openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Tensor /ComplexSharedAuxiliaryBank.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexLocalMaps | |
| import Mathlib.Logic.Equiv.Basic | |
| namespace OAI | |
| /-! Finite coefficient tensors and their algebraic transformations. -/ | |
| open scoped BigOperators | |
| namespace MatrixMultiplication.Foundation.LabelHierarchySeparation | |
| variable {K I : Type*} [CommSemiring K] [Fintype I] | |
| variable {X Y Z : I → Type*} | |
| def bankTensor (T : ∀ i, Tensor K (X i) (Y i) (Z i)) : | |
| Tensor K (∀ i, X i) (∀ i, Y i) (∀ i, Z i) := | |
| fun x y z => ∏ i, T i (x i) (y i) (z i) | |
| theorem bankTensor_rankAtMost (T : ∀ i, Tensor K (X i) (Y i) (Z i)) | |
| (r : I → ℕ) (h : ∀ i, Tensor.RankAtMost (T i) (r i)) : | |
| Tensor.RankAtMost (bankTensor T) (∏ i, r i) := by | |
| classical | |
| choose a b c hc using h | |
| let aa : (∀ i, Fin (r i)) → (∀ i, X i) → K := | |
| fun f x => ∏ i, a i (f i) (x i) | |
| let bb : (∀ i, Fin (r i)) → (∀ i, Y i) → K := | |
| fun f y => ∏ i, b i (f i) (y i) | |
| let cc : (∀ i, Fin (r i)) → (∀ i, Z i) → K := | |
| fun f z => ∏ i, c i (f i) (z i) | |
| have heq : bankTensor T = | |
| fun x y z => ∑ f, Tensor.rankOne (aa f) (bb f) (cc f) x y z := by | |
| funext x y z | |
| simp only [bankTensor, hc, Fintype.prod_sum] | |
| apply Finset.sum_congr rfl | |
| intro f hf | |
| simp only [Tensor.rankOne, aa, bb, cc, Finset.prod_mul_distrib] | |
| rw [heq] | |
| simpa only [Fintype.card_pi, Fintype.card_fin] using | |
| Tensor.rankAtMost_sum_rankOne aa bb cc | |
| theorem bankTensor_reindex {J : Type*} [Fintype J] | |
| {X' Y' Z' : J → Type*} (e : I ≃ J) | |
| (ex : ∀ i, X i ≃ X' (e i)) (ey : ∀ i, Y i ≃ Y' (e i)) | |
| (ez : ∀ i, Z i ≃ Z' (e i)) | |
| (T : ∀ i, Tensor K (X i) (Y i) (Z i)) | |
| (S : ∀ j, Tensor K (X' j) (Y' j) (Z' j)) | |
| (compatible : ∀ i x y z, S (e i) (ex i x) (ey i y) (ez i z) = T i x y z) : | |
| Tensor.pullback (e.piCongr ex) (e.piCongr ey) (e.piCongr ez) | |
| (bankTensor S) = bankTensor T := by | |
| funext x y z | |
| change (∏ j, S j ((e.piCongr ex) x j) ((e.piCongr ey) y j) | |
| ((e.piCongr ez) z j)) = ∏ i, T i (x i) (y i) (z i) | |
| rw [← e.prod_comp (fun j => S j ((e.piCongr ex) x j) | |
| ((e.piCongr ey) y j) ((e.piCongr ez) z j))] | |
| apply Finset.prod_congr rfl | |
| intro i hi | |
| simpa only [Equiv.piCongr_apply_apply] using compatible i (x i) (y i) (z i) | |
| theorem bankTensor_reindex_rankAtMost {J : Type*} [Fintype J] | |
| {X' Y' Z' : J → Type*} (e : I ≃ J) | |
| (ex : ∀ i, X i ≃ X' (e i)) (ey : ∀ i, Y i ≃ Y' (e i)) | |
| (ez : ∀ i, Z i ≃ Z' (e i)) | |
| (T : ∀ i, Tensor K (X i) (Y i) (Z i)) | |
| (S : ∀ j, Tensor K (X' j) (Y' j) (Z' j)) | |
| (compatible : ∀ i x y z, S (e i) (ex i x) (ey i y) (ez i z) = T i x y z) | |
| {r : ℕ} (hS : Tensor.RankAtMost (bankTensor S) r) : | |
| Tensor.RankAtMost (bankTensor T) r := by | |
| rw [← bankTensor_reindex e ex ey ez T S compatible] | |
| exact hS.pullback _ _ _ | |
| end MatrixMultiplication.Foundation.LabelHierarchySeparation | |
| end OAI | |