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e5a97c6 | 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 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | 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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