| import OAI.Combinatorics.ListHadwiger.RecursiveJunction |
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| namespace OAI |
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| namespace LinearListHadwiger.Recursion |
| open Finset Stage SmallList SimpleGraph |
| attribute [local instance] Classical.propDecidable |
| variable {V : Type} [Fintype V] [DecidableEq V] |
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| lemma neighbors_on {O : Type} [Fintype O] (G : SimpleGraph V) [DecidableRel G.Adj] |
| (S Z : Finset V) (port : O → V) (hp : ∀ o, port o∈S) {k : ℕ} |
| (hc : KConnectedOn G S k) (hk : Fintype.card O+Z.card≤k) : |
| ∃ n : O → V, Function.Injective n ∧ ∀ o, n o∈S\Z ∧ G.Adj (port o) (n o) := by |
| classical |
| apply Assembly.attach_ports G port (S\Z) |
| intro o |
| have hdeg := Knitting.min_degree G hc (hp o) |
| have he : Density.neighbors G S (port o) \ Z=(S\Z).filter (G.Adj (port o)) := by |
| ext v |
| simp only [Density.neighbors,mem_sdiff,mem_filter] |
| tauto |
| have hh := card_le_card (show Density.neighbors G S (port o)⊆ |
| Density.neighbors G S (port o)\Z∪Z from fun v hv => by |
| by_cases hz : v∈Z |
| · exact mem_union_right _ hz |
| · exact mem_union_left _ (mem_sdiff.mpr ⟨hv,hz⟩)) |
| have hu := card_union_le (Density.neighbors G S (port o)\Z) Z |
| rw [he] at hh hu |
| have hbound : Fintype.card O≤((S\Z).filter (G.Adj (port o))).card := by omega |
| exact hbound |
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| omit [Fintype V] in |
| lemma connected_induce_univ (G : SimpleGraph V) (J : Finset V) {k : ℕ} |
| (hc : KConnectedOn G J k) : KConnectedOn (G.induce (↑J:Set V)) univ k := by |
| have hh := under_connected G (U:=J) (A:=J) (subset_refl J) hc |
| have he : under J J=univ := by ext v; simp [under] |
| rwa [he] at hh |
|
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| theorem fresh_model : ∃ S : ℕ, 1≤S ∧ ∃ T : ℕ, 1≤T ∧ |
| ∀ a K Q : ℕ, 1≤a → 1≤K → T≤64*a → |
| (∀ (W : Type) [Fintype W] [DecidableEq W] (P : SimpleGraph W), |
| KConnectedOn P univ (K*child a) → Q*child a≤listChromaticNumber P → |
| Woven P (child a) (3*child a) ∨ FractionalObstruction P) → |
| ∀ (V : Type) [Fintype V] [DecidableEq V] (G : SimpleGraph V) |
| (H : Finset V) (r : Fin a → V), Function.Injective r → (∀ i, r i∈H) → |
| hadwigerNumber G<64*a → KConnectedOn G H (K₀*(64*a)) → |
| 3*S*(64*a)+3*a≤ell G H → |
| Q*child a+6*(K*child a)+2*S*(64*a)+3*a<ell G H → |
| (∃ M : RootedCliqueModel G r, ∀ i, M.branch i⊆(↑H:Set V)) ∨ FractionalObstruction G := by |
| obtain ⟨S,hS,T,hT,hs⟩ := three_regions |
| refine ⟨S,hS,T,hT,?_⟩ |
| intro a K Q ha hK ht ih V _ _ G H r hr hrH hm hc hsep hl |
| classical |
| let R := univ.image r |
| have hR : R.card=a := by simp only [R,card_image_of_injective _ hr,card_univ,Fintype.card_fin] |
| have hRH : R⊆H := by rintro v hv; obtain ⟨i,_,rfl⟩ := mem_image.mp hv; exact hrH i |
| obtain ⟨n,hn,hnH⟩ := neighbors_on G H R (fun o : Fin a × Fin 2 => r o.1) |
| (fun _ => hrH _) hc (by simp only [Fintype.card_prod,Fintype.card_fin,hR]; dsimp [K₀]; omega) |
| let N := univ.image n |
| have hN : N.card=2*a := by simp only [N,card_image_of_injective _ hn,card_univ,Fintype.card_prod,Fintype.card_fin]; omega |
| let B := R∪N |
| have hB : B.card≤3*a := by have := card_union_le R N; dsimp [B]; omega |
| have hdel := ell_delete G H B |
| have ha' : 1≤child a := by unfold child; omega |
| rcases hs (64*a) ht V G (H\B) (K*child a) (Q*child a) (by nlinarith) hm |
| (by omega) (by omega) with ⟨J,hJH,hJJ,hJc,hJl⟩|hb |
| swap |
| · exact Or.inr hb |
| have hchild (d : Fin 3) : WovenOn G (↑(J d):Set V) (child a) (3*child a) ∨ FractionalObstruction G := by |
| rcases ih (↑(J d):Set V) (G.induce (↑(J d):Set V)) |
| (connected_induce_univ G (J d) (hJc d)) (Nat.le_of_lt (hJl d)) with hw|hb |
| · exact Or.inl (woven_induce G (J d) hw) |
| · exact Or.inr (obstruction_induce G (J d) hb) |
| by_cases hb : FractionalObstruction G |
| · exact Or.inr hb |
| have hw (d) : WovenOn G (↑(J d):Set V) (child a) (3*child a) := (hchild d).resolve_right hb |
| have hroots (d : Fin 3) : ∃ u : Fin (child a) → V, |
| Function.Injective u ∧ ∀ i, u i∈J d := by |
| apply choose_vertices |
| have hh := (hJc d).1 |
| nlinarith |
| choose u hu huJ using hroots |
| have hconn : KConnectedOn G (H\univ.image r) (65536*(4*a)) := by |
| apply Knitting.kConnected_mono (Knitting.delete_connected G hc hRH (by rw [hR]; dsimp [K₀]; omega)) |
| rw [hR] |
| dsimp [K₀] |
| omega |
| apply Or.inl |
| apply three_district_model G ha H r hr hrH n hn |
| (fun o => (mem_sdiff.mp (hnH o).1).1) |
| (fun o i he => (mem_sdiff.mp (hnH o).1).2 (he ▸ mem_image.mpr ⟨i,mem_univ _,rfl⟩)) |
| (fun o => (hnH o).2.symm) (fun d => (↑(J d):Set V)) |
| (fun d => (hJH d).trans sdiff_subset) ?_ ?_ (fun d e hne => by exact_mod_cast hJJ d e hne) |
| hw u hu huJ hconn |
| · intro d i hi |
| exact (mem_sdiff.mp (hJH d hi)).2 (mem_union_left _ (mem_image.mpr ⟨i,mem_univ _,rfl⟩)) |
| · intro d o ho |
| exact (mem_sdiff.mp (hJH d ho)).2 (mem_union_right _ (mem_image.mpr ⟨o,mem_univ _,rfl⟩)) |
|
|
| end LinearListHadwiger.Recursion |
|
|
| namespace LinearListHadwiger.Recursion |
| open Finset Stage SmallList SimpleGraph |
| attribute [local instance] Classical.propDecidable |
|
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| theorem recursive_woven_or_obstruction : ∃ K Q : ℕ, 100≤K ∧ 1≤Q ∧ |
| ∀ a : ℕ, 1≤a → ∀ (V : Type) [Fintype V] [DecidableEq V] (G : SimpleGraph V), |
| KConnectedOn G univ (K*a) → Q*a≤listChromaticNumber G → |
| Woven G a (3*a) ∨ FractionalObstruction G := by |
| obtain ⟨A,hA,T₁,hT₁,houter⟩ := outer_region |
| obtain ⟨S,hS,T₂,hT₂,hfresh⟩ := fresh_model |
| let K := A+100 |
| let a₀ := max 24 (max T₁ T₂) |
| let B := ⌊(200000000:ℝ)*(64*a₀:ℕ)*Real.log (2*(64*a₀:ℕ))⌋₊+1 |
| let c : ℕ := 4*32768+42+5+6*K₀ |
| let Q := B+A+18*K+512*S+40+256*c+1 |
| have hK : 100≤K := by dsimp [K]; omega |
| have hQ : 1≤Q := by dsimp [Q]; omega |
| have hQA : A+7<Q := by dsimp [Q]; omega |
| have hQbig : 18*K+512*S+40+256*c<Q := by dsimp [Q]; omega |
| have hQB : B<Q := by dsimp [Q]; omega |
| refine ⟨K,Q,hK,hQ,?_⟩ |
| intro a |
| induction a using Nat.strong_induction_on with |
| | h a ih => |
| intro ha V _ _ G hc hl |
| classical |
| have hka : 100*a≤K*a := Nat.mul_le_mul_right a hK |
| by_cases hminor : 64*a≤hadwigerNumber G |
| · exact Or.inl (large_minor_woven G ha (Knitting.kConnected_mono hc hka) hminor) |
| have hm : hadwigerNumber G<64*a := by omega |
| have ha₀ : a₀<a := by |
| by_contra hn |
| have hma : hadwigerNumber G<64*a₀ := hm.trans_le (by omega) |
| have hb := CoarseDensity.coarse_list_bound G (show 2≤64*a₀ by dsimp [a₀]; omega) hma |
| change listChromaticNumber G≤B at hb |
| nlinarith only [ha,hl,hQB,hb] |
| have hac : 24<a := (le_max_left _ _).trans_lt ha₀ |
| obtain ⟨hcpos,hclt,hcshrink⟩ := child_shrink hac |
| have ht₁ : T₁≤64*a := by have := (le_max_left T₁ T₂).trans (le_max_right 24 _); omega |
| have ht₂ : T₂≤64*a := by have := (le_max_right T₁ T₂).trans (le_max_right 24 _); omega |
| by_cases hbad : FractionalObstruction G |
| · exact Or.inr hbad |
| left |
| intro roots hr _ p hp x y _ hxy |
| let O := Fin a ⊕ (Fin p × Bool) |
| let port : O → V := Sum.elim roots (fun z => if z.2 then y z.1 else x z.1) |
| let Z := univ.image port |
| have hO : Fintype.card O=a+2*p := by simp [O]; omega |
| have hZ : Z.card≤7*a := by |
| have hh := card_image_le (s:=univ) (f:=port) |
| rw [card_univ,hO] at hh |
| calc Z.card≤a+2*p := hh |
| _≤7*a := by omega |
| obtain ⟨proxy,hproxy,hadj⟩ := distinct_proxies G port Z hc |
| (show Fintype.card O+Z.card≤K*a by rw [hO]; omega) |
| let R := univ\Z |
| let P := G.induce (↑R:Set V) |
| let proxy' (o : O) : (↑R:Set V) := ⟨proxy o,mem_sdiff.mpr ⟨mem_univ _,(hadj o).1⟩⟩ |
| have hproxy' : Function.Injective proxy' := fun i j he => hproxy (congrArg Subtype.val he) |
| let e : (Fin p × Bool) ≃ Fin (2*p) := Fintype.equivOfCardEq (by simp; omega) |
| let r (i : Fin a) := proxy' (.inl i) |
| let n (j : Fin (2*p)) := proxy' (.inr (e.symm j)) |
| have hr' : Function.Injective r := hproxy'.comp Sum.inl_injective |
| have hn' : Function.Injective n := hproxy'.comp (Sum.inr_injective.comp e.symm.injective) |
| have hrn (i j) : r i≠n j := by intro he; cases hproxy' he |
| have hcR : KConnectedOn G R (30*a) := by |
| apply Knitting.kConnected_mono (Knitting.delete_connected G hc (subset_univ Z) (by omega)) |
| omega |
| have hcP : KConnectedOn P univ (30*a) := connected_induce_univ G R hcR |
| have hmP : hadwigerNumber P<64*a := (Knitting.model_induce G R).hadwiger_le.trans_lt hm |
| have hdel := ell_delete G univ Z |
| rw [ell_univ] at hdel |
| change listChromaticNumber G≤listChromaticNumber P+Z.card at hdel |
| have hQAa := Nat.mul_lt_mul_of_pos_right hQA (show 0<a by omega) |
| have hPA : A*a<listChromaticNumber P := by nlinarith only [hQAa,hdel,hl,hZ] |
| have noP : ¬FractionalObstruction P := fun hb => hbad (obstruction_induce G R hb) |
| obtain ⟨H,D,U,⟨routing⟩,hHD,hHc,hDc,hHl⟩ := |
| (houter a ha ht₁ p hp (↑R:Set V) P r n hr' hn' hrn hcP hmP hPA).resolve_right noP |
| have hcall (W : Type) [Fintype W] [DecidableEq W] (L : SimpleGraph W) |
| (hconn : KConnectedOn L univ (K*child a)) (hlist : Q*child a≤listChromaticNumber L) : |
| Woven L (child a) (3*child a) ∨ FractionalObstruction L := |
| ih (child a) hclt hcpos W L hconn hlist |
| have hsep : 3*S*(64*a)+3*a≤ell P H := by |
| change listChromaticNumber P≤ell P H+c*(64*a) at hHl |
| have hb : 192*S+10+64*c<Q := by omega |
| have hh := Nat.mul_lt_mul_of_pos_right hb (show 0<a by omega) |
| nlinarith only [hh,hHl,hdel,hl,hZ] |
| have hstep : Q*child a+6*(K*child a)+2*S*(64*a)+3*a<ell P H := by |
| change listChromaticNumber P≤ell P H+c*(64*a) at hHl |
| have hq := Nat.mul_le_mul_left Q hcshrink |
| have hk := Nat.mul_le_mul_left (6*K) hcshrink |
| have hh := Nat.mul_lt_mul_of_pos_right hQbig (show 0<a by omega) |
| nlinarith only [hh,hq,hk,hHl,hdel,hl,hZ] |
| obtain ⟨M,hMH⟩ := (hfresh a K Q ha (by omega) ht₂ hcall (↑R:Set V) P H routing.fresh |
| routing.fresh_injective routing.fresh_mem hmP hHc hsep hstep).resolve_right noP |
| have hgroup : KConnectedOn P D (65536*Fintype.card (Roles a (2*p))) := by |
| apply Knitting.kConnected_mono hDc |
| simp only [Roles,Fintype.card_sum,Fintype.card_fin] |
| dsimp [K₀] |
| omega |
| obtain ⟨C₀,hC₀,hCC₀,hE₀,hrC₀,hnC₀⟩ := junction_groups P ha r n e H D _ U hHD routing M hMH hgroup |
| let lift := Knitting.model_induce G R |
| let C (i : Fin a ⊕ Fin p) := lift.expand (C₀ i) |
| have hC (i) : (G.induce (C i)).Connected := lift.expand_connected (hC₀ i) |
| have hCC (i j) (hij : i≠j) : Disjoint (C i) (C j) := lift.expand_disjoint (hCC₀ i j hij) |
| have hCZ (i v) (hv : v∈C i) : v∉Z := by |
| obtain ⟨w,_,hw⟩ := hv |
| have he : v=w.val := hw |
| exact he ▸ (mem_sdiff.mp w.property).2 |
| have hpZ (o : O) : port o∈Z := mem_image.mpr ⟨o,mem_univ _,rfl⟩ |
| apply weaving_from_proxy_groups G roots x y hr hxy C hC hCC |
| (fun i j hv => hCZ i _ hv (hpZ (.inl j))) |
| (fun i j hv => hCZ i _ hv (hpZ (.inr (j,false)))) |
| (fun i j hv => hCZ i _ hv (hpZ (.inr (j,true)))) |
| · intro i j hij |
| obtain ⟨v,hv,w,hw,he⟩ := hE₀ i j hij |
| exact ⟨v.val,⟨v,hv,rfl⟩,w.val,⟨w,hw,rfl⟩,he⟩ |
| · intro i |
| exact ⟨proxy (.inl i),⟨r i,hrC₀ i,rfl⟩,(hadj (.inl i)).2⟩ |
| · intro i |
| have hh := hnC₀ (i,false) |
| have he : n (e (i,false))=proxy' (.inr (i,false)) := by simp [n] |
| rw [he] at hh |
| exact ⟨proxy (.inr (i,false)),⟨proxy' (.inr (i,false)),hh,rfl⟩,(hadj (.inr (i,false))).2⟩ |
| · intro i |
| have hh := hnC₀ (i,true) |
| have he : n (e (i,true))=proxy' (.inr (i,true)) := by simp [n] |
| rw [he] at hh |
| exact ⟨proxy (.inr (i,true)),⟨proxy' (.inr (i,true)),hh,rfl⟩,(hadj (.inr (i,true))).2⟩ |
|
|
| end LinearListHadwiger.Recursion |
|
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| end OAI |
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