| import Mathlib |
| import OAI.LinearAlgebra.MatrixFields.Tensors.TerminalRelabel |
|
|
| namespace OAI |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| section |
| open scoped BigOperators |
| open MatrixMultiplication.Foundation |
|
|
| namespace MatrixMultiplication.CommonDimensions |
|
|
| variable {K H : Type*} [CommSemiring K] [Fintype H] [DecidableEq H] |
|
|
| def familyProduct {X Y Z : H → Type*} (T : ∀ h, Tensor K (X h) (Y h) (Z h)) : |
| Tensor K (∀ h, X h) (∀ h, Y h) (∀ h, Z h) := |
| fun x y z => ∏ h, T h (x h) (y h) (z h) |
|
|
| def pairWords (A B : H → Type*) : |
| ((∀ h, A h) × (∀ h, B h)) ≃ (∀ h, A h × B h) where |
| toFun x h := (x.1 h, x.2 h) |
| invFun x := (fun h => (x h).1, fun h => (x h).2) |
| left_inv _ := rfl |
| right_inv _ := rfl |
|
|
| omit [DecidableEq H] in |
| theorem matrix_familyProduct (A B C : H → Type*) |
| [∀ h, DecidableEq (A h)] [∀ h, DecidableEq (B h)] [∀ h, DecidableEq (C h)] : |
| Tensor.pullback (pairWords A B) (pairWords B C) (pairWords C A) |
| (familyProduct (fun h => Tensor.matrixCoefficients (K := K) (A h) (B h) (C h))) = |
| Tensor.matrixCoefficients (∀ h, A h) (∀ h, B h) (∀ h, C h) := by |
| classical |
| funext x y z |
| simp only [Tensor.pullback, pairWords, familyProduct, Tensor.matrixCoefficients, |
| Equiv.coe_fn_mk] |
| by_cases matched : x.2 = y.1 ∧ y.2 = z.1 ∧ z.2 = x.1 |
| · simp [matched.1, matched.2.1, matched.2.2] |
| · have mismatch : ¬ ∀ index : H, |
| x.2 index = y.1 index ∧ y.2 index = z.1 index ∧ z.2 index = x.1 index := by |
| intro pointwise |
| exact matched ⟨funext (fun index => (pointwise index).1), |
| funext (fun index => (pointwise index).2.1), |
| funext (fun index => (pointwise index).2.2)⟩ |
| obtain ⟨index, mismatch⟩ := not_forall.mp mismatch |
| simp only [matched, ite_false] |
| apply Finset.prod_eq_zero (Finset.mem_univ index) |
| simp only [mismatch, ite_false] |
|
|
| theorem matrix_familyProduct_restrict (A B C : H → Type*) |
| [∀ h, Fintype (A h)] [∀ h, Fintype (B h)] [∀ h, Fintype (C h)] |
| [∀ h, DecidableEq (A h)] [∀ h, DecidableEq (B h)] [∀ h, DecidableEq (C h)] : |
| ∃ (a : ((∀ h, A h) × (∀ h, B h)) → (∀ h, A h × B h) → K) |
| (b : ((∀ h, B h) × (∀ h, C h)) → (∀ h, B h × C h) → K) |
| (c : ((∀ h, C h) × (∀ h, A h)) → (∀ h, C h × A h) → K), |
| Tensor.restrict a b c |
| (familyProduct (fun h => Tensor.matrixCoefficients (K := K) (A h) (B h) (C h))) = |
| Tensor.matrixCoefficients (∀ h, A h) (∀ h, B h) (∀ h, C h) := by |
| classical |
| refine ⟨(fun x s => if s = pairWords A B x then 1 else 0), |
| (fun y s => if s = pairWords B C y then 1 else 0), |
| (fun z s => if s = pairWords C A z then 1 else 0), ?_⟩ |
| rw [← Tensor.pullback_eq_restrict] |
| exact matrix_familyProduct A B C |
|
|
| theorem dimension_card (a : H → ℕ) : |
| Fintype.card (∀ h, Fin (a h)) = ∏ h, a h := by simp |
|
|
| theorem volume_card (a b c : H → ℕ) : |
| Fintype.card (∀ h, Fin (a h)) * Fintype.card (∀ h, Fin (b h)) * |
| Fintype.card (∀ h, Fin (c h)) = ∏ h, a h * b h * c h := by |
| simp [Finset.prod_mul_distrib] |
|
|
| theorem common_oriented_dimensions {S : Type*} |
| (a b c : S → H → ℕ) (a₀ b₀ c₀ : H → ℕ) |
| (ha : ∀ s h, a s h = a₀ h) (hb : ∀ s h, b s h = b₀ h) |
| (hc : ∀ s h, c s h = c₀ h) (s : S) : |
| (Fintype.card (∀ h, Fin (a s h)), Fintype.card (∀ h, Fin (b s h)), |
| Fintype.card (∀ h, Fin (c s h))) = |
| (∏ h, a₀ h, ∏ h, b₀ h, ∏ h, c₀ h) := by |
| simp [ha, hb, hc] |
|
|
| end MatrixMultiplication.CommonDimensions |
|
|
| end |
|
|
| end MatrixAllFields |
|
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| end OAI |
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|
