File size: 4,256 Bytes
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 | import OAI.LinearAlgebra.MatrixMultiplication.Recovery.EquivOrbitTransport
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
namespace MatrixMultiplication.JointOrbitFibers
attribute [local instance] Classical.propDecidable
variable {G X Y : Type*} [Group G] [MulAction G X] [Fintype X]
noncomputable def fullFiber (q : MulAction.orbitRel.Quotient G X) : Finset X :=
Finset.univ.filter fun x => (Quotient.mk'' x : MulAction.orbitRel.Quotient G X) = q
@[simp] theorem mem_fullFiber {q : MulAction.orbitRel.Quotient G X} {x : X} :
x ∈ fullFiber q ↔ (Quotient.mk'' x : MulAction.orbitRel.Quotient G X) = q := by
simp only [fullFiber, Finset.mem_filter, Finset.mem_univ, true_and]
@[simp] theorem mem_fullFiber_mk {x y : X} :
y ∈ fullFiber (Quotient.mk'' x : MulAction.orbitRel.Quotient G X) ↔
∃ g : G, g • x = y := by
rw [mem_fullFiber]
constructor
· intro h
exact Quotient.exact h
· intro h
exact Quotient.sound h
theorem fullFiber_nonempty (q : MulAction.orbitRel.Quotient G X) :
(fullFiber q).Nonempty := by
induction q using Quotient.inductionOn with
| h x => exact ⟨x, mem_fullFiber.mpr rfl⟩
theorem fullFiber_card_pos (q : MulAction.orbitRel.Quotient G X) :
0 < (fullFiber q).card :=
Finset.card_pos.mpr (fullFiber_nonempty q)
section FiniteGroup
variable [Fintype G] [DecidableEq X]
theorem fullFiber_mk (x : X) :
fullFiber (Quotient.mk'' x : MulAction.orbitRel.Quotient G X) =
OrbitCounting.orbitSet (G := G) x := by
ext y
rw [mem_fullFiber_mk]
simp only [OrbitCounting.orbitSet, Finset.mem_image, Finset.mem_univ, true_and]
theorem fullFiber_eq_orbitSet (q : MulAction.orbitRel.Quotient G X) (x : X)
(hx : x ∈ fullFiber q) :
fullFiber q = OrbitCounting.orbitSet (G := G) x := by
rw [← mem_fullFiber.mp hx, fullFiber_mk]
end FiniteGroup
noncomputable def transportedFiber (e : Y ≃ X)
(q : MulAction.orbitRel.Quotient G X) : Finset Y :=
(fullFiber q).map e.symm.toEmbedding
@[simp] theorem mem_transportedFiber (e : Y ≃ X)
{q : MulAction.orbitRel.Quotient G X} {y : Y} :
y ∈ transportedFiber e q ↔
(Quotient.mk'' (e y) : MulAction.orbitRel.Quotient G X) = q := by
constructor
· intro h
obtain ⟨x, hx, hxy⟩ := Finset.mem_map.mp h
have he : e y = x := by
rw [← hxy]
exact e.apply_symm_apply x
rw [he]
exact mem_fullFiber.mp hx
· intro h
exact Finset.mem_map.mpr ⟨e y, mem_fullFiber.mpr h, e.symm_apply_apply y⟩
@[simp] theorem transportedFiber_card (e : Y ≃ X)
(q : MulAction.orbitRel.Quotient G X) :
(transportedFiber e q).card = (fullFiber q).card :=
Finset.card_map _
theorem transportedFiber_card_pos (e : Y ≃ X)
(q : MulAction.orbitRel.Quotient G X) :
0 < (transportedFiber e q).card := by
rw [transportedFiber_card]
exact fullFiber_card_pos q
theorem transportedFiber_mk [Fintype G] [DecidableEq X] [DecidableEq Y]
(e : Y ≃ X) (y : Y) :
letI : MulAction G Y := EquivOrbitTransport.action e
transportedFiber e
(Quotient.mk'' (e y) : MulAction.orbitRel.Quotient G X) =
OrbitCounting.orbitSet (G := G) y := by
let : MulAction G Y := EquivOrbitTransport.action e
ext z
rw [mem_transportedFiber]
constructor
· intro h
have hz : e z ∈ fullFiber
(Quotient.mk'' (e y) : MulAction.orbitRel.Quotient G X) :=
mem_fullFiber.mpr h
obtain ⟨g, hg⟩ := mem_fullFiber_mk.mp hz
apply Finset.mem_image.mpr
refine ⟨g, Finset.mem_univ _, ?_⟩
change e.symm (g • e y) = z
rw [hg, e.symm_apply_apply]
· intro h
obtain ⟨g, _, hg⟩ := Finset.mem_image.mp h
apply Quotient.sound
change ∃ g : G, g • e y = e z
refine ⟨g, ?_⟩
have he := congrArg e hg
change e (e.symm (g • e y)) = e z at he
simpa only [e.apply_symm_apply] using he
theorem transportedFiber_filter_card (e : Y ≃ X)
(q : MulAction.orbitRel.Quotient G X) (bad : Y → Prop) [DecidablePred bad] :
((transportedFiber e q).filter bad).card =
((fullFiber q).filter (bad ∘ e.symm)).card := by
rw [transportedFiber, Finset.filter_map]
exact Finset.card_map _
end MatrixMultiplication.JointOrbitFibers
end OAI
|