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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 | import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.Data.Fintype.Sigma
import Mathlib.Logic.Equiv.Basic
import Mathlib.Basic.Real.Basic
namespace OAI
/-! Finite orbit symmetries, masks and exact recovery operations. -/
open scoped BigOperators
namespace MatrixMultiplication.OrbitCounting
variable {G X : Type*} [Group G] [MulAction G X] [Fintype G] [DecidableEq X]
def orbitSet (x : X) : Finset X := Finset.univ.image fun g : G => g • x
@[simp] theorem smul_mem_orbitSet (x : X) (g : G) : g • x ∈ orbitSet (G := G) x := by
exact Finset.mem_image.mpr ⟨g, Finset.mem_univ _, rfl⟩
theorem card_fiber_eq (x y : X) (hy : y ∈ orbitSet (G := G) x) :
Fintype.card {g : G // g • x = y} = Fintype.card {g : G // g • x = x} := by
classical
obtain ⟨r, _, hr⟩ := Finset.mem_image.mp hy
let e : {g : G // g • x = y} ≃ {g : G // g • x = x} :=
{ toFun := fun g => ⟨r⁻¹ * g.1, by
rw [mul_smul, g.2, ← hr, inv_smul_smul]⟩
invFun := fun g => ⟨r * g.1, by
rw [mul_smul, g.2, hr]⟩
left_inv := by intro g; apply Subtype.ext; simp
right_inv := by intro g; apply Subtype.ext; simp }
exact Fintype.card_congr e
theorem card_group (x : X) :
Fintype.card G = (orbitSet (G := G) x).card *
Fintype.card {g : G // g • x = x} := by
classical
let e := Equiv.sigmaSubtypeFiberEquiv (fun g : G => g • x)
(fun y => y ∈ orbitSet (G := G) x) (smul_mem_orbitSet x)
calc
Fintype.card G = Fintype.card (Σ y : {y // y ∈ orbitSet (G := G) x},
{g : G // g • x = y.1}) := (Fintype.card_congr e).symm
_ = ∑ y : {y // y ∈ orbitSet (G := G) x},
Fintype.card {g : G // g • x = y.1} := Fintype.card_sigma
_ = ∑ _y : {y // y ∈ orbitSet (G := G) x},
Fintype.card {g : G // g • x = x} := by
apply Finset.sum_congr rfl
intro y _
exact card_fiber_eq x y.1 y.2
_ = _ := by simp
theorem bad_shift_count (x : X) (bad : X → Prop) [DecidablePred bad] :
(Finset.univ.filter fun g : G => bad (g • x)).card =
((orbitSet (G := G) x).filter bad).card *
Fintype.card {g : G // g • x = x} := by
classical
let e := Equiv.sigmaSubtypeFiberEquivSubtype (fun g : G => g • x)
(p := fun g => bad (g • x))
(q := fun y => y ∈ (orbitSet (G := G) x).filter bad)
(fun g => by simp)
calc
(Finset.univ.filter fun g : G => bad (g • x)).card =
Fintype.card {g : G // bad (g • x)} := (Fintype.card_subtype _).symm
_ = Fintype.card (Σ y : {y // y ∈ (orbitSet (G := G) x).filter bad},
{g : G // g • x = y.1}) := (Fintype.card_congr e).symm
_ = ∑ y : {y // y ∈ (orbitSet (G := G) x).filter bad},
Fintype.card {g : G // g • x = y.1} := Fintype.card_sigma
_ = ∑ _y : {y // y ∈ (orbitSet (G := G) x).filter bad},
Fintype.card {g : G // g • x = x} := by
apply Finset.sum_congr rfl
intro y _
exact card_fiber_eq x y.1 (Finset.mem_filter.mp y.2).1
_ = _ := by simp
theorem bad_shift_bound (x : X) (bad : X → Prop) [DecidablePred bad] (N C : ℕ)
(h : N * ((orbitSet (G := G) x).filter bad).card ≤ C * (orbitSet (G := G) x).card) :
N * (Finset.univ.filter fun g : G => bad (g • x)).card ≤ C * Fintype.card G := by
rw [bad_shift_count, card_group (G := G) x]
simpa only [Nat.mul_assoc] using Nat.mul_le_mul_right
(Fintype.card {g : G // g • x = x}) h
theorem bad_shift_fraction (x : X) (bad : X → Prop) [DecidablePred bad] :
((Finset.univ.filter fun g : G => bad (g • x)).card : ℝ) / Fintype.card G =
(((orbitSet (G := G) x).filter bad).card : ℝ) / (orbitSet (G := G) x).card := by
have hsNat : 0 < Fintype.card {g : G // g • x = x} :=
Fintype.card_pos_iff.mpr ⟨⟨1, one_smul G x⟩⟩
have hs : (Fintype.card {g : G // g • x = x} : ℝ) ≠ 0 :=
Nat.cast_ne_zero.mpr (Nat.ne_of_gt hsNat)
rw [bad_shift_count, card_group (G := G) x]
simp only [Nat.cast_mul]
exact mul_div_mul_right _ _ hs
end MatrixMultiplication.OrbitCounting
end OAI
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