82 lines
2.4 KiB
Plaintext
82 lines
2.4 KiB
Plaintext
import Mathlib.Analysis.InnerProductSpace.Basic
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import Mathlib.Analysis.InnerProductSpace.PiL2
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import Mathlib.Algebra.BigOperators.Basic
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import Mathlib.Analysis.Calculus.ContDiff.Bounds
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import Mathlib.Analysis.Calculus.FDeriv.Symmetric
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import Mathlib.LinearAlgebra.Basis
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open BigOperators
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open Finset
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variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
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variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
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lemma vectorPresentation
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[Fintype ι]
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(b : Basis ι ℝ E)
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(hb : Orthonormal ℝ b)
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(v : E) :
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v = ∑ i, ⟪b i, v⟫_ℝ • (b i) := by
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nth_rw 1 [← (b.sum_repr v)]
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apply Fintype.sum_congr
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intro i
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rw [← Orthonormal.inner_right_finsupp hb (b.repr v) i]
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simp
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theorem LaplaceIndep
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[Fintype ι]
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(v₁ : Basis ι ℝ E)
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(hv₁ : Orthonormal ℝ v₁)
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(v₂ : Basis ι ℝ E)
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(hv₂ : Orthonormal ℝ v₂)
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(f : E → F) :
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∑ i, iteratedFDeriv ℝ 2 f z ![v₁ i, v₁ i] = ∑ i, iteratedFDeriv ℝ 2 f z ![v₂ i, v₂ i] := by
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have (v : E) : v = ∑ j, ⟪v₁ j, v⟫_ℝ • (v₁ j) :=
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sorry
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conv =>
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right
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arg 2
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intro i
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rw [this (v₂ i)]
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rw [this (v₂ i)]
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conv =>
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right
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arg 2
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intro i
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--rw [ContinuousMultilinearMap.map_sum_finset]
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have v : E := by sorry
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--let t := ![∑ j, ⟪v₁ j, v⟫_ℝ • (v₁ j), ∑ j, ⟪v₁ j, v⟫_ℝ • (v₁ j)]
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--simp at t
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--have L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F := by exact iteratedFDeriv ℝ 2 f z
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--have α : Fin 2 → Type* := by exact fun _ ↦ ι
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--have g : (i : Fin 2) → ι → E := by exact fun _ ↦ (fun j ↦ ⟪v₁ j, v⟫_ℝ • (v₁ j))
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--have A : (i : Fin 2) → Finset ι := by exact fun _ ↦ Finset.univ
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let X := ContinuousMultilinearMap.map_sum
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(iteratedFDeriv ℝ 2 f z)
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(fun _ ↦ (fun j ↦ ⟪v₁ j, v⟫_ℝ • (v₁ j)))
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--
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-- (fun _ ↦ Finset.univ)
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simp at X
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sorry
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noncomputable def Laplace₁ (n : ℕ) (f : EuclideanSpace ℝ (Fin n) → F) : EuclideanSpace ℝ (Fin n) → F := by
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let e : Fin n → EuclideanSpace ℝ (Fin n) := fun i ↦ EuclideanSpace.single i (1 : ℝ)
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exact fun z ↦ ∑ i, iteratedFDeriv ℝ 2 f z ![e i, e i]
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noncomputable def Laplace₂
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[Fintype ι]
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(v : Basis ι ℝ E)
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(hv : Orthonormal ℝ v)
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(f : E → F) :
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E → F :=
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fun z ↦ ∑ i, iteratedFDeriv ℝ 2 f z ![v i, v i]
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