| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Graphs.RelabelMore |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.Enlargement |
| open Transport |
|
|
| variable {V : Type*} [Fintype V] [LinearOrder V] [Nontrivial V] |
| noncomputable local instance relabelWeightsSourceLabelDecidableEq (w : V → V → ℝ) (K : ℕ) : DecidableEq (SourceLabel w K) := Classical.decEq _ |
| noncomputable local instance relabelWeightsSourceBagEdgeFintype (w : V → V → ℝ) (K : ℕ) : Fintype (BagEdge (sourceBags w K)) := Fintype.ofFinite _ |
| lemma realPortion_on (w : V → V → ℝ) (K : ℕ) (q : SourceRealEdge V K) : |
| sourceRealPortion w K (sourceColoredReal w K q)=sourceRealActivity w K q := by |
| classical |
| exact edgeExtension_on (sourceRealEmbedding w K) (sourceRealActivity w K) q |
| lemma realPortion_off (w : V → V → ℝ) (K : ℕ) (e : BagEdge (sourceBags w K)) |
| (he : ¬∃ q,sourceColoredReal w K q=e) : sourceRealPortion w K e=0 := by |
| classical |
| exact edgeExtension_off (sourceRealEmbedding w K) (sourceRealActivity w K) e he |
| end MatchingFPRAS.Enlargement |
| namespace MatchingFPRAS.RationalSource |
| open MatchingFPRAS Transport Enlargement Finset |
|
|
| variable {V : Type*} [Fintype V] [LinearOrder V] [Nontrivial V] |
| attribute [local instance] NumberedHierarchy.labelOrder sym2Order bagEdgeOrder bagEdgeFintype |
| attribute [local instance] UnitRefresh.Computable.allowedDec prodOrder sumOrder |
| noncomputable local instance (w : V → V → ℚ) (K : ℕ) : |
| DecidableEq (SourceLabel (fun x y=>(w x y:ℝ)) K) := Classical.decEq _ |
| noncomputable local instance (w : V → V → ℚ) (K : ℕ) : |
| Fintype (BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K)) := Fintype.ofFinite _ |
| noncomputable def sourceEdgeEquiv (w : V → V → ℚ) (K : ℕ) : |
| BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K) ≃ BagEdge (bags w K) := |
| relabelEdgeEquiv (labelsEquiv w K) (source_bags_equiv w K) |
| lemma sourceEdgeEquiv_real (w : V → V → ℚ) (K : ℕ) (q : SourceRealEdge V K) : |
| sourceEdgeEquiv w K (sourceColoredReal (fun x y=>(w x y:ℝ)) K q)=coloredReal w K q := by |
| apply Subtype.ext |
| apply Subtype.ext |
| apply Prod.ext |
| · rfl |
| · exact labelsEquiv_edgeLabel w K q |
| lemma sourceEdgeEquiv_color (w : V → V → ℚ) (K : ℕ) |
| (e : BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K)) : |
| (sourceEdgeEquiv w K e).color=labelsEquiv w K e.color := rfl |
| lemma bagHeight_cast (w : V → V → ℚ) (K : ℕ) (l : SourceLabel (fun x y=>(w x y:ℝ)) K) : |
| (bagHeight w K (labelsEquiv w K l):ℝ)=Enlargement.bagHeight (fun x y=>(w x y:ℝ)) K l := by |
| simp only [bagHeight,Enlargement.bagHeight,labelsEquiv_level] |
| norm_cast |
| lemma realPortion_cast (w : V → V → ℚ) (K : ℕ) |
| (e : BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K)) : |
| (realPortion w K (sourceEdgeEquiv w K e):ℝ)=sourceRealPortion (fun x y=>(w x y:ℝ)) K e := by |
| classical |
| by_cases he : ∃ q,sourceColoredReal (fun x y=>(w x y:ℝ)) K q=e |
| · obtain ⟨q,rfl⟩:=he |
| rw [sourceEdgeEquiv_real,realPortion_on,Enlargement.realPortion_on] |
| exact pathActivity_cast w K q.1 q.2 |
| · rw [realPortion_off w K _ ?_,Enlargement.realPortion_off _ _ _ he] |
| norm_num |
| rintro ⟨q,hq⟩ |
| apply he |
| refine ⟨q,(sourceEdgeEquiv w K).injective ?_⟩ |
| rwa [sourceEdgeEquiv_real] |
| lemma bagWeight_cast (w : V → V → ℚ) (K D : ℕ) |
| (e : BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K)) : |
| (bagWeight w K D (sourceEdgeEquiv w K e):ℝ)=sourceBagWeight (fun x y=>(w x y:ℝ)) K D e := by |
| simp only [bagWeight,Rat.cast_add,Rat.cast_div,Rat.cast_natCast,realPortion_cast, |
| sourceEdgeEquiv_color,bagHeight_cast,sourceBagWeight] |
| rfl |
| end MatchingFPRAS.RationalSource |
|
|
| end OAI |
|
|
