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f18e7de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 | 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
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