import Mathlib import OAI.Combinatorics.MatchingCount.Graphs.Incident import OAI.Combinatorics.MatchingCount.Graphs.Height namespace OAI namespace MatchingFPRAS.Transport section open Finset MatchingFPRAS.Switching variable {V E : Type*} [Fintype V] [LinearOrder V] variable (edge : E → Sym2 V) (hnd : ∀ e, ¬(edge e).IsDiag) local notation "inc" => (fun (e : E) (x : V) => x ∈ edge e) noncomputable def edgeLow (s : Sym2 V) : V := Sym2.lift ⟨min,min_comm⟩ s noncomputable def edgeHigh (s : Sym2 V) : V := Sym2.lift ⟨max,max_comm⟩ s omit [Fintype V] in @[simp] lemma edgeLow_mk (x y : V) : edgeLow s(x,y)=min x y := rfl omit [Fintype V] in @[simp] lemma edgeHigh_mk (x y : V) : edgeHigh s(x,y)=max x y := rfl omit [Fintype V] in lemma edgeLow_mem (s : Sym2 V) : edgeLow s ∈ s := by induction s using Sym2.inductionOn with | hf x y => by_cases h : x≤y <;> simp [min_def,h] omit [Fintype V] in lemma edgeHigh_mem (s : Sym2 V) : edgeHigh s ∈ s := by induction s using Sym2.inductionOn with | hf x y => by_cases h : x≤y <;> simp [max_def,h] omit [Fintype V] in lemma edge_low_high (s : Sym2 V) : s(edgeLow s,edgeHigh s)=s := by induction s using Sym2.inductionOn with | hf x y => by_cases h : x≤y <;> simp [min_def,max_def,h,Sym2.eq_swap] omit [Fintype V] in lemma edgeLow_lt_high (s : Sym2 V) (hs : ¬s.IsDiag) : edgeLow s < edgeHigh s := by induction s using Sym2.inductionOn with | hf x y => have hxy : x ≠ y := by simpa using hs rcases hxy.lt_or_gt with h | h <;> simp [h.le,h] variable (Q : Partial (fun (e : E) (x : V) => x ∈ edge e)) omit [Fintype V] in lemma edgeLow_ne_high (s : Sym2 V) (hs : ¬s.IsDiag) : edgeLow s ≠ edgeHigh s := (edgeLow_lt_high s hs).ne noncomputable def pickedEdge (x : V) (hx : x ∈ support inc Q.val) : E := Classical.choose (Q.property.2 x hx) omit [LinearOrder V] in lemma pickedEdge_spec (x : V) (hx : x ∈ support inc Q.val) : pickedEdge edge Q x hx ∈ Q.val ∧ x ∈ edge (pickedEdge edge Q x hx) := (Classical.choose_spec (Q.property.2 x hx)).1 omit [LinearOrder V] in lemma pickedEdge_unique (x : V) (hx : x ∈ support inc Q.val) (e : E) (he : e∈Q.val) (hxe : x ∈ edge e) : e=pickedEdge edge Q x hx := (Classical.choose_spec (Q.property.2 x hx)).2 e ⟨he,hxe⟩ noncomputable def partialMateFun (x : V) : V := if hx : x∈support inc Q.val then Sym2.Mem.other (pickedEdge_spec edge Q x hx).2 else x lemma partialMateFun_out (x : V) (hx : x∉support inc Q.val) : partialMateFun edge Q x=x := by simp [partialMateFun,hx] include hnd in lemma partialMateFun_ne (x : V) (hx : x∈support inc Q.val) : partialMateFun edge Q x≠x := by simp only [partialMateFun,dite_eq_left hx] exact Sym2.other_ne (hnd _) _ lemma partialMateFun_pair (x : V) (hx : x∈support inc Q.val) : s(x,partialMateFun edge Q x)=edge (pickedEdge edge Q x hx) := by simp only [partialMateFun,dite_eq_left hx] exact Sym2.other_spec _ lemma partialMateFun_incident (e : E) (he : e∈Q.val) (x : V) (hx : x∈edge e) : s(x,partialMateFun edge Q x)=edge e := by have hxQ := (mem_support inc Q.val x).mpr ⟨e,he,hx⟩ rw [partialMateFun_pair edge Q x hxQ,←pickedEdge_unique edge Q x hxQ e he hx] lemma partialMateFun_covered (x : V) (hx : x∈support inc Q.val) : partialMateFun edge Q x ∈ support inc Q.val := by apply (mem_support inc Q.val _).mpr refine ⟨pickedEdge edge Q x hx,(pickedEdge_spec edge Q x hx).1,?_⟩ rw [←partialMateFun_pair edge Q x hx] simp include hnd in lemma partialMateFun_involutive : Function.Involutive (partialMateFun edge Q) := by intro x by_cases hx : x∈support inc Q.val · have h₁ := partialMateFun_pair edge Q x hx have hi : partialMateFun edge Q x ∈ edge (pickedEdge edge Q x hx) := by rw [←h₁]; simp have h₂ := partialMateFun_incident edge Q (pickedEdge edge Q x hx) (pickedEdge_spec edge Q x hx).1 _ hi rcases Sym2.eq_iff.mp (h₂.trans h₁.symm) with h | h · exact False.elim (partialMateFun_ne edge hnd Q x hx h.1) · exact h.2 · rw [partialMateFun_out edge Q x hx,partialMateFun_out edge Q x hx] noncomputable def partialMate : Mate V := ⟨partialMateFun edge Q,partialMateFun_involutive edge hnd Q⟩ lemma partialMate_holes : HasHoles (partialMate edge hnd Q) (support inc Q.val)ᶜ := by intro x change partialMateFun edge Q x=x ↔ _ rw [Finset.mem_compl] constructor · intro h hx; exact partialMateFun_ne edge hnd Q x hx h · exact partialMateFun_out edge Q x lemma partialMate_low (e : E) (he : e∈Q.val) : partialMateFun edge Q (edgeLow (edge e))=edgeHigh (edge e) := by have h := partialMateFun_incident edge Q e he _ (edgeLow_mem _) exact Sym2.congr_right.mp (h.trans (edge_low_high (edge e)).symm) lemma partial_low_mem (e : E) (he : e∈Q.val) : edgeLow (edge e) ∈ univ.filter (fun x => x < (partialMate edge hnd Q).val x) := by simp only [Finset.mem_filter,Finset.mem_univ,true_and] change edgeLow (edge e) < partialMateFun edge Q (edgeLow (edge e)) rw [partialMate_low edge Q e he] exact edgeLow_lt_high _ (hnd e) lemma partial_low_injective (e : E) (he : e∈Q.val) (f : E) (hf : f∈Q.val) (h : edgeLow (edge e)=edgeLow (edge f)) : e=f := by have hx := edgeLow_mem (edge e) exact Q.property.unique (Q.property.1 e he _ hx) he hx hf (h.symm ▸ edgeLow_mem (edge f)) lemma partial_low_surjective (x : V) (hx : x ∈ univ.filter (fun x => x < (partialMate edge hnd Q).val x)) : ∃ e, ∃ _ : e∈Q.val, edgeLow (edge e)=x := by have hx' : x < partialMateFun edge Q x := (Finset.mem_filter.mp hx).2 have hcov : x∈support inc Q.val := by by_contra hn rw [partialMateFun_out edge Q x hn] at hx' exact lt_irrefl _ hx' refine ⟨pickedEdge edge Q x hcov,(pickedEdge_spec edge Q x hcov).1,?_⟩ rw [←partialMateFun_pair edge Q x hcov,edgeLow_mk,min_eq_left hx'.le] include hnd in lemma partial_card : (support inc Q.val).card=2*Q.val.card := by have hcard := Finset.card_bij (fun e _ => edgeLow (edge e)) (partial_low_mem edge hnd Q) (partial_low_injective edge Q) (partial_low_surjective edge hnd Q) have hmate := mate_edge_count (partialMate edge hnd Q) (support inc Q.val) (partialMate_holes edge hnd Q) omega include hnd in lemma partial_even : Even (support inc Q.val).card := by rw [partial_card edge hnd Q] exact even_two_mul _ lemma partial_weight (B : V → V → ℝ) : (∏ e ∈ Q.val, B (edgeLow (edge e)) (edgeHigh (edge e)))=weight B (partialMate edge hnd Q) := by classical unfold weight rw [←Finset.prod_filter] apply Finset.prod_bij (fun e _ => edgeLow (edge e)) (partial_low_mem edge hnd Q) (partial_low_injective edge Q) (partial_low_surjective edge hnd Q) intro e he change B _ _ = B _ (partialMateFun edge Q _) rw [partialMate_low edge Q e he] theorem partial_height_le_capacity (B : V → V → ℝ) (height : E → ℝ) (h0 : ∀ e, 0≤height e) (hh : ∀ e, height e ≤ B (edgeLow (edge e)) (edgeHigh (edge e))) : (∏ e ∈ Q.val, height e) ≤ pairingCapacity B (support inc Q.val) (partial_even edge hnd Q) := by calc _ ≤ ∏ e ∈ Q.val, B (edgeLow (edge e)) (edgeHigh (edge e)) := Finset.prod_le_prod₀ (fun e _ => h0 e) (fun e _ => hh e) _ = weight B (partialMate edge hnd Q) := partial_weight edge hnd Q B _ ≤ _ := Finset.le_sup' (f := weight B) ((mem_matchings _ _).mpr (partialMate_holes edge hnd Q)) end section open Finset variable {V L : Type*} [Fintype V] [Fintype L] [LinearOrder V] [DecidableEq L] variable (bags : V → Finset L) noncomputable local instance edgeLowCardinalityBagEdgeFintype : Fintype (BagEdge bags) := Fintype.ofFinite _ noncomputable def bagEdgeSigmaEquiv : BagEdge bags ≃ Σ s : {s : Sym2 V // ¬s.IsDiag}, {l : L // ∀ x, x∈s.val → l∈bags x} where toFun e := ⟨⟨e.val.val.1,e.val.property⟩,⟨e.val.val.2,e.property⟩⟩ invFun a := ⟨⟨(a.1.val,a.2.val),a.1.property⟩,a.2.property⟩ left_inv _ := rfl right_inv _ := rfl lemma bag_color_card_le_two (hb : ∀ x,(bags x).card≤2) (s : {s : Sym2 V // ¬s.IsDiag}) : Fintype.card {l : L // ∀ x, x∈s.val → l∈bags x} ≤ 2 := by let f : {l : L // ∀ x, x∈s.val → l∈bags x} → {l : L // l∈bags (edgeLow s.val)} := fun l => ⟨l.val,l.property _ (edgeLow_mem _)⟩ have hi : Function.Injective f := by intro a b h apply Subtype.ext exact congrArg (fun z : {l : L // l∈bags (edgeLow s.val)} => z.val) h have := Fintype.card_le_of_injective f hi rw [Fintype.card_coe] at this exact this.trans (hb _) theorem bagEdge_card_le_square (hb : ∀ x,(bags x).card≤2) : Fintype.card (BagEdge bags) ≤ (Fintype.card V)^2 := by classical rw [Fintype.card_congr (bagEdgeSigmaEquiv bags),Fintype.card_sigma] calc _ ≤ ∑ _s : {s : Sym2 V // ¬s.IsDiag}, 2 := Finset.sum_le_sum (fun s _ => bag_color_card_le_two bags hb s) _ = 2*(Fintype.card V).choose 2 := by rw [Finset.sum_const,Finset.card_univ,smul_eq_mul,Sym2.card_subtype_not_diag,mul_comm] _ ≤ Fintype.card V*(Fintype.card V-1) := by rw [Nat.choose_two_right]; omega _ ≤ (Fintype.card V)^2 := by rw [pow_two] exact Nat.mul_le_mul_left _ (Nat.sub_le _ _) end open Finset MatchingFPRAS.Switching variable {V L : Type*} [Fintype V] [Fintype L] [LinearOrder V] [DecidableEq L] variable (bags : V → Finset L) noncomputable local instance edgeLowInflationBagEdgeFintype : Fintype (BagEdge bags) := Fintype.ofFinite _ def inflationScale (D₀ : ℝ) : ℝ := 100*(Fintype.card V : ℝ)^2*D₀ theorem bag_inflation_of_hole_estimate (hN : 0 r e+height e.val.val.2/D) univ / partitionWeight BagEdge.Incident r univ ∧ partitionWeight BagEdge.Incident (fun e => r e+height e.val.val.2/D) univ / partitionWeight BagEdge.Incident r univ ≤ 2 := by classical let D := inflationScale (V := V) D₀ have hNp : 0<(Fintype.card V:ℝ) := Nat.cast_pos.mpr hN have hD : 0 height e.val.val.2/D) hr (fun e => div_nonneg (hheight _) hD.le) (fun e => ⟨edgeLow e.val.val.1,edgeLow_mem _⟩) hz (D₀/D) (div_nonneg hD₀.le hD.le) · have hcard : (Fintype.card (BagEdge bags):ℝ) ≤ (Fintype.card V:ℝ)^2 := by exact_mod_cast bagEdge_card_le_square bags hb rw [←mul_div_assoc] apply (div_le_iff₀ hD).mpr change (Fintype.card (BagEdge bags):ℝ)*D₀ ≤ 1/100*D dsimp [D,inflationScale] nlinarith [mul_le_mul_of_nonneg_right hcard hD₀.le] · intro Q _ let edge : BagEdge bags → Sym2 V := fun e => e.val.val.1 have hnd : ∀ e,¬(edge e).IsDiag := fun e => e.val.property have heven := partial_even edge hnd Q let g := partitionWeight BagEdge.Incident r (univ \ support BagEdge.Incident Q.val) / partitionWeight BagEdge.Incident r univ have hg : 0≤g := div_nonneg (partitionWeight_nonneg _ r hr _) hz.le have hc := partial_height_le_capacity edge hnd Q B (fun e => height e.val.val.2) (fun e => hheight _) hcommon have hh := hhole _ heven have hcard := partial_card edge hnd Q change (support BagEdge.Incident Q.val).card=2*Q.val.card at hcard change g * pairingCapacity B (support BagEdge.Incident Q.val) heven ≤ 2*D₀^((support BagEdge.Incident Q.val).card/2) at hh have hpow : (support BagEdge.Incident Q.val).card/2=Q.val.card := by omega rw [hpow] at hh rw [Finset.prod_div_distrib,Finset.prod_const] change (∏ e∈Q.val,height e.val.val.2) / D^Q.val.card * partitionWeight BagEdge.Incident r (univ \ support BagEdge.Incident Q.val) / partitionWeight BagEdge.Incident r univ ≤ _ calc _ = g*(∏ e∈Q.val,height e.val.val.2) / D^Q.val.card := by dsimp [g]; ring _ ≤ g*pairingCapacity B (support BagEdge.Incident Q.val) heven / D^Q.val.card := div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hc hg) (pow_nonneg hD.le _) _ ≤ (2*D₀^Q.val.card)/D^Q.val.card := div_le_div_of_nonneg_right hh (pow_nonneg hD.le _) _ = 2*(D₀/D)^Q.val.card := by rw [div_pow]; ring end MatchingFPRAS.Transport namespace MatchingFPRAS.RealGeometry open Finset MatchingFPRAS.Subdivision MatchingFPRAS.Transport variable {V E : Type*} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] variable (u v : E → V) (m : ℕ) omit [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] in lemma edge_incident_internal (q : E × Fin (m+1)) (e : E) (i : Fin m) : Sum.inr (e,i) ∈ edge u v m q.1 q.2 ↔ q=(e,i.castSucc) ∨ q=(e,i.succ) := by simp only [edge,Sym2.mem_iff,eq_comm (a := Sum.inr _) ] rw [atVertex_eq_inr_iff,atVertex_eq_inr_iff] simp only [Fin.val_castSucc,Fin.val_succ,Prod.ext_iff] constructor · rintro (⟨h,h'⟩|⟨h,h'⟩) · exact Or.inr ⟨h,Fin.ext h'⟩ · exact Or.inl ⟨h,Fin.ext (show q.2.val=i.val by omega)⟩ · rintro (⟨h,h'⟩|⟨h,h'⟩) · exact Or.inr ⟨h,by have := congrArg Fin.val h'; simpa using this⟩ · exact Or.inl ⟨h,by have := congrArg Fin.val h'; simpa using this⟩ omit [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] in lemma edge_incident_terminal (q : E × Fin (m+1)) (x : V) : Sum.inl x ∈ edge u v m q.1 q.2 ↔ (q.2=0 ∧ u q.1=x) ∨ (q.2=Fin.last m ∧ v q.1=x) := by simp only [edge,Sym2.mem_iff,eq_comm (a := Sum.inl _)] rw [first_vertex_terminal,second_vertex_terminal] simp only [Fin.ext_iff,Fin.val_zero,Fin.val_last] noncomputable def selectedSet (b : E → Selection m) : Finset (E × Fin (m+1)) := by classical exact univ.filter fun q => b q.1 q.2=true omit [DecidableEq E] in @[simp] lemma mem_selectedSet (b : E → Selection m) (q : E × Fin (m+1)) : q∈selectedSet m b ↔ b q.1 q.2=true := by simp [selectedSet] noncomputable def survivors (R : Finset V) (U : E → Finset (Fin m)) : Finset (Vertex V E m) := by classical exact univ.filter fun x => match x with | Sum.inl x => x∉R | Sum.inr (e,i) => i∉U e omit [DecidableEq V] [DecidableEq E] in @[simp] lemma mem_survivors_terminal (R : Finset V) (U : E → Finset (Fin m)) (x : V) : Sum.inl x ∈ survivors m R U ↔ x∉R := by simp [survivors] omit [DecidableEq V] [DecidableEq E] in @[simp] lemma mem_survivors_internal (R : Finset V) (U : E → Finset (Fin m)) (e : E) (i : Fin m) : Sum.inr (e,i) ∈ survivors m R U ↔ i∉U e := by simp [survivors] omit [Fintype V] [DecidableEq V] [DecidableEq E] in lemma internal_unique_iff (b : E → Selection m) (e : E) (i : Fin m) : (∃! q, q∈selectedSet m b ∧ Sum.inr (e,i)∈edge u v m q.1 q.2) ↔ b e i.castSucc ≠ b e i.succ := by simp only [mem_selectedSet,edge_incident_internal] have hne : i.castSucc≠i.succ := by intro h; have := congrArg Fin.val h; simp at this cases h₁ : b e i.castSucc <;> cases h₂ : b e i.succ <;> simp only [ne_eq,Bool.false_eq_true, not_false_eq_true,Bool.true_eq_false,not_true_eq_false,iff_false,iff_true] · rintro ⟨q,hq,_⟩; rcases hq.2 with rfl|rfl <;> simp_all · refine ⟨(e,i.succ),⟨h₂,Or.inr rfl⟩,?_⟩ rintro ⟨f,j⟩ ⟨hs,hq⟩ rcases hq with hq|hq · cases hq; simp_all · exact hq · refine ⟨(e,i.castSucc),⟨h₁,Or.inl rfl⟩,?_⟩ rintro ⟨f,j⟩ ⟨hs,hq⟩ rcases hq with hq|hq · exact hq · cases hq; simp_all · rintro ⟨q,hq,hu⟩ have h := (hu (e,i.castSucc) ⟨h₁,Or.inl rfl⟩).trans (hu (e,i.succ) ⟨h₂,Or.inr rfl⟩).symm exact hne (congrArg Prod.snd h) omit [Fintype V] [DecidableEq V] [DecidableEq E] in lemma terminal_selected_iff (b : E → Selection m) (q : E × Fin (m+1)) (x : V) : q∈selectedSet m b ∧ Sum.inl x∈edge u v m q.1 q.2 ↔ (q.2=0 ∧ b q.1 0=true ∧ u q.1=x) ∨ (q.2=Fin.last m ∧ b q.1 (Fin.last m)=true ∧ v q.1=x) := by rw [mem_selectedSet,edge_incident_terminal] constructor · rintro ⟨hs, (⟨h₀,hu⟩|⟨hm,hv⟩)⟩ · exact Or.inl ⟨h₀,h₀ ▸ hs,hu⟩ · exact Or.inr ⟨hm,hm ▸ hs,hv⟩ · rintro (⟨h₀,hs,hu⟩|⟨hm,hs,hv⟩) · exact ⟨h₀ ▸ hs,Or.inl ⟨h₀,hu⟩⟩ · exact ⟨hm ▸ hs,Or.inr ⟨hm,hv⟩⟩ omit [Fintype V] [DecidableEq V] [DecidableEq E] in lemma terminal_unique_iff (huv : ∀ e,u e≠v e) (b : E → Selection m) (x : V) : (∃! q, q∈selectedSet m b ∧ Sum.inl x∈edge u v m q.1 q.2) ↔ (∃! e, Uses u v b e x) := by constructor · rintro ⟨q,hq,hunique⟩ rw [terminal_selected_iff] at hq have huse : Uses u v b q.1 x := by rcases hq with h|h; exact Or.inl h.2; exact Or.inr h.2 refine ⟨q.1,huse,?_⟩ intro f hf rcases hf with hf|hf · exact congrArg Prod.fst (hunique (f,0) ((terminal_selected_iff _ _ _ _ _ _).mpr (Or.inl ⟨rfl,hf⟩))) · exact congrArg Prod.fst (hunique (f,Fin.last m) ((terminal_selected_iff _ _ _ _ _ _).mpr (Or.inr ⟨rfl,hf⟩))) · rintro ⟨e,he,hu⟩ have uniq (i : Fin (m+1)) (hi : (e,i)∈selectedSet m b ∧ Sum.inl x∈edge u v m e i) (q : E × Fin (m+1)) (hq : q∈selectedSet m b ∧ Sum.inl x∈edge u v m q.1 q.2) : q=(e,i) := by rw [terminal_selected_iff] at hi hq have hf : q.1=e := hu q.1 (by rcases hq with h|h; exact Or.inl h.2; exact Or.inr h.2) apply Prod.ext hf rcases hi with hi|hi <;> rcases hq with hq|hq · exact hq.1.trans hi.1.symm · exact False.elim (huv e (hi.2.2.trans (hf ▸ hq.2.2).symm)) · exact False.elim (huv e ((hf ▸ hq.2.2).trans hi.2.2.symm)) · exact hq.1.trans hi.1.symm rcases he with he|he · have hi := (terminal_selected_iff u v m b (e,0) x).mpr (Or.inl ⟨rfl,he⟩) exact ⟨(e,0),hi,uniq 0 hi⟩ · have hi := (terminal_selected_iff u v m b (e,Fin.last m) x).mpr (Or.inr ⟨rfl,he⟩) exact ⟨(e,Fin.last m),hi,uniq _ hi⟩ omit [DecidableEq V] [DecidableEq E] in theorem family_iff_matching (huv : ∀ e,u e≠v e) (R : Finset V) (U : E → Finset (Fin m)) (b : E → Selection m) : FamilyHoles u v m R U b ↔ MatchingOn (fun (q : E × Fin (m+1)) x => x∈edge u v m q.1 q.2) (selectedSet m b) (survivors m R U) := by constructor · intro hb constructor · intro q hq x hx cases x with | inl x => rw [mem_survivors_terminal] intro hr have huse : Uses u v b q.1 x := by rcases (terminal_selected_iff u v m b q x).mp ⟨hq,hx⟩ with h|h · exact Or.inl h.2 · exact Or.inr h.2 exact hb.2.1 x hr q.1 huse | inr ei => rcases ei with ⟨e,i⟩ rw [mem_survivors_internal] intro hi have hd := hb.1 e i simp only [hi,ite_eq_left] at hd rcases (edge_incident_internal u v m q e i).mp hx with rfl|rfl · have h := (mem_selectedSet m b _).mp hq; simp_all · have h := (mem_selectedSet m b _).mp hq; simp_all · intro x hx cases x with | inl x => exact (terminal_unique_iff u v m huv b x).mpr (hb.2.2 x ((mem_survivors_terminal m R U x).mp hx)) | inr ei => rcases ei with ⟨e,i⟩ apply (internal_unique_iff u v m b e i).mpr simpa only [Valid,ite_eq_right ((mem_survivors_internal m R U e i).mp hx)] using hb.1 e i · intro hM refine ⟨?_,?_,?_⟩ · intro e i split_ifs with hi · constructor · cases h : b e i.castSucc · rfl · have hx := hM.1 (e,i.castSucc) ((mem_selectedSet m b _).mpr h) (Sum.inr (e,i)) ((edge_incident_internal u v m _ e i).mpr (Or.inl rfl)) exact False.elim (((mem_survivors_internal m R U e i).mp hx) hi) · cases h : b e i.succ · rfl · have hx := hM.1 (e,i.succ) ((mem_selectedSet m b _).mpr h) (Sum.inr (e,i)) ((edge_incident_internal u v m _ e i).mpr (Or.inr rfl)) exact False.elim (((mem_survivors_internal m R U e i).mp hx) hi) · exact (internal_unique_iff u v m b e i).mp (hM.2 (Sum.inr (e,i)) ((mem_survivors_internal m R U e i).mpr hi)) · intro x hx e he have hq : ∃ q, q∈selectedSet m b ∧ Sum.inl x∈edge u v m q.1 q.2 := by rcases he with he|he · exact ⟨(e,0),(terminal_selected_iff u v m b _ x).mpr (Or.inl ⟨rfl,he⟩)⟩ · exact ⟨(e,Fin.last m),(terminal_selected_iff u v m b _ x).mpr (Or.inr ⟨rfl,he⟩)⟩ obtain ⟨q,hq,hqx⟩ := hq exact ((mem_survivors_terminal m R U x).mp (hM.1 q hq _ hqx)) hx · intro x hx exact (terminal_unique_iff u v m huv b x).mp (hM.2 (Sum.inl x) ((mem_survivors_terminal m R U x).mpr hx)) noncomputable def selectionEquiv : (E → Selection m) ≃ Finset (E × Fin (m+1)) where toFun := selectedSet m invFun S e i := decide ((e,i)∈S) left_inv b := by funext e i simp [mem_selectedSet] right_inv S := by ext ⟨e,i⟩ simp noncomputable def realMatchingEquiv (huv : ∀ e,u e≠v e) (R : Finset V) (U : E → Finset (Fin m)) : {b : E → Selection m // FamilyHoles u v m R U b} ≃ Matchings (fun (q : E × Fin (m+1)) x => x∈edge u v m q.1 q.2) (survivors m R U) := (selectionEquiv (E := E) m).subtypeEquiv (family_iff_matching u v m huv R U) omit [DecidableEq E] in lemma selectedSet_weight (w : E → Fin (m+1) → ℝ) (b : E → Selection m) : (∏ q ∈ selectedSet m b,w q.1 q.2)=∏ e,selectedWeight (w e) (b e) := by classical simp only [selectedSet,Finset.prod_filter,selectedWeight,Fintype.prod_prod_type] omit [DecidableEq V] in theorem realMatching_partition (huv : ∀ e,u e≠v e) (R : Finset V) (U : E → Finset (Fin m)) (w : E → Fin (m+1) → ℝ) : partitionWeight (fun (q : E × Fin (m+1)) x => x∈edge u v m q.1 q.2) (fun q => w q.1 q.2) (survivors m R U)= realHolePartition u v m R U w := by classical let := Fintype.ofFinite {b : E → Selection m // FamilyHoles u v m R U b} unfold partitionWeight realHolePartition rw [←(realMatchingEquiv u v m huv R U).sum_comp] apply Finset.sum_congr rfl intro b _ exact selectedSet_weight m w b.val end MatchingFPRAS.RealGeometry end OAI