| import Mathlib |
| import OAI.Computability.MaxCut.Model |
|
|
| namespace OAI |
|
|
| /-! |
| Equal kernels give the same orbit under invertible postcomposition. |
| The canonical range equivalence comes from the first isomorphism theorem; |
| Mathlib's finite-dimensional subspace extension theorem then extends it to an |
| automorphism of the ambient codomain. No orbit or counting premise is assumed. |
| -/ |
|
|
| namespace MaxCutGames.Inverse.KMSKernelOrbits |
|
|
| noncomputable section |
|
|
| variable {K E F : Type*} [DivisionRing K] |
| [AddCommGroup E] [Module K E] [AddCommGroup F] [Module K F] |
|
|
| /-- The range identification induced by a common kernel. -/ |
| def rangeEquivOfKerEq (S T : F →ₗ[K] E) (hker : S.ker = T.ker) : |
| S.range ≃ₗ[K] T.range := |
| S.quotKerEquivRange.symm ≪≫ₗ |
| Submodule.quotEquivOfEq S.ker T.ker hker ≪≫ₗ T.quotKerEquivRange |
|
|
| /-- This identification respects every original vector, including vectors in |
| the kernel and maps of rank zero. -/ |
| theorem rangeEquivOfKerEq_apply (S T : F →ₗ[K] E) (hker : S.ker = T.ker) (x : F) : |
| (rangeEquivOfKerEq S T hker ⟨S x, ⟨x, rfl⟩⟩ : E) = T x := by |
| change (T.quotKerEquivRange ((Submodule.quotEquivOfEq S.ker T.ker hker) |
| (S.quotKerEquivRange.symm ⟨S x, ⟨x, rfl⟩⟩)) : E) = T x |
| have hs : (⟨S x, ⟨x, rfl⟩⟩ : S.range) = |
| S.quotKerEquivRange (Submodule.Quotient.mk x) := rfl |
| rw [hs, LinearEquiv.symm_apply_apply] |
| rfl |
|
|
| /-- Finite-dimensional image suffices: the ambient codomain and domain may |
| otherwise be infinite-dimensional. -/ |
| theorem exists_postcomp_equiv_of_ker_eq (S T : F →ₗ[K] E) |
| [FiniteDimensional K S.range] (hker : S.ker = T.ker) : |
| ∃ g : E ≃ₗ[K] E, g.toLinearMap.comp S = T := by |
| obtain ⟨g, hg⟩ := Submodule.exists_linearEquiv_restrict_eq |
| (rangeEquivOfKerEq S T hker) |
| refine ⟨g, ?_⟩ |
| ext x |
| change g (S x) = T x |
| exact (hg ⟨S x, ⟨x, rfl⟩⟩).symm.trans (rangeEquivOfKerEq_apply S T hker x) |
|
|
| /-- Invertible postcomposition never changes the kernel. Thus the kernel is |
| an exact orbit invariant, rather than only a necessary condition. -/ |
| theorem ker_eq_iff_exists_postcomp_equiv (S T : F →ₗ[K] E) |
| [FiniteDimensional K S.range] : |
| S.ker = T.ker ↔ ∃ g : E ≃ₗ[K] E, g.toLinearMap.comp S = T := by |
| constructor |
| · exact exists_postcomp_equiv_of_ker_eq S T |
| · rintro ⟨g, rfl⟩ |
| ext x |
| simp |
|
|
| end |
| end MaxCutGames.Inverse.KMSKernelOrbits |
|
|
| end OAI |
|
|
