import Mathlib import OAI.Combinatorics.MatchingCount.Sampling.LadderL import OAI.Combinatorics.MatchingCount.Graphs.MatchingMap namespace OAI namespace MatchingFPRAS.Enlargement open MatchingFPRAS.Switching MatchingFPRAS.Subdivision MatchingFPRAS.RealGeometry open MatchingFPRAS.Hierarchy MatchingFPRAS.Transport Finset open MatchingFPRAS.Replica MatchingFPRAS.Replica.Ladder variable {V : Type*} [Fintype V] [LinearOrder V] [Nontrivial V] noncomputable local instance (w : V → V → ℝ) (K : ℕ) : DecidableEq (SourceLabel w K) := Classical.decEq _ noncomputable local instance (w : V → V → ℝ) (K : ℕ) : Fintype (BagEdge (sourceBags w K)) := Fintype.ofFinite _ abbrev SourcePM (w : V → V → ℝ) (K : ℕ) := PM (BagEdge.Incident (bags:=sourceBags w K)) noncomputable local instance (w : V → V → ℝ) (K : ℕ) : DecidableEq (SourcePM w K) := Classical.decEq _ omit [Nontrivial V] in lemma sourceVertex_card_ge (K : ℕ) : Fintype.card V ≤ Fintype.card (SourceVertex V K) := by exact Fintype.card_le_of_injective (Sum.inl : V → SourceVertex V K) Sum.inl_injective theorem source_ladder_sampling (hn : 2 ≤ Fintype.card V) (w : V → V → ℝ) (hw : ∀ x y, 0 < w x y) (K : ℕ) (hK : ∀ x y, x ≠ y → edgeActivity w x y ≤ (4:ℝ)^K) (M₀ : SourcePM w K) (T : ℕ) (z : Slot T (ladderQ (Fintype.card (SourceVertex V K)) (sourceD (V:=V) K)) → SourcePM w K) (ξ : ℝ) (hξ : 0 < ξ) (hξ1 : ξ < 1) : let D := sourceD (V:=V) K let N := Fintype.card (SourceVertex V K) let wt := sourceBagWeight w K D let ν := ladderLaw wt T (ladderQ N D) let C := ladderFlow (fun x a ha b hb => sourceBag_tree_adjacent w K hK x a b ha hb) (sourceTree_isTree w K).2 wt T (ladderQ N D) (∑ s, |MatchingFPRAS.FiniteChain.walk ν C z (ladderSteps N D K T ξ) s-ν s|)/2 ≤ ξ := by classical dsimp only apply ideal_ladder_sampling (fun x a ha b hb => sourceBag_tree_adjacent w K hK x a b ha hb) (sourceTree_isTree w K).2 (bagHeight w K) (bagHeight_pos w K) (bagHeight_adjacent w K) (sourceD (V:=V) K) K (sourceD_one K) (hn.trans (sourceVertex_card_ge K)) (sourceBagWeight w K (sourceD (V:=V) K)) (sourceBagWeight_range w K _ (sourceD_one K) hw hK) (sourceBagWeight_global w K _ (sourceD_one K) hw hK) M₀ T z ξ hξ hξ1 end MatchingFPRAS.Enlargement namespace MatchingFPRAS.Subdivision open Finset def probeHoles (p : ℕ) (hp : 1≤p) : Finset (Fin (4*p)) := {⟨2*p-1,by omega⟩,⟨2*p,by omega⟩} def probeWord (p : ℕ) : Selection (4*p) := fun i => if i.val=2*p then false else alternating true i.val lemma probeWord_valid (p : ℕ) (hp : 1≤p) : Valid (probeHoles p hp) (probeWord p) := by intro i by_cases h : i.val=2*p-1 ∨ i.val=2*p · have hm : i∈probeHoles p hp := by simpa [probeHoles,Fin.ext_iff] using h rw [ite_eq_left hm] rcases h with h | h · have _hm : i.val%2=1 := by omega simp [probeWord,Fin.val_castSucc,Fin.val_succ,h,show 2*p-1+1=2*p by omega, alternating,show (2*p-1)%2=1 by omega,show 2*p-1≠2*p by omega] · simp [probeWord,Fin.val_castSucc,Fin.val_succ,h,alternating,Nat.add_mod] · have hm : i∉probeHoles p hp := by simpa [probeHoles,Fin.ext_iff] using h rw [ite_eq_right hm] have h₁ : i.val≠2*p := by omega have h₂ : i.val+1≠2*p := by omega simp only [probeWord,Fin.val_castSucc,Fin.val_succ,ite_eq_right h₁,ite_eq_right h₂,alternating_succ] cases alternating true i.val <;> decide @[simp] lemma probeWord_first (p : ℕ) (hp : 1≤p) : probeWord p 0=true := by simp [probeWord,show 0≠2*p by omega] @[simp] lemma probeWord_last (p : ℕ) (hp : 1≤p) : probeWord p (Fin.last (4*p))=true := by simp [probeWord,show 4*p≠2*p by omega,alternating,Nat.mul_mod] lemma probeHoles_nonempty (p : ℕ) (hp : 1≤p) : (probeHoles p hp).Nonempty := by simp [probeHoles] lemma probeWord_unique (p : ℕ) (hp : 1≤p) (b : Selection (4*p)) (hb : Valid (probeHoles p hp) b) : b=probeWord p := broken_unique (probeHoles_nonempty p hp) hb (probeWord_valid p hp) lemma probe_weight (p : ℕ) (B Hu Hv lam : ℝ) : selectedWeight (profileActivity p B Hu Hv lam) (probeWord p)= (∏ r : Fin p,max B (Hu/2^r.val))*(∏ r : Fin p,max B (Hv/2^r.val)) := by have he : selectedWeight (profileActivity p B Hu Hv lam) (probeWord p)= selectedWeight (profileActivity p B Hu Hv 1) (fun i => alternating true i.val) := by unfold selectedWeight apply Finset.prod_congr rfl intro i _ by_cases hi : i.val=2*p · have hfi : i=⟨2*p,by omega⟩ := Fin.ext hi rw [hfi] simp [probeWord,alternating] · simp only [probeWord,ite_eq_right hi] congr 1 rcases lt_or_gt_of_ne hi with h | h · rw [profile_left _ _ _ _ _ _ h,profile_left _ _ _ _ _ _ h] · rw [profile_right _ _ _ _ _ _ h,profile_right _ _ _ _ _ _ h] rw [he,exact_profile_weight] simp variable {V E : Type*} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] variable (u v : E → V) (p : ℕ) (hp : 1≤p) (e₀ : E) def singleProbeHoles (e : E) : Finset (Fin (4*p)) := if e=e₀ then probeHoles p hp else ∅ @[simp] lemma broken_singleProbe : broken (singleProbeHoles p hp e₀)={e₀} := by ext e by_cases he : e=e₀ <;> simp [singleProbeHoles,he,probeHoles_nonempty] omit [Fintype V] [Fintype E] [DecidableEq V] in lemma singleProbe_word (b : E → Selection (4*p)) (hb : FamilyHoles u v (4*p) ∅ (singleProbeHoles p hp e₀) b) : b e₀=probeWord p := by exact probeWord_unique p hp (b e₀) (by simpa [singleProbeHoles] using hb.1 e₀) lemma residual_singleProbe (b : E → Selection (4*p)) (hb : FamilyHoles u v (4*p) ∅ (singleProbeHoles p hp e₀) b) : residualHoles u v ∅ (singleProbeHoles p hp e₀) b={u e₀,v e₀} := by ext x simp only [mem_residualHoles,notMem_empty,broken_singleProbe,mem_singleton, exists_eq_left,false_or,Uses,singleProbe_word u v p hp e₀ b hb, probeWord_first p hp,probeWord_last p hp,true_and,mem_insert,mem_singleton] exact or_congr eq_comm eq_comm def insertProbe (s : E → Bool) (e : E) : Selection (4*p) := if e=e₀ then probeWord p else fun i => alternating (s e) i.val include hp in omit [Fintype V] [Fintype E] [DecidableEq V] in lemma insertProbe_uses (s : E → Bool) (e : E) (x : V) : Uses u v (insertProbe p e₀ s) e x ↔ if e=e₀ then u e₀=x ∨ v e₀=x else s e=true ∧ (u e=x ∨ v e=x) := by by_cases he : e=e₀ · subst e; simp [Uses,insertProbe,probeWord_first p hp,probeWord_last p hp] · simp [Uses,insertProbe,he,alternating,and_or_left,Nat.mul_mod] omit [Fintype V] [Fintype E] in lemma insertProbe_family (_hne : u e₀≠v e₀) (s : E → Bool) (hs : LogicalHoles u v {u e₀,v e₀} {e₀} s) : FamilyHoles u v (4*p) ∅ (singleProbeHoles p hp e₀) (insertProbe p e₀ s) := by refine ⟨?_,by simp,?_⟩ · intro e by_cases he : e=e₀ · subst e; simpa [singleProbeHoles,insertProbe] using probeWord_valid p hp · simpa [singleProbeHoles,insertProbe,he] using alternating_valid (4*p) (s e) · intro x _ by_cases hx : x∈({u e₀,v e₀} : Finset V) · refine ⟨e₀,?_,?_⟩ · change Uses u v (insertProbe p e₀ s) e₀ x rw [insertProbe_uses u v p hp e₀,ite_eq_left rfl] simpa only [mem_insert,mem_singleton,eq_comm] using hx · intro e he by_contra hen rw [insertProbe_uses u v p hp e₀,ite_eq_right hen] at he exact hs.2.1 x hx e he · obtain ⟨e,he,hu⟩ := hs.2.2 x hx have he₀ : e≠e₀ := by intro h subst e exact (Bool.noConfusion (he.1.symm.trans (hs.1 e₀ (mem_singleton_self _)))) refine ⟨e,(insertProbe_uses u v p hp e₀ s e x).mpr (by simpa [he₀] using he),?_⟩ intro f hf by_cases hf₀ : f=e₀ · subst f rw [insertProbe_uses u v p hp e₀,ite_eq_left rfl] at hf exact False.elim (hx (by simpa only [mem_insert,mem_singleton,eq_comm] using hf)) · rw [insertProbe_uses u v p hp e₀,ite_eq_right hf₀] at hf exact hu f hf omit [Fintype V] [DecidableEq V] in lemma singleProbe_deletionConstant (b : E → Selection (4*p)) (hb : FamilyHoles u v (4*p) ∅ (singleProbeHoles p hp e₀) b) (B Hu Hv lam : E → ℝ) : deletionConstant (singleProbeHoles p hp e₀) b (fun e => profileActivity p (B e) (Hu e) (Hv e) (lam e)) (fun e => (∏ r : Fin p,max (B e) (Hu e/2^r.val))*(∏ r : Fin p,max (B e) (Hv e/2^r.val))) = ∏ e,(∏ r : Fin p,max (B e) (Hu e/2^r.val))*(∏ r : Fin p,max (B e) (Hv e/2^r.val)) := by unfold deletionConstant apply Finset.prod_congr rfl intro e _ by_cases he : e=e₀ · subst e simp only [singleProbeHoles,probeHoles_nonempty,ite_true, singleProbe_word u v p hp e₀ b hb,probe_weight] · simp [singleProbeHoles,he] theorem singleProbe_partition (hne : u e₀≠v e₀) (B Hu Hv lam : E → ℝ) : realHolePartition u v (4*p) ∅ (singleProbeHoles p hp e₀) (fun e => profileActivity p (B e) (Hu e) (Hv e) (lam e)) = (∏ e,(∏ r : Fin p,max (B e) (Hu e/2^r.val))*(∏ r : Fin p,max (B e) (Hv e/2^r.val))) * logicalHolePartition u v {u e₀,v e₀} {e₀} lam := by classical by_cases h : ∃ b,FamilyHoles u v (4*p) ∅ (singleProbeHoles p hp e₀) b · obtain ⟨b,hb⟩ := h have hid := deletion_partition_identity (show Even (4*p) from ⟨2*p,by omega⟩) u v ∅ (singleProbeHoles p hp e₀) b hb (fun e => profileActivity p (B e) (Hu e) (Hv e) (lam e)) (fun e => List.ofFn fun r : Fin p => max (B e) (Hu e/2^r.val)) (fun e => (List.ofFn fun r : Fin p => max (B e) (Hv e/2^r.val)).reverse) lam (fun e => profileActivity_ofFn p (B e) (Hu e) (Hv e) (lam e)) simp only [List.prod_reverse,List.prod_ofFn] at hid rw [singleProbe_deletionConstant u v p hp e₀ b hb,residual_singleProbe u v p hp e₀ b hb, broken_singleProbe] at hid exact hid · have hl : ¬∃ s,LogicalHoles u v {u e₀,v e₀} {e₀} s := by rintro ⟨s,hs⟩ exact h ⟨insertProbe p e₀ s,insertProbe_family u v p hp e₀ hne s hs⟩ have hz : realHolePartition u v (4*p) ∅ (singleProbeHoles p hp e₀) (fun e => profileActivity p (B e) (Hu e) (Hv e) (lam e))=0 := by unfold realHolePartition apply Finset.sum_eq_zero intro b _ exact False.elim (h ⟨b.val,b.property⟩) have hz' : logicalHolePartition u v {u e₀,v e₀} {e₀} lam=0 := by unfold logicalHolePartition apply Finset.sum_eq_zero intro s _ exact False.elim (hl ⟨s.val,s.property⟩) rw [hz,hz',mul_zero] end MatchingFPRAS.Subdivision namespace MatchingFPRAS.Representation open MatchingFPRAS.Switching MatchingFPRAS.Subdivision MatchingFPRAS.Enlargement Finset variable {V : Type*} [Fintype V] [LinearOrder V] omit [Fintype V] in lemma forbidden_endpoint_redundant (R : Finset V) (e₀ : LogicalEdge V) (he : source e₀∈R) (s : LogicalEdge V → Bool) : LogicalHoles source target R {e₀} s ↔ LogicalHoles source target R ∅ s := by constructor · intro h; exact ⟨by simp,h.2⟩ · intro h refine ⟨?_,h.2⟩ intro e hem have heq : e=e₀ := mem_singleton.mp hem subst e cases hs : s e₀ with | false => rfl | true => exact False.elim (h.2.1 (source e₀) he e₀ ⟨hs,Or.inl rfl⟩) lemma logical_partition_redundant (w : V → V → ℝ) (R : Finset V) (e₀ : LogicalEdge V) (he : source e₀∈R) : logicalHolePartition source target R {e₀} (logicalActivity w)=partition w R := by classical let := Fintype.ofFinite {s : LogicalEdge V → Bool // LogicalHoles source target R {e₀} s} let := Fintype.ofFinite {s : LogicalEdge V → Bool // LogicalHoles source target R ∅ s} rw [← logical_partition_eq] unfold logicalHolePartition apply Fintype.sum_equiv (Equiv.subtypeEquivRight (forbidden_endpoint_redundant R e₀ he)) intro s rfl end MatchingFPRAS.Representation namespace MatchingFPRAS.Enlargement open MatchingFPRAS.Switching MatchingFPRAS.Subdivision MatchingFPRAS.RealGeometry open MatchingFPRAS.Transport MatchingFPRAS.Representation Finset variable {V : Type*} [Fintype V] [LinearOrder V] [Nontrivial V] noncomputable local instance (w : V → V → ℝ) (K : ℕ) : Fintype (BagEdge (sourceBags w K)) := Fintype.ofFinite _ theorem source_probe_partition (w : V → V → ℝ) (K : ℕ) (e₀ : LogicalEdge V) : partitionWeight BagEdge.Incident (sourceRealPortion w K) (survivors (4*(2*K+2)) ∅ (singleProbeHoles (2*K+2) (by omega) e₀)) = (∏ e, pathConstant (2*K+2) (bottleneck w (source e) (target e)) (sourceHeight w (source e)) (sourceHeight w (target e))) * partition w {source e₀,target e₀} := by rw [sourceRealPortion_partition] change realHolePartition source target (4*(2*K+2)) ∅ (singleProbeHoles (2*K+2) (by omega) e₀) (fun e => profileActivity (2*K+2) (bottleneck w (source e) (target e)) (sourceHeight w (source e)) (sourceHeight w (target e)) (logicalActivity w e)) = _ rw [singleProbe_partition _ _ _ _ _ (source_ne_target e₀)] rw [logical_partition_redundant w _ e₀ (by simp)] rfl theorem source_probe_ratio (w : V → V → ℝ) (hz : 0 pathConstant_pos _ _ _ _ (zero_lt_one.trans_le (one_le_bottleneck w _ _))) unfold holeRatio field_simp end MatchingFPRAS.Enlargement end OAI