File size: 3,653 Bytes
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 61 62 63 64 65 66 67 68 69 70 71 | 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
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