`BookProof.TensorCore.derPow_symm_tmul` (hA : SymmetricOn D₂ A) (n : ℕ) (ih : ∀ x y : ((domSpace Hs D₂).pow n), (inner ℂ (derPow Hs D₂ A n x) (inclPow Hs D₂ n y) : ℂ) = inner ℂ (in

Proved

BookProof.TensorCore.derPow_symm_tmul

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterTensorGraphCore.

BookProof.TensorCore.derPow_symm_tmul (hA : SymmetricOn D₂ A) (n : ℕ) (ih : ∀ x y : ((domSpace Hs D₂).pow n), (inner ℂ (derPow Hs D₂ A n x) (inclPow Hs D₂ n y) : ℂ) = inner ℂ (inclPow Hs D₂ n x) (derPow Hs D₂ A n y)) (a c : D₂) (b d : ((domSpace Hs D₂).pow n)) : (inner ℂ (derPow Hs D₂ A (n + 1) (a ⊗ₜ[ℂ] b)) (inclPow Hs D₂ (n + 1) (c ⊗ₜ[ℂ] d)) : ℂ) = inner ℂ (inclPow Hs D₂ (n + 1) (a ⊗ₜ[ℂ] b)) (derPow Hs D₂ A (n + 1) (c ⊗ₜ[ℂ] d))

Formalization note: Lean 4 identifier BookProof.TensorCore.derPow_symm_tmul.

Preamble

-- Generated from ChapterTensorGraphCore.lean — theorem BookProof.TensorCore.derPow_symm_tmul
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterGraphCoreTransfer
import Mathlib
import Definitions.Def_ChapterTensorGraphCore
import Definitions.Def_ChapterFarisLavineCore
open BookProof.TensorCore



open scoped TensorProduct
open BookProof.FarisLavine BookProof.GraphCore

noncomputable section

variable (Hs : IPSpace) (D₂ : Submodule ℂ Hs.carrier)
variable (A : D₂ →ₗ[ℂ] Hs.carrier)
variable (D : Submodule ℂ Hs.carrier)

Formal statement

theorem BookProof.TensorCore.derPow_symm_tmul (hA : SymmetricOn D₂ A) (n : ℕ)
    (ih : ∀ x y : ((domSpace Hs D₂).pow n),
      (inner ℂ (derPow Hs D₂ A n x) (inclPow Hs D₂ n y) : ℂ)
        = inner ℂ (inclPow Hs D₂ n x) (derPow Hs D₂ A n y))
    (a c : D₂) (b d : ((domSpace Hs D₂).pow n)) :
    (inner ℂ (derPow Hs D₂ A (n + 1) (a ⊗ₜ[ℂ] b)) (inclPow Hs D₂ (n + 1) (c ⊗ₜ[ℂ] d)) : ℂ)
      = inner ℂ (inclPow Hs D₂ (n + 1) (a ⊗ₜ[ℂ] b)) (derPow Hs D₂ A (n + 1) (c ⊗ₜ[ℂ] d)) := by sorry

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