openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Recovery /InheritedOrbitProfiles.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMaskAction | |
| import Mathlib.SetTheory.Cardinal.Finite | |
| namespace OAI | |
| /-! Finite orbit symmetries, masks and exact recovery operations. -/ | |
| namespace MatrixMultiplication.PermutationMatching | |
| open scoped BigOperators | |
| open Classical | |
| noncomputable section | |
| variable {C : Type*} {P X Y : C → Type*} | |
| [∀ c, Fintype (P c)] [∀ c, DecidableEq (X c)] [∀ c, DecidableEq (Y c)] | |
| abbrev PairProfile (P X Y : C → Type*) [∀ c, Fintype (P c)] := | |
| InheritedMasks.Profile P X × InheritedMasks.Profile P Y | |
| def pairProfile (w : CompleteWordPair P X Y) : PairProfile P X Y := | |
| (InheritedMasks.profile P X w.left, InheritedMasks.profile P Y w.right) | |
| theorem pairProfile_smul (g : HalfClassPermutations P) | |
| (w : CompleteWordPair P X Y) : pairProfile (g • w) = pairProfile w := by | |
| apply Prod.ext | |
| · apply (InheritedMasks.profile_eq_iff P X _ _).mpr | |
| exact ⟨fun c => g.1 c, fun c => by | |
| funext i | |
| simp⟩ | |
| · apply (InheritedMasks.profile_eq_iff P Y _ _).mpr | |
| exact ⟨fun c => g.2 c, fun c => by | |
| funext i | |
| simp⟩ | |
| omit [∀ c, Fintype (P c)] [∀ c, DecidableEq (X c)] [∀ c, DecidableEq (Y c)] in | |
| private theorem completeWordPair_ext {w v : CompleteWordPair P X Y} | |
| (hl : w.left = v.left) (hr : w.right = v.right) : w = v := by | |
| cases w | |
| cases v | |
| cases hl | |
| cases hr | |
| rfl | |
| theorem pairProfile_eq_iff (w v : CompleteWordPair P X Y) : | |
| pairProfile w = pairProfile v ↔ | |
| ∃ g : HalfClassPermutations P, g • w = v := by | |
| constructor | |
| · intro h | |
| have hl := (InheritedMasks.profile_eq_iff P X w.left v.left).mp | |
| (congrArg Prod.fst h) | |
| have hr := (InheritedMasks.profile_eq_iff P Y w.right v.right).mp | |
| (congrArg Prod.snd h) | |
| obtain ⟨el, hel⟩ := hl | |
| obtain ⟨er, her⟩ := hr | |
| refine ⟨(fun c => (el c)⁻¹, fun c => (er c)⁻¹), ?_⟩ | |
| apply completeWordPair_ext | |
| · funext c i | |
| simpa [Function.comp_def, Equiv.Perm.inv_def] using congrFun (hel c) i | |
| · funext c i | |
| simpa [Function.comp_def, Equiv.Perm.inv_def] using congrFun (her c) i | |
| · rintro ⟨g, rfl⟩ | |
| exact (pairProfile_smul g w).symm | |
| theorem pairProfile_eq_iff_orbitRel (w v : CompleteWordPair P X Y) : | |
| pairProfile w = pairProfile v ↔ | |
| MulAction.orbitRel (HalfClassPermutations P) (CompleteWordPair P X Y) w v := by | |
| constructor | |
| · intro h | |
| obtain ⟨g, hg⟩ := (pairProfile_eq_iff v w).mp h.symm | |
| exact ⟨g, hg⟩ | |
| · rintro ⟨g, rfl⟩ | |
| exact pairProfile_smul g v | |
| abbrev PairOrbit (P X Y : C → Type*) := | |
| MulAction.orbitRel.Quotient (HalfClassPermutations P) (CompleteWordPair P X Y) | |
| def pairOrbitProfile : PairOrbit P X Y → PairProfile P X Y := | |
| Quotient.lift pairProfile fun w v h => (pairProfile_eq_iff_orbitRel w v).mpr h | |
| theorem pairOrbitProfile_injective : | |
| Function.Injective (pairOrbitProfile (P := P) (X := X) (Y := Y)) := by | |
| intro q r | |
| induction q using Quotient.inductionOn with | |
| | h w => | |
| induction r using Quotient.inductionOn with | |
| | h v => | |
| intro h | |
| apply Quotient.sound | |
| exact (pairProfile_eq_iff_orbitRel w v).mp h | |
| variable [Fintype C] [∀ c, Fintype (X c)] [∀ c, Fintype (Y c)] | |
| theorem card_pairOrbits_le : | |
| Nat.card (PairOrbit P X Y) ≤ | |
| (∏ c, (Fintype.card (P c) + 1) ^ Fintype.card (X c)) * | |
| (∏ c, (Fintype.card (P c) + 1) ^ Fintype.card (Y c)) := by | |
| have : Finite (PairOrbit P X Y) := | |
| Finite.of_injective pairOrbitProfile pairOrbitProfile_injective | |
| let := Fintype.ofFinite (PairOrbit P X Y) | |
| rw [Nat.card_eq_fintype_card] | |
| calc | |
| _ ≤ Fintype.card (PairProfile P X Y) := | |
| Fintype.card_le_of_injective pairOrbitProfile pairOrbitProfile_injective | |
| _ = _ := by | |
| rw [Fintype.card_prod, InheritedMasks.card_profile, InheritedMasks.card_profile] | |
| theorem card_pairOrbits_le_common_bound (m : ℕ) | |
| (hm : ∀ c, Fintype.card (P c) ≤ m) : | |
| Nat.card (PairOrbit P X Y) ≤ (m + 1) ^ | |
| ((∑ c, Fintype.card (X c)) + ∑ c, Fintype.card (Y c)) := by | |
| refine (card_pairOrbits_le (P := P) (X := X) (Y := Y)).trans ?_ | |
| calc | |
| _ ≤ (∏ c, (m + 1) ^ Fintype.card (X c)) * | |
| (∏ c, (m + 1) ^ Fintype.card (Y c)) := by | |
| apply Nat.mul_le_mul | |
| · exact Finset.prod_le_prod fun c _ => | |
| Nat.pow_le_pow_left (Nat.add_le_add_right (hm c) 1) _ | |
| · exact Finset.prod_le_prod fun c _ => | |
| Nat.pow_le_pow_left (Nat.add_le_add_right (hm c) 1) _ | |
| _ = _ := by rw [Finset.prod_pow_eq_pow_sum, Finset.prod_pow_eq_pow_sum, pow_add] | |
| end | |
| end MatrixMultiplication.PermutationMatching | |
| end OAI | |