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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 | 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
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