| 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 |
| classical |
| obtain ⟨r, _, hr⟩ := Finset.mem_image.mp hy |
| let e : {g : G |
| { 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 |
| 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 |
| {g : G |
| _ = ∑ y : {y |
| Fintype.card {g : G |
| _ = ∑ _y : {y |
| Fintype.card {g : G |
| 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 |
| 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 |
| _ = Fintype.card (Σ y : {y |
| {g : G |
| _ = ∑ y : {y |
| Fintype.card {g : G |
| _ = ∑ _y : {y |
| Fintype.card {g : G |
| 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 |
|
|
| 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 |
| Fintype.card_pos_iff.mpr ⟨⟨1, one_smul G x⟩⟩ |
| have hs : (Fintype.card {g : G |
| 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 |
|
|
