| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Sampling.Relabel |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.Transport |
| open Finset |
|
|
| variable {V E F I : Type*} [Fintype V] [DecidableEq V] |
| variable [Fintype E] [DecidableEq E] [Fintype F] [DecidableEq F] [Fintype I] |
| variable {inc : E → V → Prop} {j : F → V → Prop} |
| def selectedEquiv (e : E ≃ F) (he : ∀ a v,j (e a) v ↔ inc a v) (M : PM inc) : |
| ↥M.val ≃ ↥(pmEquiv e he M).val where |
| toFun a:=⟨e a.val,mem_image.mpr ⟨a.val,a.property,rfl⟩⟩ |
| invFun a:=⟨e.symm a.val,by |
| obtain ⟨b,hb,heq⟩:=mem_image.mp a.property |
| simpa [←heq] using hb⟩ |
| left_inv a:=by apply Subtype.ext; exact e.symm_apply_apply _ |
| right_inv a:=by apply Subtype.ext; exact e.apply_symm_apply _ |
| def markedEquiv (e : E ≃ F) (he : ∀ a v,j (e a) v ↔ inc a v) : |
| Marking.Marked (I:=I) inc ≃ Marking.Marked (I:=I) j := |
| Equiv.sigmaCongr (pmEquiv e he) (fun M=>Equiv.arrowCongr (selectedEquiv e he M) (Equiv.refl I)) |
| omit [DecidableEq V] [Fintype E] [Fintype F] [Fintype I] in |
| lemma markedEquiv_fst (e : E ≃ F) (he : ∀ a v,j (e a) v ↔ inc a v) |
| (a : Marking.Marked (I:=I) inc) : (markedEquiv e he a).1=pmEquiv e he a.1 := rfl |
| omit [DecidableEq V] [Fintype E] [Fintype F] [Fintype I] in |
| lemma markedEquiv_mark (e : E ≃ F) (he : ∀ a v,j (e a) v ↔ inc a v) |
| (a : Marking.Marked (I:=I) inc) (x : a.1.val) : |
| (markedEquiv e he a).2 (selectedEquiv e he a.1 x)=a.2 x := by |
| change a.2 ((selectedEquiv e he a.1).symm (selectedEquiv e he a.1 x))=a.2 x |
| rw [Equiv.symm_apply_apply] |
| omit [DecidableEq V] [Fintype E] [Fintype F] [Fintype I] in |
| lemma mass_markedEquiv (e : E ≃ F) (he : ∀ a v,j (e a) v ↔ inc a v) |
| (part : F → I → ℝ) (a : Marking.Marked (I:=I) inc) : |
| Marking.mass j part (markedEquiv e he a)=Marking.mass inc (part ∘ e) a := by |
| apply Fintype.prod_equiv (selectedEquiv e he a.1).symm |
| intro x |
| change part x.val ((markedEquiv e he a).2 x)=part (e ((selectedEquiv e he a.1).symm x).val) (a.2 ((selectedEquiv e he a.1).symm x)) |
| rw [←markedEquiv_mark e he a] |
| simp only [Equiv.apply_symm_apply] |
| congr 1 |
| exact (e.apply_symm_apply x.val).symm |
| lemma partitionWeight_pmEquiv (e : E ≃ F) (he : ∀ a v,j (e a) v ↔ inc a v) |
| (w : F → ℝ) : partitionWeight j w univ=partitionWeight inc (w ∘ e) univ := by |
| classical |
| unfold partitionWeight |
| symm |
| apply Fintype.sum_equiv (pmEquiv e he) |
| intro M |
| exact (matchingWeight_pmEquiv e he w M).symm |
| lemma law_markedEquiv (e : E ≃ F) (he : ∀ a v,j (e a) v ↔ inc a v) |
| (part : F → I → ℝ) (a : Marking.Marked (I:=I) inc) : |
| Marking.law j part (markedEquiv e he a)=Marking.law inc (part ∘ e) a := by |
| unfold Marking.law |
| rw [mass_markedEquiv,partitionWeight_pmEquiv e he] |
| rfl |
| end MatchingFPRAS.Transport |
|
|
| end OAI |
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|
