openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /OrbitFibers.lean
| 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 | |