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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 | import OAI.LinearAlgebra.MatrixMultiplication.Entropy.FiniteAverage
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Fintype.Perm
namespace OAI
/-! Finite orbit symmetries, masks and exact recovery operations. -/
noncomputable section
namespace MatrixMultiplication.PermutationMatching
open scoped BigOperators
variable {P : Type*} [DecidableEq P]
def membership (B : Finset P) (p : Equiv.Perm P) (i : P) : ℝ :=
if p i ∈ B then 1 else 0
@[simp] theorem membership_sq (B : Finset P) (p : Equiv.Perm P) (i : P) :
membership B p i * membership B p i = membership B p i := by
simp only [membership]
split_ifs <;> norm_num
theorem membership_nonneg (B : Finset P) (p : Equiv.Perm P) (i : P) :
0 ≤ membership B p i := by
simp only [membership]
split_ifs <;> norm_num
theorem membership_le_one (B : Finset P) (p : Equiv.Perm P) (i : P) :
membership B p i ≤ 1 := by
simp only [membership]
split_ifs <;> norm_num
variable [Fintype P]
theorem sum_membership (B : Finset P) (p : Equiv.Perm P) :
(∑ i : P, membership B p i) = (B.card : ℝ) := by
unfold membership
rw [Equiv.sum_comp p (fun i => if i ∈ B then (1 : ℝ) else 0)]
simp
theorem sum_perm_right (f : Equiv.Perm P → ℝ) (q : Equiv.Perm P) :
(∑ p : Equiv.Perm P, f (p * q)) = ∑ p : Equiv.Perm P, f p := by
exact Equiv.sum_comp (Equiv.mulRight q) f
theorem sum_membership_position_eq (B : Finset P) (i j : P) :
(∑ p : Equiv.Perm P, membership B p i) =
∑ p : Equiv.Perm P, membership B p j := by
calc
_ = ∑ p : Equiv.Perm P, membership B (p * Equiv.swap i j) j := by
simp [membership, Equiv.Perm.mul_apply]
_ = _ := sum_perm_right (fun p => membership B p j) _
theorem card_mul_sum_membership (B : Finset P) (i : P) :
(Fintype.card P : ℝ) * (∑ p : Equiv.Perm P, membership B p i) =
(Fintype.card (Equiv.Perm P) : ℝ) * B.card := by
calc
_ = ∑ j : P, ∑ p : Equiv.Perm P, membership B p i := by simp
_ = ∑ j : P, ∑ p : Equiv.Perm P, membership B p j := by
apply Finset.sum_congr rfl
intro j _
exact sum_membership_position_eq B i j
_ = ∑ p : Equiv.Perm P, ∑ j : P, membership B p j := Finset.sum_comm
_ = _ := by simp [sum_membership]
theorem average_membership [Nonempty P] (B : Finset P) (i : P) :
average (fun p : Equiv.Perm P => membership B p i) =
(B.card : ℝ) / Fintype.card P := by
have hn : (Fintype.card P : ℝ) ≠ 0 := by
exact_mod_cast Fintype.card_ne_zero
have hp : (Fintype.card (Equiv.Perm P) : ℝ) ≠ 0 := by
exact_mod_cast Fintype.card_ne_zero
unfold average
apply (div_eq_div_iff hp hn).2
simpa only [mul_comm] using card_mul_sum_membership B i
theorem sum_membership_mul_position_eq (B : Finset P) {i j k : P}
(hij : i ≠ j) (hik : i ≠ k) :
(∑ p : Equiv.Perm P, membership B p i * membership B p j) =
∑ p : Equiv.Perm P, membership B p i * membership B p k := by
calc
_ = ∑ p : Equiv.Perm P,
membership B (p * Equiv.swap j k) i *
membership B (p * Equiv.swap j k) k := by
simp [membership, Equiv.Perm.mul_apply,
Equiv.swap_apply_of_ne_of_ne hij hik]
_ = _ := sum_perm_right (fun p => membership B p i * membership B p k) _
theorem card_sub_one_mul_sum_membership_mul (B : Finset P) {i j : P}
(hij : i ≠ j) :
((Fintype.card P : ℝ) - 1) *
(∑ p : Equiv.Perm P, membership B p i * membership B p j) =
((B.card : ℝ) - 1) * (∑ p : Equiv.Perm P, membership B p i) := by
have hcard : (((Finset.univ : Finset P).erase i).card : ℝ) =
(Fintype.card P : ℝ) - 1 := by
have h := congrArg (fun n : ℕ => (n : ℝ))
(Finset.card_erase_add_one (Finset.mem_univ i))
simp only [Nat.cast_add, Nat.cast_one, Finset.card_univ] at h
linarith
have herase :
(∑ k ∈ (Finset.univ : Finset P).erase i,
∑ p : Equiv.Perm P, membership B p i * membership B p k) =
((Fintype.card P : ℝ) - 1) *
(∑ p : Equiv.Perm P, membership B p i * membership B p j) := by
calc
_ = ∑ k ∈ (Finset.univ : Finset P).erase i,
∑ p : Equiv.Perm P, membership B p i * membership B p j := by
apply Finset.sum_congr rfl
intro k hk
exact sum_membership_mul_position_eq B (Finset.mem_erase.mp hk).1.symm hij
_ = _ := by simp only [Finset.sum_const, nsmul_eq_mul, hcard]
have htotal :
(∑ k : P, ∑ p : Equiv.Perm P, membership B p i * membership B p k) =
(B.card : ℝ) * (∑ p : Equiv.Perm P, membership B p i) := by
rw [Finset.sum_comm]
simp_rw [← Finset.mul_sum, sum_membership]
rw [← Finset.sum_mul, mul_comm]
have hsplit := Finset.sum_erase_add (Finset.univ : Finset P)
(fun k => ∑ p : Equiv.Perm P, membership B p i * membership B p k)
(Finset.mem_univ i)
rw [herase, htotal] at hsplit
simp only [membership_sq] at hsplit
nlinarith
theorem average_membership_mul_of_ne (B : Finset P) {i j : P} (hij : i ≠ j) :
average (fun p : Equiv.Perm P => membership B p i * membership B p j) =
(B.card : ℝ) * ((B.card : ℝ) - 1) /
((Fintype.card P : ℝ) * ((Fintype.card P : ℝ) - 1)) := by
have hnlarge : 1 < Fintype.card P := Fintype.one_lt_card_iff.mpr ⟨i, j, hij⟩
have hnlarge' : (1 : ℝ) < Fintype.card P := by exact_mod_cast hnlarge
have hn : (Fintype.card P : ℝ) ≠ 0 := by linarith
have hn1 : (Fintype.card P : ℝ) - 1 ≠ 0 := by linarith
have hp : (Fintype.card (Equiv.Perm P) : ℝ) ≠ 0 := by
exact_mod_cast Fintype.card_ne_zero
unfold average
apply (div_eq_div_iff hp (mul_ne_zero hn hn1)).2
calc
_ = (Fintype.card P : ℝ) * (((Fintype.card P : ℝ) - 1) *
(∑ p : Equiv.Perm P, membership B p i * membership B p j)) := by ring
_ = (Fintype.card P : ℝ) * (((B.card : ℝ) - 1) *
(∑ p : Equiv.Perm P, membership B p i)) := by
rw [card_sub_one_mul_sum_membership_mul B hij]
_ = ((B.card : ℝ) - 1) * ((Fintype.card P : ℝ) *
(∑ p : Equiv.Perm P, membership B p i)) := by ring
_ = ((B.card : ℝ) - 1) * ((Fintype.card (Equiv.Perm P) : ℝ) * B.card) := by
rw [card_mul_sum_membership B i]
_ = _ := by ring
end MatrixMultiplication.PermutationMatching
end
end OAI
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