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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 | import OAI.LinearAlgebra.MatrixMultiplication.Recovery.PermutationMatchingMoments
namespace OAI
/-! Finite orbit symmetries, masks and exact recovery operations. -/
noncomputable section
namespace MatrixMultiplication.PermutationMatching
open scoped BigOperators
open Classical
variable {P : Type*} [Fintype P] [DecidableEq P]
def twoHalfCount (A B : Finset P) (left right : Equiv.Perm P) : ℝ :=
∑ i : P, if left i ∈ A ∧ right i ∈ B then 1 else 0
theorem twoHalfCount_eq_matchingCount (A B : Finset P)
(left right : Equiv.Perm P) :
twoHalfCount A B left right =
matchingCount A B (left.symm.trans right) := by
classical
unfold twoHalfCount matchingCount membership
calc
_ = ∑ i : P, if i ∈ A ∧ right (left.symm i) ∈ B then (1 : ℝ) else 0 := by
simpa only [Equiv.symm_apply_apply] using
Equiv.sum_comp left (fun i =>
if i ∈ A ∧ right (left.symm i) ∈ B then (1 : ℝ) else 0)
_ = ∑ i : P, if i ∈ A then
(if right (left.symm i) ∈ B then (1 : ℝ) else 0) else 0 := by
apply Finset.sum_congr rfl
intro i _
split_ifs <;> simp_all
_ = _ := by
simp only [Equiv.trans_apply, Fintype.sum_ite_mem]
apply Finset.sum_congr rfl
intro i _
apply if_congr <;> rfl
def relativePairEquiv :
(Equiv.Perm P × Equiv.Perm P) ≃ (Equiv.Perm P × Equiv.Perm P) where
toFun lr := (lr.1, lr.1.symm.trans lr.2)
invFun lq := (lq.1, lq.1.trans lq.2)
left_inv := by
rintro ⟨left, right⟩
dsimp only
apply Prod.ext
· rfl
· ext i
simp
right_inv := by
rintro ⟨left, relative⟩
dsimp only
apply Prod.ext
· rfl
· ext i
simp
theorem average_relativePerm (f : Equiv.Perm P → ℝ) :
average (fun lr : Equiv.Perm P × Equiv.Perm P =>
f (lr.1.symm.trans lr.2)) = average f := by
calc
_ = average (fun lq : Equiv.Perm P × Equiv.Perm P => f lq.2) :=
average_equiv relativePairEquiv (fun lq => f lq.2)
_ = _ := average_prod_snd f
theorem average_twoHalfCount (A B : Finset P) (f : ℝ → ℝ) :
average (fun lr : Equiv.Perm P × Equiv.Perm P =>
f (twoHalfCount A B lr.1 lr.2)) =
average (fun p : Equiv.Perm P => f (matchingCount A B p)) := by
simp_rw [twoHalfCount_eq_matchingCount]
exact average_relativePerm (fun p => f (matchingCount A B p))
theorem twoHalfCount_statisticPositions {X Y S T : Type*}
(leftWord : P → X) (rightWord : P → Y) (sl : X → S) (sr : Y → T)
(s : S) (t : T) (left right : Equiv.Perm P) :
twoHalfCount (statisticPositions leftWord sl s)
(statisticPositions rightWord sr t) left right =
∑ i : P, if sl (leftWord (left i)) = s ∧ sr (rightWord (right i)) = t
then (1 : ℝ) else 0 := by
classical
simp [twoHalfCount, statisticPositions]
section Families
variable {C : Type*} (Q : C → Type*)
def relativeFamily
(left right : ∀ c, Equiv.Perm (Q c)) : ∀ c, Equiv.Perm (Q c) :=
fun c => (left c).symm.trans (right c)
def relativeFamilyPairEquiv :
((∀ c, Equiv.Perm (Q c)) × (∀ c, Equiv.Perm (Q c))) ≃
((∀ c, Equiv.Perm (Q c)) × (∀ c, Equiv.Perm (Q c))) where
toFun lr := (lr.1, relativeFamily Q lr.1 lr.2)
invFun lq := (lq.1, fun c => (lq.1 c).trans (lq.2 c))
left_inv := by
rintro ⟨left, right⟩
dsimp only
apply Prod.ext
· rfl
· funext c
ext i
simp [relativeFamily]
right_inv := by
rintro ⟨left, relative⟩
dsimp only
apply Prod.ext
· rfl
· funext c
ext i
simp [relativeFamily]
variable [Fintype C] [DecidableEq C]
[∀ c, Fintype (Q c)] [∀ c, DecidableEq (Q c)]
theorem average_relativeFamily (f : (∀ c, Equiv.Perm (Q c)) → ℝ) :
average (fun lr :
(∀ c, Equiv.Perm (Q c)) × (∀ c, Equiv.Perm (Q c)) =>
f (relativeFamily Q lr.1 lr.2)) = average f := by
calc
_ = average (fun lq :
(∀ c, Equiv.Perm (Q c)) × (∀ c, Equiv.Perm (Q c)) => f lq.2) :=
average_equiv (relativeFamilyPairEquiv Q) (fun lq => f lq.2)
_ = _ := average_prod_snd f
omit [DecidableEq C] in
theorem sum_twoHalfCount_eq_matchingCount
(A B : ∀ c, Finset (Q c)) (left right : ∀ c, Equiv.Perm (Q c)) :
(∑ c, twoHalfCount (A c) (B c) (left c) (right c)) =
∑ c, matchingCount (A c) (B c) (relativeFamily Q left right c) := by
simp only [twoHalfCount_eq_matchingCount, relativeFamily]
end Families
end MatrixMultiplication.PermutationMatching
end
end OAI
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