Nevanlinna/bilinear.lean

@@ -1,89 +0,0 @@
-/-
-Copyright (c) 2024 Stefan Kebekus. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Stefan Kebekus
--/
-
-import Mathlib.Analysis.InnerProductSpace.PiL2
-
-/-!
-
-# Canoncial Elements in Tensor Powers of Real Inner Product Spaces
-
-Given an `InnerProductSpace ℝ E`, this file defines two canonical tensors, which
-are relevant when for a coordinate-free definition of differential operators
-such as the Laplacian.
-
-* `InnerProductSpace.canonicalContravariantTensor E : E ⊗[ℝ] E →ₗ[ℝ] ℝ`. This is
-  the element corresponding to the inner product.
-
-* If `E` is finite-dimensional, then `E ⊗[ℝ] E` is canonically isomorphic to its
-  dual. Accordingly, there exists an element
-  `InnerProductSpace.canonicalCovariantTensor E : E ⊗[ℝ] E` that corresponds to
-  `InnerProductSpace.canonicalContravariantTensor E` under this identification.
-
-The theorem `InnerProductSpace.canonicalCovariantTensorRepresentation` shows
-that `InnerProductSpace.canonicalCovariantTensor E` can be computed from any
-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 ⊗[ℝ] E := ∑ i, ((stdOrthonormalBasis ℝ E) i) ⊗ₜ[ℝ] ((stdOrthonormalBasis ℝ E) i)
-
-
-theorem InnerProductSpace.canonicalCovariantTensorRepresentation
-  (E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
-  [Fintype ι]
-  (v : OrthonormalBasis ι ℝ E)
-  : InnerProductSpace.canonicalCovariantTensor E = ∑ i, (v i) ⊗ₜ[ℝ] (v i) := by
-
-  let w := stdOrthonormalBasis ℝ E
-  conv =>
-    right
-    arg 2
-    intro 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]
-    arg 2
-    intro y
-    rw [Finset.sum_comm]
-    arg 2
-    intro x
-    rw [← Finset.sum_smul]
-    arg 1
-    arg 2
-    intro i
-    rw [← real_inner_comm (w x)]
-  simp_rw [OrthonormalBasis.sum_inner_mul_inner v]
-
-  have {i} : ∑ j, ⟪w i, w j⟫_ℝ • w i ⊗ₜ[ℝ] w j = w i ⊗ₜ[ℝ] w i := by
-    rw [Fintype.sum_eq_single i, orthonormal_iff_ite.1 w.orthonormal]; simp
-    intro _ _; rw [orthonormal_iff_ite.1 w.orthonormal]; simp; tauto
-  simp_rw [this]
-  rfl

Nevanlinna/laplace2.lean

@@ -1,294 +0,0 @@
-import Mathlib.Analysis.InnerProductSpace.Basic
-import Mathlib.Analysis.InnerProductSpace.Dual
-import Mathlib.Analysis.InnerProductSpace.PiL2
---import Mathlib.Algebra.BigOperators.Basic
-import Mathlib.Analysis.Calculus.ContDiff.Bounds
-import Mathlib.Analysis.Calculus.FDeriv.Symmetric
-import Mathlib.LinearAlgebra.Basis
-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]
-
-noncomputable def realInnerAsElementOfDualTensorprod
-  {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
-  : TensorProduct ℝ E E →ₗ[ℝ] ℝ := TensorProduct.lift bilinFormOfRealInner
-
-
-instance 
-  {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]  [FiniteDimensional ℝ E]
-  : NormedAddCommGroup (TensorProduct ℝ E E) where 
-    norm := by 
-      
-      sorry
-    dist_self := by 
-      
-      sorry
-
-  sorry
-
-/-
-instance 
-  {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]  [FiniteDimensional ℝ E]
-  : InnerProductSpace ℝ (TensorProduct ℝ E E) where
-    smul := _
-    one_smul := _
-    mul_smul := _
-    smul_zero := _
-    smul_add := _
-    add_smul := _
-    zero_smul := _
-    norm_smul_le := _
-    inner := _
-    norm_sq_eq_inner := _
-    conj_symm := _
-    add_left := _
-    smul_left := _
-  -/
-
-
-
-noncomputable def dual'
-  {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]  [FiniteDimensional ℝ E]
-  : (TensorProduct ℝ E E →ₗ[ℝ] ℝ) ≃ₗ[ℝ] TensorProduct ℝ E E := by
-
-  let d := InnerProductSpace.toDual ℝ E
-
-
-  let e := d.toLinearEquiv
-
-  let a := TensorProduct.congr e e
-
-  let b := homTensorHomEquiv ℝ E E ℝ ℝ
-
-  sorry
-
-
-variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
-variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
-
-
-theorem InvariantTensor
-  [Fintype ι]
-  (v : Basis ι ℝ E)
-  (c : ι → ℝ)
-  (L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F) :
-  L (fun _ => ∑ j : ι, c j • v j) = ∑ x : Fin 2 → ι, (c (x 0) * c (x 1)) • L ((fun i => v (x i))) := by
-
-  rw [L.map_sum]
-  simp_rw [L.map_smul_univ]
-  sorry
-
-
-
-theorem BilinearCalc
-  [Fintype ι]
-  (v : Basis ι ℝ E)
-  (c : ι → ℝ)
-  (L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F) :
-  L (fun _ => ∑ j : ι, c j • v j) = ∑ x : Fin 2 → ι, (c (x 0) * c (x 1)) • L ((fun i => v (x i))) := by
-
-  rw [L.map_sum]
-  simp_rw [L.map_smul_univ]
-  sorry
-
-/-
-lemma c2
-  [Fintype ι]
-  (b : Basis ι ℝ E)
-  (hb : Orthonormal ℝ b)
-  (x y : E) :
-  ⟪x, y⟫_ℝ = ∑ i : ι, ⟪x, b i⟫_ℝ * ⟪y, b i⟫_ℝ := by
-  rw [vectorPresentation b hb x]
-  rw [vectorPresentation b hb y]
-  rw [Orthonormal.inner_sum hb]
-  simp
-  conv =>
-    right
-    arg 2
-    intro i'
-    rw [Orthonormal.inner_left_fintype hb]
-    rw [Orthonormal.inner_left_fintype hb]
--/
-
-lemma fin_sum
-  [Fintype ι]
-  (f : ι → ι → F) :
-  ∑ r : Fin 2 → ι, f (r 0) (r 1) = ∑ r₀ : ι, (∑ r₁ : ι, f r₀ r₁) := by
-
-  rw [← Fintype.sum_prod_type']
-  apply Fintype.sum_equiv (finTwoArrowEquiv ι)
-  intro x
-  dsimp
-
-
-theorem LaplaceIndep
-  [Fintype ι] [DecidableEq ι]
-  (v₁ : OrthonormalBasis ι ℝ E)
-  (v₂ : OrthonormalBasis ι ℝ E)
-  (L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F) :
-  ∑ i, L (fun _ ↦ v₁ i) = ∑ i, L (fun _ => v₂ i) := by
-
-  have vector_vs_function
-    {y : Fin 2 → ι}
-    {v : ι → E}
-    : (fun i => v (y i)) = ![v (y 0), v (y 1)] := by
-    funext i
-    by_cases h : i = 0
-    · rw [h]
-      simp
-    · rw [Fin.eq_one_of_neq_zero i h]
-      simp
-
-  conv =>
-    right
-    arg 2
-    intro i
-    rw [v₁.sum_repr' (v₂ i)]
-    rw [BilinearCalc]
-  rw [Finset.sum_comm]
-  conv =>
-    right
-    arg 2
-    intro y
-    rw [← Finset.sum_smul]
-    rw [← c2 v₂ hv₂ (v₁ (y 0)) (v₁ (y 1))]
-    rw [vector_vs_function]
-  simp
-
-  rw [fin_sum (fun i₀ ↦ (fun i₁ ↦ ⟪v₁ i₀, v₁ i₁⟫_ℝ • L ![v₁ i₀, v₁ i₁]))]
-
-  have xx {r₀ : ι} : ∀ r₁ : ι, r₁ ≠ r₀ → ⟪v₁ r₀, v₁ r₁⟫_ℝ • L ![v₁ r₀, v₁ r₁] = 0 := by
-    intro r₁ hr₁
-    rw [orthonormal_iff_ite.1 hv₁]
-    simp
-    tauto
-
-  conv =>
-    right
-    arg 2
-    intro r₀
-    rw [Fintype.sum_eq_single r₀ xx]
-    rw [orthonormal_iff_ite.1 hv₁]
-  apply sum_congr
-  rfl
-  intro x _
-  rw [vector_vs_function]
-  simp
-
-
-noncomputable def Laplace_wrt_basis
-  [Fintype ι]
-  (v : Basis ι ℝ E)
-  (_ : Orthonormal ℝ v)
-  (f : E → F) :
-  E → F :=
-  fun z ↦ ∑ i, iteratedFDeriv ℝ 2 f z ![v i, v i]
-
-
-theorem LaplaceIndep'
-  [Fintype ι] [DecidableEq ι]
-  (v₁ : OrthonormalBasis ι ℝ E)
-  (v₂ : OrthonormalBasis ι ℝ E)
-  (f : E → F)
-  : (Laplace_wrt_basis v₁ hv₁ f) = (Laplace_wrt_basis v₂ hv₂ f) := by
-
-  funext z
-  unfold Laplace_wrt_basis
-  let XX := LaplaceIndep v₁ hv₁ v₂ hv₂ (iteratedFDeriv ℝ 2 f z)
-  have vector_vs_function
-    {v : E}
-    : ![v, v] = (fun _ => v) := by
-    funext i
-    by_cases h : i = 0
-    · rw [h]
-      simp
-    · rw [Fin.eq_one_of_neq_zero i h]
-      simp
-  conv =>
-    left
-    arg 2
-    intro i
-    rw [vector_vs_function]
-  conv =>
-    right
-    arg 2
-    intro i
-    rw [vector_vs_function]
-  assumption
-
-
-theorem LaplaceIndep''
-  [Fintype ι₁] [DecidableEq ι₁]
-  (v₁ : Basis ι₁ ℝ E)
-  (hv₁ : Orthonormal ℝ v₁)
-  [Fintype ι₂] [DecidableEq ι₂]
-  (v₂ : Basis ι₂ ℝ E)
-  (hv₂ : Orthonormal ℝ v₂)
-  (f : E → F)
-  : (Laplace_wrt_basis v₁ hv₁ f) = (Laplace_wrt_basis v₂ hv₂ f) := by
-
-  have b : ι₁ ≃ ι₂ := by
-    apply Fintype.equivOfCardEq
-    rw [← FiniteDimensional.finrank_eq_card_basis v₁]
-    rw [← FiniteDimensional.finrank_eq_card_basis v₂]
-
-  let v'₁ := Basis.reindex v₁ b
-  have hv'₁ : Orthonormal ℝ v'₁ := by
-    let A := Basis.reindex_apply v₁ b
-    have : ⇑v'₁ = v₁ ∘ b.symm := by
-      funext i
-      exact A i
-    rw [this]
-    let B := Orthonormal.comp hv₁ b.symm
-    apply B
-    exact Equiv.injective b.symm
-  rw [← LaplaceIndep' v'₁ hv'₁ v₂ hv₂ f]
-
-  unfold Laplace_wrt_basis
-  simp
-  funext z
-
-  rw [← Equiv.sum_comp b.symm]
-  apply Fintype.sum_congr
-  intro i₂
-  congr
-  rw [Basis.reindex_apply v₁ b i₂]
-
-
-noncomputable def Laplace
-  (f : E → F)
-  : E → F := by
-  exact Laplace_wrt_basis (stdOrthonormalBasis ℝ E).toBasis (stdOrthonormalBasis ℝ E).orthonormal f
-
-
-theorem LaplaceIndep'''
-  [Fintype ι] [DecidableEq ι]
-  (v : Basis ι ℝ E)
-  (hv : Orthonormal ℝ v)
-  (f : E → F)
-  : (Laplace f) = (Laplace_wrt_basis v hv f) := by
-
-  unfold Laplace
-  apply LaplaceIndep'' (stdOrthonormalBasis ℝ E).toBasis (stdOrthonormalBasis ℝ E).orthonormal v hv f
-
-
-theorem Complex.Laplace'
-  (f : ℂ → F)
-  : (Laplace f) = fun z ↦ (iteratedFDeriv ℝ 2 f z) ![1, 1] + (iteratedFDeriv ℝ 2 f z) ![Complex.I, Complex.I] := by
-
-  rw [LaplaceIndep''' Complex.orthonormalBasisOneI.toBasis Complex.orthonormalBasisOneI.orthonormal f]
-  unfold Laplace_wrt_basis
-  simp

Nevanlinna/tensor.lean

@@ -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

lake-manifest.json

@@ -1,11 +1,11 @@
-{"version": "1.0.0",
+{"version": 7,
  "packagesDir": ".lake/packages",
  "packages":
- [{"url": "https://github.com/leanprover-community/batteries",
+ [{"url": "https://github.com/leanprover/std4",
    "type": "git",
    "subDir": null,
-   "rev": "555ec79bc6effe7009036184a5f73f7d8d4850ed",
+   "rev": "e840c18f7334c751efbd4cfe531476e10c943cdb",
-   "name": "batteries",
+   "name": "std",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
    "inherited": true,
@@ -13,7 +13,7 @@
   {"url": "https://github.com/leanprover-community/quote4",
    "type": "git",
    "subDir": null,
-   "rev": "a7bfa63f5dddbcab2d4e0569c4cac74b2585e2c6",
+   "rev": "64365c656d5e1bffa127d2a1795f471529ee0178",
    "name": "Qq",
    "manifestFile": "lake-manifest.json",
    "inputRev": "master",
@@ -22,25 +22,25 @@
   {"url": "https://github.com/leanprover-community/aesop",
    "type": "git",
    "subDir": null,
-   "rev": "30619d94ce4a3d69cdb87bb1771562ca2e687cfa",
+   "rev": "5fefb40a7c9038a7150e7edd92e43b1b94c49e79",
    "name": "aesop",
    "manifestFile": "lake-manifest.json",
    "inputRev": "master",
    "inherited": true,
-   "configFile": "lakefile.toml"},
+   "configFile": "lakefile.lean"},
   {"url": "https://github.com/leanprover-community/ProofWidgets4",
    "type": "git",
    "subDir": null,
-   "rev": "87c1e7a427d8e21b6eaf8401f12897f52e2c3be9",
+   "rev": "fb65c476595a453a9b8ffc4a1cea2db3a89b9cd8",
    "name": "proofwidgets",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "v0.0.38",
+   "inputRev": "v0.0.30",
    "inherited": true,
    "configFile": "lakefile.lean"},
   {"url": "https://github.com/leanprover/lean4-cli",
    "type": "git",
    "subDir": null,
-   "rev": "a11566029bd9ec4f68a65394e8c3ff1af74c1a29",
+   "rev": "be8fa79a28b8b6897dce0713ef50e89c4a0f6ef5",
    "name": "Cli",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -49,16 +49,16 @@
   {"url": "https://github.com/leanprover-community/import-graph.git",
    "type": "git",
    "subDir": null,
-   "rev": "1588be870b9c76fe62286e8f42f0b4dafa154c96",
+   "rev": "61a79185b6582573d23bf7e17f2137cd49e7e662",
    "name": "importGraph",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
    "inherited": true,
-   "configFile": "lakefile.toml"},
+   "configFile": "lakefile.lean"},
   {"url": "https://github.com/leanprover-community/mathlib4.git",
    "type": "git",
    "subDir": null,
-   "rev": "dcc73cfb2ce3763f830c52042fb8617e762dbf60",
+   "rev": "e408da18380ef42acf7b8337c6b0536bfac09105",
    "name": "mathlib",
    "manifestFile": "lake-manifest.json",
    "inputRev": null,

lean-toolchain

@@ -1 +1 @@
-leanprover/lean4:v4.9.0-rc3
+leanprover/lean4:v4.7.0