openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Recovery /RecoveryOrbitDecisions.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.GroupOrbitProjection | |
| namespace OAI | |
| /-! Finite orbit symmetries, masks and exact recovery operations. -/ | |
| namespace MatrixMultiplication.RecoveryOrbitDecisions | |
| theorem orbitSet_instances {G X : Type*} [Group G] [MulAction G X] | |
| (f₁ f₂ : Fintype G) (d₁ d₂ : DecidableEq X) (x : X) : | |
| @OrbitCounting.orbitSet G X _ _ f₁ d₁ x = | |
| @OrbitCounting.orbitSet G X _ _ f₂ d₂ x := by | |
| apply Finset.ext | |
| intro y | |
| simp only [OrbitCounting.orbitSet, Finset.mem_image, Finset.mem_univ, true_and] | |
| theorem orbitSet_eq_transportedOrbit {G X Y : Type*} [Group G] [MulAction G Y] | |
| (fRaw fCanonical : Fintype G) (d : DecidableEq X) (e : X ≃ Y) (x : X) : | |
| @OrbitCounting.orbitSet G X _ (EquivOrbitTransport.action e) fRaw d x = | |
| @GroupOrbitProjection.transportedOrbit G X Y _ fCanonical _ e x := by | |
| unfold GroupOrbitProjection.transportedOrbit | |
| exact @orbitSet_instances G X _ (EquivOrbitTransport.action e) | |
| fRaw fCanonical d _ x | |
| theorem transportedOrbit_eq_orbitSet {G X Y : Type*} [Group G] [MulAction G Y] | |
| (fCanonical fRaw : Fintype G) (d : DecidableEq X) (e : X ≃ Y) (x : X) : | |
| @GroupOrbitProjection.transportedOrbit G X Y _ fCanonical _ e x = | |
| @OrbitCounting.orbitSet G X _ (EquivOrbitTransport.action e) fRaw d x := | |
| (orbitSet_eq_transportedOrbit fRaw fCanonical d e x).symm | |
| end MatrixMultiplication.RecoveryOrbitDecisions | |
| end OAI | |