ₗ[ℝ] ℝ := TensorProduct.lift bilinFormOfRealInner
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@ -64,7 +53,7 @@ theorem InnerProductSpace.canonicalCovariantTensorRepresentation
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right
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arg 2
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intro i
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rw [w.sum_repr' (v i)]
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rw [← w.sum_repr' (v i)]
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simp_rw [TensorProduct.sum_tmul, TensorProduct.tmul_sum, TensorProduct.smul_tmul_smul]
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conv =>
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@ -58,13 +58,13 @@ theorem CauchyRiemann₄
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simp
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rw [← mul_one Complex.I]
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rw [← smul_eq_mul]
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rw [ContinuousLinearMap.map_smul_of_tower (fderiv ℂ f w) Complex.I 1]
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conv =>
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right
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right
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intro w
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rw [DifferentiableAt.fderiv_restrictScalars ℝ (h w)]
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funext w
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simp
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theorem CauchyRiemann₅ {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : ℂ → F} {z : ℂ} : (DifferentiableAt ℂ f z)
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→ partialDeriv ℝ Complex.I f z = Complex.I • partialDeriv ℝ 1 f z := by
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@ -77,8 +77,8 @@ theorem CauchyRiemann₅ {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
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simp
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rw [← mul_one Complex.I]
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rw [← smul_eq_mul]
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rw [ContinuousLinearMap.map_smul_of_tower (fderiv ℂ f z) Complex.I 1]
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conv =>
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right
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right
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rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
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simp
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@ -33,7 +33,7 @@ theorem HarmonicAt_iff
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· constructor
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· exact Set.mem_inter h₂s₁ h₃s₂
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· constructor
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· exact h₃s₁.mono (Set.inter_subset_left s₁ s₂)
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· exact h₃s₁.mono Set.inter_subset_left
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· intro z hz
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exact h₂t₂ (h₁s₂ hz.2)
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· intro hyp
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@ -137,12 +137,11 @@ theorem CauchyRiemann'₅
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simp
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rw [← mul_one Complex.I]
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rw [← smul_eq_mul]
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rw [ContinuousLinearMap.map_smul_of_tower (fderiv ℂ f z) Complex.I 1]
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conv =>
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right
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right
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rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
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simp
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theorem CauchyRiemann'₆
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{f : ℂ → F}
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@ -10,15 +10,6 @@ import Mathlib.LinearAlgebra.Contraction
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open BigOperators
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open Finset
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lemma OrthonormalBasis.sum_repr'
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{𝕜 : Type*} [RCLike 𝕜]
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{E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
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[Fintype ι]
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(b : OrthonormalBasis ι 𝕜 E)
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(v : E) :
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v = ∑ i, ⟪b i, v⟫_𝕜 • (b i) := by
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nth_rw 1 [← (b.sum_repr v)]
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simp_rw [b.repr_apply_apply v]
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variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
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@ -44,6 +44,8 @@ theorem partialDeriv_smul₁ {f : E → F} {a : 𝕜} {v : E} : partialDeriv
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left
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intro w
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rw [map_smul]
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funext w
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simp
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theorem partialDeriv_add₁ {f : E → F} {v₁ v₂ : E} : partialDeriv 𝕜 (v₁ + v₂) f = (partialDeriv 𝕜 v₁ f) + (partialDeriv 𝕜 v₂ f) := by
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@ -52,6 +54,8 @@ theorem partialDeriv_add₁ {f : E → F} {v₁ v₂ : E} : partialDeriv 𝕜 (v
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left
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intro w
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rw [map_add]
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funext w
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simp
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theorem partialDeriv_smul₂ {f : E → F} {a : 𝕜} {v : E} : partialDeriv 𝕜 v (a • f) = a • partialDeriv 𝕜 v f := by
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@ -90,6 +94,8 @@ theorem partialDeriv_add₂ {f₁ f₂ : E → F} (h₁ : Differentiable 𝕜 f
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intro w
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left
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rw [fderiv_add (h₁ w) (h₂ w)]
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funext w
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simp
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theorem partialDeriv_add₂_differentiableAt
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@ -1,40 +0,0 @@
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import Mathlib.Analysis.Complex.CauchyIntegral
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--import Mathlib.Analysis.Complex.Module
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example simplificationTest₁
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{E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [IsScalarTower ℝ ℂ E]
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{v : E}
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{z : ℂ} :
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z • v = z.re • v + Complex.I • z.im • v := by
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/-
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An attempt to write "rw [add_smul]" will fail with "did not find instance of
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the pattern in the target -- expression (?r + ?s) • ?x".
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-/
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sorry
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theorem add_smul'
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(𝕜₁ : Type*) [NontriviallyNormedField ℝ]
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{𝕜₂ : Type*} [NontriviallyNormedField ℂ] [NormedAlgebra ℝ ℂ]
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{E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] [IsScalarTower ℝ ℂ E]
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{v : E}
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{r s : ℂ} :
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(r + s) • v = r • v + s • v :=
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Module.add_smul r s v
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theorem smul_add' (a : M) (b₁ b₂ : A) : a • (b₁ + b₂) = a • b₁ + a • b₂ :=
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DistribSMul.smul_add _ _ _
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#align smul_add smul_add
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example simplificationTest₂
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{v : E}
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{z : ℂ} :
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z • v = z.re • v + Complex.I • z.im • v := by
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sorry
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@ -1,47 +0,0 @@
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import Mathlib.Analysis.InnerProductSpace.PiL2
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open RCLike Real Filter
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open Topology ComplexConjugate
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open LinearMap (BilinForm)
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open TensorProduct
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open InnerProductSpace
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open Inner
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example
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{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
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{F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
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: 1 = 0 := by
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let e : E →ₗ[ℝ] E →ₗ[ℝ] ℝ := innerₗ E
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let f : F →ₗ[ℝ] F →ₗ[ℝ] ℝ := innerₗ F
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let e₂ := TensorProduct.map₂ e f
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let l := TensorProduct.lid ℝ ℝ
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have : ∀ e1 e2 : E, ∀ f1 f2 : F, e₂ (e1 ⊗ f1) (e2 ⊗ f2) =
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let X : InnerProductSpace.Core ℝ (E ⊗[ℝ] F) := {
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inner := by
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intro a b
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exact TensorProduct.lid ℝ ℝ ((TensorProduct.map₂ (innerₗ E) (innerₗ F)) a b)
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conj_symm := by
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simp
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intro x y
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unfold innerₗ
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simp
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sorry
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nonneg_re := _
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definite := _
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add_left := _
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smul_left := _
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}
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let inner : E ⊗[𝕜] E → E ⊗[𝕜] E → 𝕜 :=
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--let x := TensorProduct.lift
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--E.inner
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--TensorProduct.map₂
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sorry
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sorry
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@ -1,130 +0,0 @@
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import Mathlib.Analysis.Complex.CauchyIntegral
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--import Mathlib.Analysis.Complex.Module
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-- test
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open ComplexConjugate
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#check DiffContOnCl.circleIntegral_sub_inv_smul
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open Real
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theorem CauchyIntegralFormula :
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∀
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{R : ℝ} -- Radius of the ball
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{w : ℂ} -- Point in the interior of the ball
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{f : ℂ → ℂ}, -- Holomorphic function
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DiffContOnCl ℂ f (Metric.ball 0 R)
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→ w ∈ Metric.ball 0 R
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→ (∮ (z : ℂ) in C(0, R), (z - w)⁻¹ • f z) = (2 * π * Complex.I) • f w := by
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exact DiffContOnCl.circleIntegral_sub_inv_smul
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#check CauchyIntegralFormula
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#check HasDerivAt.continuousAt
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#check Real.log
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#check ComplexConjugate.conj
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#check Complex.exp
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theorem SimpleCauchyFormula :
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∀
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{R : ℝ} -- Radius of the ball
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{w : ℂ} -- Point in the interior of the ball
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{f : ℂ → ℂ}, -- Holomorphic function
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Differentiable ℂ f
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→ w ∈ Metric.ball 0 R
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→ (∮ (z : ℂ) in C(0, R), (z - w)⁻¹ • f z) = (2 * Real.pi * Complex.I) • f w := by
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intro R w f fHyp
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apply DiffContOnCl.circleIntegral_sub_inv_smul
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constructor
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· exact Differentiable.differentiableOn fHyp
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· suffices Continuous f from by
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exact Continuous.continuousOn this
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rw [continuous_iff_continuousAt]
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intro x
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exact DifferentiableAt.continuousAt (fHyp x)
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theorem JensenFormula₂ :
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∀
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{R : ℝ} -- Radius of the ball
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{f : ℂ → ℂ}, -- Holomorphic function
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Differentiable ℂ f
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→ (∀ z ∈ Metric.ball 0 R, f z ≠ 0)
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→ (∫ (θ : ℝ) in Set.Icc 0 (2 * π), Complex.log ‖f (circleMap 0 R θ)‖ ) = 2 * π * Complex.log ‖f 0‖ := by
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intro r f fHyp₁ fHyp₂
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/- We treat the trivial case r = 0 separately. -/
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by_cases rHyp : r = 0
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rw [rHyp]
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simp
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left
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unfold ENNReal.ofReal
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simp
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rw [max_eq_left (mul_nonneg zero_le_two pi_nonneg)]
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simp
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/- From hereon, we assume that r ≠ 0. -/
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/- Replace the integral over 0 … 2π by a circle integral -/
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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
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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
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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
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funext θ
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rw [← smul_assoc, smul_eq_mul, smul_eq_mul, mul_inv_cancel, one_mul]
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simp
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exact rHyp
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rw [this]
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simp
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let tmp := circleIntegral_def_Icc (fun z ↦ -(Complex.I * z⁻¹ * (Complex.log ↑‖f z‖))) 0 r
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simp at tmp
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rw [← tmp]
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rw [this]
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have : ∀ z : ℂ, log (Complex.abs z) = 1/2 * Complex.log z + 1/2 * Complex.log (conj z) := by
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intro z
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by_cases argHyp : Complex.arg z = π
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· rw [Complex.log, argHyp, Complex.log]
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let ZZ := Complex.arg_conj z
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rw [argHyp] at ZZ
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simp at ZZ
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have : conj z = z := by
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apply?
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sorry
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let ZZ := Complex.log_conj z
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nth_rewrite 1 [Complex.log]
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simp
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let φ := Complex.arg z
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let r := Complex.abs z
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have : z = r * Complex.exp (φ * Complex.I) := by
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exact (Complex.abs_mul_exp_arg_mul_I z).symm
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have : Complex.log z = Complex.log r + r*Complex.I := by
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apply?
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sorry
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simp at XX
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sorry
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sorry
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@ -1,10 +1,11 @@
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{"version": "1.0.0",
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{"version": "1.1.0",
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"packagesDir": ".lake/packages",
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"packages":
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[{"url": "https://github.com/leanprover-community/batteries",
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"type": "git",
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"subDir": null,
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"rev": "555ec79bc6effe7009036184a5f73f7d8d4850ed",
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"scope": "leanprover-community",
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"rev": "d2b1546c5fc05a06426e3f6ee1cb020e71be5592",
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"name": "batteries",
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"manifestFile": "lake-manifest.json",
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"inputRev": "main",
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@ -13,7 +14,8 @@
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{"url": "https://github.com/leanprover-community/quote4",
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"type": "git",
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"subDir": null,
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"rev": "a7bfa63f5dddbcab2d4e0569c4cac74b2585e2c6",
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"scope": "leanprover-community",
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"rev": "01ad33937acd996ee99eb74eefb39845e4e4b9f5",
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"name": "Qq",
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"manifestFile": "lake-manifest.json",
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"inputRev": "master",
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@ -22,7 +24,8 @@
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{"url": "https://github.com/leanprover-community/aesop",
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"type": "git",
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"subDir": null,
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"rev": "30619d94ce4a3d69cdb87bb1771562ca2e687cfa",
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"scope": "leanprover-community",
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"rev": "622d52c803db99ff4ea4fb442c1db9e91aed944c",
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"name": "aesop",
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"manifestFile": "lake-manifest.json",
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"inputRev": "master",
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@ -31,25 +34,28 @@
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{"url": "https://github.com/leanprover-community/ProofWidgets4",
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"type": "git",
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"subDir": null,
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"rev": "87c1e7a427d8e21b6eaf8401f12897f52e2c3be9",
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"scope": "leanprover-community",
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"rev": "d1b33202c3a29a079f292de65ea438648123b635",
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"name": "proofwidgets",
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"manifestFile": "lake-manifest.json",
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"inputRev": "v0.0.38",
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"inputRev": "v0.0.39",
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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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"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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{"url": "https://github.com/leanprover-community/import-graph.git",
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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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||||
"rev": "1588be870b9c76fe62286e8f42f0b4dafa154c96",
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||||
"scope": "leanprover-community",
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"rev": "68b518c9b352fbee16e6d632adcb7a6d0760e2b7",
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"name": "importGraph",
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"manifestFile": "lake-manifest.json",
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||||
"inputRev": "main",
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@ -58,7 +64,8 @@
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{"url": "https://github.com/leanprover-community/mathlib4.git",
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"type": "git",
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"subDir": null,
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||||
"rev": "dcc73cfb2ce3763f830c52042fb8617e762dbf60",
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||||
"scope": "",
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"rev": "cc495260156b40dcbd55b947c047061e15344000",
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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 @@
|
|||
leanprover/lean4:v4.9.0-rc3
|
||||
leanprover/lean4:v4.10.0-rc2
|
||||
|
|
|
|||
Loading…
Reference in New Issue