| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Probability.Marking |
| import OAI.Combinatorics.MatchingCount.Algorithm.RelabelWeights |
| import OAI.Combinatorics.MatchingCount.Sampling.RelabelMarking |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.RationalSource |
| open Finset Transport Variation Enlargement |
|
|
| 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 local instance (w : V → V → ℚ) (K : ℕ) : |
| DecidableEq (BagPM (bags w K)) := Classical.decEq _ |
| noncomputable local instance (w : V → V → ℚ) (K : ℕ) : |
| Fintype (BagPM (bags w K)) := Fintype.ofFinite _ |
| noncomputable local instance (w : V → V → ℚ) (K : ℕ) : |
| Fintype (Marked w K) := Fintype.ofFinite _ |
| noncomputable local instance (w : V → V → ℚ) (K : ℕ) : |
| Fintype (SourceMarked (fun x y=>(w x y:ℝ)) K) := Fintype.ofFinite _ |
| lemma sourceEdgeEquiv_central (w : V → V → ℚ) (K : ℕ) (q : LogicalEdge V) : |
| sourceEdgeEquiv w K (Enlargement.centralEdge (fun x y=>(w x y:ℝ)) K q)=centralEdge w K q := |
| sourceEdgeEquiv_real w K _ |
| lemma probePortion_cast (w : V → V → ℚ) (K D : ℕ) |
| (e : BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K)) : |
| (probePortion w K D (sourceEdgeEquiv w K e):ℝ)=Enlargement.probePortion (fun x y=>(w x y:ℝ)) K D e := by |
| classical |
| have he : (∃ q,centralEdge w K q=sourceEdgeEquiv w K e) ↔ |
| ∃ q,Enlargement.centralEdge (fun x y=>(w x y:ℝ)) K q=e := by |
| constructor |
| · rintro ⟨q,hq⟩ |
| exact ⟨q,(sourceEdgeEquiv w K).injective ((sourceEdgeEquiv_central w K q).trans hq)⟩ |
| · rintro ⟨q,rfl⟩ |
| exact ⟨q,(sourceEdgeEquiv_central w K q).symm⟩ |
| simp only [probePortion,Enlargement.probePortion,he] |
| split_ifs <;> push_cast <;> rfl |
| lemma portions_cast (w : V → V → ℚ) (K D : ℕ) |
| (e : BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K)) (i : Fin 3) : |
| (portions w K D (sourceEdgeEquiv w K e) i:ℝ)=sourcePortions (fun x y=>(w x y:ℝ)) K D e i := by |
| fin_cases i |
| · exact realPortion_cast w K e |
| · change ((bagHeight w K (sourceEdgeEquiv w K e).color/D-probePortion w K D (sourceEdgeEquiv w K e):ℚ):ℝ)=_ |
| rw [Rat.cast_sub,Rat.cast_div,Rat.cast_natCast,probePortion_cast,sourceEdgeEquiv_color,bagHeight_cast] |
| rfl |
| · exact probePortion_cast w K D e |
| noncomputable def sourceMarkedEquiv (w : V → V → ℚ) (K : ℕ) : |
| SourceMarked (fun x y=>(w x y:ℝ)) K ≃ Marked w K := |
| markedEquiv (inc:=BagEdge.Incident (bags:=sourceBags (fun x y=>(w x y:ℝ)) K)) |
| (j:=BagEdge.Incident (bags:=bags w K)) (sourceEdgeEquiv w K) (fun _ _=>Iff.rfl) |
| lemma portions_collapse (w : V → V → ℚ) (K D : ℕ) : |
| Marking.collapse (fun e i=>(portions w K D e i:ℝ))=fun e=>(bagWeight w K D e:ℝ) := by |
| funext e |
| change (∑ i : Fin 3,(portions w K D e i:ℝ))=(bagWeight w K D e:ℝ) |
| exact_mod_cast portions_sum w K D e |
| lemma markedLaw_eq (w : V → V → ℚ) (hw : ∀ x y,0<w x y) (K D : ℕ) (hD : 0<D) |
| (a : Marked w K) : |
| markedLaw w K D a=Marking.law BagEdge.Incident (fun e i=>(portions w K D e i:ℝ)) a := by |
| have hc : ∀ e,Marking.collapse (fun e i=>(portions w K D e i:ℝ)) e≠0 := by |
| intro e |
| rw [portions_collapse] |
| dsimp only |
| exact_mod_cast (bagWeight_pos w hw K D hD e).ne' |
| have h:=Marking.law_from_reveal BagEdge.Incident (fun e i=>(portions w K D e i:ℝ)) hc a |
| rw [portions_collapse] at h |
| convert h using 1 |
| simp only [markedLaw,Variation.sigmaLift,product,gibbs,matchingWeight,partition,Marking.conditional, |
| portions_collapse,Rat.cast_div,partitionWeight] |
| congr 3 |
| ext |
| simp |
| lemma markedLaw_equiv (w : V → V → ℚ) (hw : ∀ x y,0<w x y) (K D : ℕ) (hD : 0<D) |
| (a : SourceMarked (fun x y=>(w x y:ℝ)) K) : |
| markedLaw w K D (sourceMarkedEquiv w K a)=sourceLaw (fun x y=>(w x y:ℝ)) K D a := by |
| rw [markedLaw_eq w hw K D hD] |
| change Marking.law BagEdge.Incident (fun e i=>(portions w K D e i:ℝ)) |
| (markedEquiv (inc:=BagEdge.Incident (bags:=sourceBags (fun x y=>(w x y:ℝ)) K)) |
| (j:=BagEdge.Incident (bags:=bags w K)) (sourceEdgeEquiv w K) (fun _ _=>Iff.rfl) a)=_ |
| rw [law_markedEquiv (inc:=BagEdge.Incident (bags:=sourceBags (fun x y=>(w x y:ℝ)) K)) |
| (j:=BagEdge.Incident (bags:=bags w K)) (sourceEdgeEquiv w K) (fun _ _=>Iff.rfl)] |
| have hp : ((fun e i=>(portions w K D e i:ℝ)) ∘ sourceEdgeEquiv w K)= |
| sourcePortions (fun x y=>(w x y:ℝ)) K D := by |
| funext e i |
| exact portions_cast w K D e i |
| rw [hp] |
| rfl |
| end MatchingFPRAS.RationalSource |
|
|
| end OAI |
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