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Nevanlinna/laplace2.lean

@@ -1,51 +1,60 @@
 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]
+
+noncomputable def realInnerAsElementOfDualTensorprod
+  {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  : TensorProduct ℝ E E →ₗ[ℝ] ℝ := TensorProduct.lift bilinFormOfRealInner
+
+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]
 
 
-lemma vectorPresentation'
+theorem InvariantTensor
   [Fintype ι]
-  (b : OrthonormalBasis ι ℝ E)
-  --(hb : Orthonormal ℝ b)
-  (v : E) :
-  v = ∑ i, ⟪b i, v⟫_ℝ • (b i) := by
+  (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
 
-  let A :=  b.sum_repr v
-  let i : ι := by sorry
-  let B := b.repr v i
+  rw [L.map_sum]
+  simp_rw [L.map_smul_univ]
+  sorry
 
 
-  nth_rw 1 [← (b.sum_repr v)]
-  apply Fintype.sum_congr
-  intro i
-  --let A := b.repr v
-  --have : (b.repr v) = ((OrthonormalBasis.toBasis b).repr v) := by tauto
-  rw [← Orthonormal.inner_right_finsupp hb (b.repr v) i]
-  simp
-
-
-
-lemma vectorPresentation
-  [Fintype ι]
-  (b : Basis ι ℝ E)
-  (hb : Orthonormal ℝ b)
-  (v : E) :
-  v = ∑ i, ⟪b i, v⟫_ℝ • (b i) := by
-  nth_rw 1 [← (b.sum_repr v)]
-  apply Fintype.sum_congr
-  intro i
-  rw [← Orthonormal.inner_right_finsupp hb (b.repr v) i]
-  simp
-
 
 theorem BilinearCalc
   [Fintype ι]
@@ -55,14 +64,10 @@ theorem BilinearCalc
   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]
-  conv =>
-    left
-    arg 2
-    intro r
-    rw [L.map_smul_univ]
-  simp
-
+  simp_rw [L.map_smul_univ]
+  sorry
 
+/-
 lemma c2
   [Fintype ι]
   (b : Basis ι ℝ E)
@@ -79,7 +84,7 @@ lemma c2
     intro i'
     rw [Orthonormal.inner_left_fintype hb]
     rw [Orthonormal.inner_left_fintype hb]
-
+-/
 
 lemma fin_sum
   [Fintype ι]
@@ -94,10 +99,8 @@ lemma fin_sum
 
 theorem LaplaceIndep
   [Fintype ι] [DecidableEq ι]
-  (v₁ : Basis ι ℝ E)
-  (hv₁ : Orthonormal ℝ v₁)
-  (v₂ : Basis ι ℝ E)
-  (hv₂ : Orthonormal ℝ v₂)
+  (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
 
@@ -116,7 +119,7 @@ theorem LaplaceIndep
     right
     arg 2
     intro i
-    rw [vectorPresentation v₁ hv₁ (v₂ i)]
+    rw [v₁.sum_repr' (v₂ i)]
     rw [BilinearCalc]
   rw [Finset.sum_comm]
   conv =>
@@ -160,10 +163,8 @@ noncomputable def Laplace_wrt_basis
 
 theorem LaplaceIndep'
   [Fintype ι] [DecidableEq ι]
-  (v₁ : Basis ι ℝ E)
-  (hv₁ : Orthonormal ℝ v₁)
-  (v₂ : Basis ι ℝ E)
-  (hv₂ : Orthonormal ℝ v₂)
+  (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