openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Recovery /HistorySymmetry.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.CompleteWordPair | |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMasks | |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.ExactRecovery | |
| namespace OAI | |
| /-! Finite orbit symmetries, masks and exact recovery operations. -/ | |
| open scoped BigOperators | |
| open MatrixMultiplication.Foundation | |
| open MatrixMultiplication.PermutationMatching | |
| open MatrixMultiplication.InheritedMasks | |
| namespace MatrixMultiplication.HistorySymmetry | |
| variable {C : Type*} {P X Y : C → Type*} | |
| def completeWordPairEquiv : CompleteWordPair P X Y ≃ | |
| ((∀ c, P c → X c) × (∀ c, P c → Y c)) where | |
| toFun w := (w.left, w.right) | |
| invFun w := ⟨w.1, w.2⟩ | |
| left_inv _ := rfl | |
| right_inv _ := rfl | |
| noncomputable instance completeWordPairFintype | |
| [Fintype C] [DecidableEq C] [∀ c, Fintype (P c)] [∀ c, DecidableEq (P c)] | |
| [∀ c, Fintype (X c)] [∀ c, Fintype (Y c)] : Fintype (CompleteWordPair P X Y) := | |
| Fintype.ofEquiv _ completeWordPairEquiv.symm | |
| theorem card_completeWordPair | |
| [Fintype C] [DecidableEq C] [∀ c, Fintype (P c)] [∀ c, DecidableEq (P c)] | |
| [∀ c, Fintype (X c)] [∀ c, Fintype (Y c)] : | |
| Fintype.card (CompleteWordPair P X Y) = | |
| (∏ c, Fintype.card (X c) ^ Fintype.card (P c)) * | |
| ∏ c, Fintype.card (Y c) ^ Fintype.card (P c) := by | |
| rw [Fintype.card_congr completeWordPairEquiv] | |
| simp | |
| section Coefficients | |
| variable {K : Type*} [CommSemiring K] [Fintype C] [∀ c, Fintype (P c)] | |
| {XL YL ZL XR YR ZR : C → Type*} | |
| def pairClassProduct | |
| (TL : ∀ c, Tensor K (XL c) (YL c) (ZL c)) | |
| (TR : ∀ c, Tensor K (XR c) (YR c) (ZR c)) : | |
| Tensor K (CompleteWordPair P XL XR) (CompleteWordPair P YL YR) (CompleteWordPair P ZL ZR) := | |
| fun x y z => classProduct TL x.left y.left z.left * classProduct TR x.right y.right z.right | |
| theorem pairClassProduct_smul | |
| (TL : ∀ c, Tensor K (XL c) (YL c) (ZL c)) | |
| (TR : ∀ c, Tensor K (XR c) (YR c) (ZR c)) | |
| (g : HalfClassPermutations P) | |
| (x : CompleteWordPair P XL XR) (y : CompleteWordPair P YL YR) | |
| (z : CompleteWordPair P ZL ZR) : | |
| pairClassProduct TL TR (g • x) (g • y) (g • z) = pairClassProduct TL TR x y z := by | |
| change classProduct TL (fun c => x.left c ∘ (g.1 c).symm) | |
| (fun c => y.left c ∘ (g.1 c).symm) (fun c => z.left c ∘ (g.1 c).symm) * | |
| classProduct TR (fun c => x.right c ∘ (g.2 c).symm) | |
| (fun c => y.right c ∘ (g.2 c).symm) (fun c => z.right c ∘ (g.2 c).symm) = _ | |
| rw [classProduct_permutation, classProduct_permutation] | |
| rfl | |
| end Coefficients | |
| section Windows | |
| variable [Fintype C] [∀ c, Fintype (P c)] [∀ c, DecidableEq (P c)] | |
| {SL SR : C → Type*} [∀ c, Fintype (SL c)] [∀ c, Fintype (SR c)] | |
| [∀ c, DecidableEq (SL c)] [∀ c, DecidableEq (SR c)] | |
| def childWindows (sl : ∀ c, X c → SL c) (sr : ∀ c, Y c → SR c) | |
| (νl : ∀ c, SL c → ℝ) (νr : ∀ c, SR c → ℝ) (ηl ηr : C → ℝ) | |
| (w : CompleteWordPair P X Y) : Prop := | |
| ∀ c, typeWindow (νl c) (ηl c) (sl c ∘ w.left c) ∧ | |
| typeWindow (νr c) (ηr c) (sr c ∘ w.right c) | |
| omit [Fintype C] [∀ index, DecidableEq (P index)] | |
| [∀ index, Fintype (SL index)] [∀ index, Fintype (SR index)] in | |
| theorem childWindows_smul | |
| (sl : ∀ c, X c → SL c) (sr : ∀ c, Y c → SR c) | |
| (νl : ∀ c, SL c → ℝ) (νr : ∀ c, SR c → ℝ) (ηl ηr : C → ℝ) | |
| (g : HalfClassPermutations P) (w : CompleteWordPair P X Y) : | |
| childWindows sl sr νl νr ηl ηr (g • w) ↔ childWindows sl sr νl νr ηl ηr w := by | |
| unfold childWindows | |
| change (∀ c, typeWindow (νl c) (ηl c) ((sl c ∘ w.left c) ∘ (g.1 c).symm) ∧ | |
| typeWindow (νr c) (ηr c) ((sr c ∘ w.right c) ∘ (g.2 c).symm)) ↔ _ | |
| simp only [typeWindow_permutation] | |
| end Windows | |
| theorem delete_invariant {K G U V W : Type*} [CommSemiring K] [Group G] | |
| [MulAction G U] [MulAction G V] [MulAction G W] | |
| (Q : Tensor K U V W) (px : U → Prop) (py : V → Prop) (pz : W → Prop) | |
| [DecidablePred px] [DecidablePred py] [DecidablePred pz] | |
| (hQ : ∀ (g : G) x y z, Q (g • x) (g • y) (g • z) = Q x y z) | |
| (hx : ∀ (g : G) x, px (g • x) ↔ px x) | |
| (hy : ∀ (g : G) y, py (g • y) ↔ py y) | |
| (hz : ∀ (g : G) z, pz (g • z) ↔ pz z) | |
| (g : G) (x : U) (y : V) (z : W) : | |
| ExactRecovery.delete Q px py pz (g • x) (g • y) (g • z) = | |
| ExactRecovery.delete Q px py pz x y z := by | |
| simp only [ExactRecovery.delete, hx, hy, hz, hQ] | |
| theorem delete_used_left {K U V W : Type*} [CommSemiring K] | |
| (Q : Tensor K U V W) (px : U → Prop) (py : V → Prop) (pz : W → Prop) | |
| [DecidablePred px] [DecidablePred py] [DecidablePred pz] (x : U) | |
| (h : ∃ y z, ExactRecovery.delete Q px py pz x y z ≠ 0) : px x := by | |
| obtain ⟨y, z, h⟩ := h | |
| by_contra hn | |
| simp [ExactRecovery.delete, hn] at h | |
| theorem delete_used_middle {K U V W : Type*} [CommSemiring K] | |
| (Q : Tensor K U V W) (px : U → Prop) (py : V → Prop) (pz : W → Prop) | |
| [DecidablePred px] [DecidablePred py] [DecidablePred pz] (y : V) | |
| (h : ∃ x z, ExactRecovery.delete Q px py pz x y z ≠ 0) : py y := by | |
| obtain ⟨x, z, h⟩ := h | |
| by_contra hn | |
| simp [ExactRecovery.delete, hn] at h | |
| theorem delete_used_right {K U V W : Type*} [CommSemiring K] | |
| (Q : Tensor K U V W) (px : U → Prop) (py : V → Prop) (pz : W → Prop) | |
| [DecidablePred px] [DecidablePred py] [DecidablePred pz] (z : W) | |
| (h : ∃ x y, ExactRecovery.delete Q px py pz x y z ≠ 0) : pz z := by | |
| obtain ⟨x, y, h⟩ := h | |
| by_contra hn | |
| simp [ExactRecovery.delete, hn] at h | |
| end MatrixMultiplication.HistorySymmetry | |
| end OAI | |