| import OAI.Combinatorics.ListHadwiger.RecursiveDistricts |
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| namespace OAI |
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| namespace LinearListHadwiger.Recursion |
| open SimpleGraph Finset Stage |
| attribute [local instance] Classical.propDecidable |
| variable {V : Type} [Fintype V] [DecidableEq V] |
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| theorem junction_groups (G : SimpleGraph V) {a p : ℕ} (ha : 1≤a) |
| (r : Fin a → V) (n : Fin (2*p) → V) (e : (Fin p × Bool) ≃ Fin (2*p)) (H D : Finset V) (F U : Set V) |
| (hHD : Disjoint H D) |
| (R : RoleRouting G r n F (↑H:Set V) D U) |
| (M : RootedCliqueModel G R.fresh) (hMH : ∀ i, M.branch i⊆(↑H:Set V)) |
| (hc : KConnectedOn G D (65536*Fintype.card (Roles a (2*p)))) : |
| ∃ C : (Fin a ⊕ Fin p) → Set V, |
| (∀ i, (G.induce (C i)).Connected) ∧ |
| (∀ i j, i≠j → Disjoint (C i) (C j)) ∧ |
| (∀ i j, i≠j → ∃ v∈C (.inl i), ∃ w∈C (.inl j), G.Adj v w) ∧ |
| (∀ i, r i∈C (.inl i)) ∧ |
| (∀ q, n (e q)∈C (.inr q.1)) := by |
| classical |
| let owner : Roles a (2*p) → Fin a ⊕ Fin p := |
| Sum.elim Sum.inl (Sum.elim (fun j => Sum.inr (e.symm j).1) Sum.inl) |
| let start := roleStart r n R.fresh |
| let target (o : Roles a (2*p)) := (R.target o).val |
| have htar : Function.Injective target := Subtype.val_injective.comp R.target_injective |
| have hs : Function.Surjective owner := by |
| intro i |
| cases i with |
| | inl i => exact ⟨.inl i,rfl⟩ |
| | inr i => exact ⟨.inr (.inl (e (i,false))),by simp [owner]⟩ |
| have hcpos : 1≤Fintype.card (Roles a (2*p)) := by simp only [Roles,Fintype.card_sum,Fintype.card_fin]; omega |
| obtain ⟨E,hE,hEE,hED,htE⟩ := grouped_ends G D target htar (fun o => (R.target o).property) |
| owner hs hcpos hc |
| let B : Roles a (2*p) → Set V := Sum.elim (fun i => {r i}) (Sum.elim (fun j => {n j}) M.branch) |
| have hB (o) : (G.induce (B o)).Connected := by |
| rcases o with i|(j|i) |
| · change (G.induce ({r i}:Set V)).Connected |
| have : Nonempty ({r i}:Set V) := ⟨⟨r i,rfl⟩⟩ |
| exact Connected.of_subsingleton |
| · change (G.induce ({n j}:Set V)).Connected |
| have : Nonempty ({n j}:Set V) := ⟨⟨n j,rfl⟩⟩ |
| exact Connected.of_subsingleton |
| · exact M.connected i |
| have hxB (o) : start o∈B o := by |
| rcases o with i|(j|i) |
| · rfl |
| · rfl |
| · exact M.root_mem i |
| have hBP (o q) (hne : owner o≠owner q) : |
| Disjoint (B o) {v | v∈(R.route q).val.support} := by |
| apply Set.disjoint_left.mpr |
| intro v hv hw |
| rcases o with i|(j|i) |
| · have he : v=r i := hv |
| subst v |
| have hneq : Sum.inl i≠q := fun h => hne (congrArg owner h) |
| exact Set.disjoint_left.mp (R.disjoint _ _ hneq) (R.route (.inl i)).val.start_mem_support hw |
| · have he : v=n j := hv |
| subst v |
| have hneq : Sum.inr (Sum.inl j)≠q := fun h => hne (congrArg owner h) |
| exact Set.disjoint_left.mp (R.disjoint _ _ hneq) (R.route (.inr (.inl j))).val.start_mem_support hw |
| · obtain ⟨j,rfl,hv'⟩ := R.new_only q v hw (hMH i hv) |
| have hij : i≠j := by intro h; subst j; exact hne rfl |
| exact Set.disjoint_left.mp (M.disjoint i j hij) hv (hv'.symm ▸ M.root_mem j) |
| have hBB (o q) (hne : owner o≠owner q) : Disjoint (B o) (B q) := by |
| apply Set.disjoint_left.mpr |
| intro v hv hw |
| rcases q with j|(j|j) |
| · have he : v=r j := hw |
| exact Set.disjoint_left.mp (hBP o (.inl j) hne) hv (he.symm ▸ (R.route (.inl j)).val.start_mem_support) |
| · have he : v=n j := hw |
| exact Set.disjoint_left.mp (hBP o (.inr (.inl j)) hne) hv (he.symm ▸ (R.route (.inr (.inl j))).val.start_mem_support) |
| · rcases o with i|(i|i) |
| · have he : v=r i := hv |
| exact Set.disjoint_left.mp (hBP (.inr (.inr j)) (.inl i) (Ne.symm hne)) hw |
| (he.symm ▸ (R.route (.inl i)).val.start_mem_support) |
| · have he : v=n i := hv |
| exact Set.disjoint_left.mp (hBP (.inr (.inr j)) (.inr (.inl i)) (Ne.symm hne)) hw |
| (he.symm ▸ (R.route (.inr (.inl i))).val.start_mem_support) |
| · exact Set.disjoint_left.mp (M.disjoint i j (fun h => hne (congrArg Sum.inl h))) hv hw |
| have hEP (i o) (hne : i≠owner o) : Disjoint (E i) {v | v∈(R.route o).val.support} := by |
| apply Set.disjoint_left.mpr |
| intro v hv hw |
| have he := R.target_only o v hw (hED i hv) |
| exact Set.disjoint_left.mp (hEE i (owner o) hne) hv (he.symm ▸ htE o) |
| have hEB (i o) (hne : i≠owner o) : Disjoint (E i) (B o) := by |
| apply Set.disjoint_left.mpr |
| intro v hv hw |
| rcases o with j|(j|j) |
| · have he : v=r j := hw |
| exact Set.disjoint_left.mp (hEP i (.inl j) hne) hv (he.symm ▸ (R.route (.inl j)).val.start_mem_support) |
| · have he : v=n j := hw |
| exact Set.disjoint_left.mp (hEP i (.inr (.inl j)) hne) hv (he.symm ▸ (R.route (.inr (.inl j))).val.start_mem_support) |
| · exact Finset.disjoint_left.mp hHD (hMH j hw) (hED i hv) |
| let C := extended E B owner start target R.route |
| have hC (i) : (G.induce (C i)).Connected := extended_connected E B owner start target R.route |
| hE hB hxB htE i |
| have hCC : ∀ i j, i≠j → Disjoint (C i) (C j) := |
| extended_disjoint E B owner start target R.route hEE hBB hEB hEP hBP |
| (fun o q hne => R.disjoint o q (fun h => hne (congrArg owner h))) |
| have hBC (o) : B o⊆C (owner o) := fun _ hv => Or.inr ⟨o,rfl,Or.inl hv⟩ |
| refine ⟨C,hC,hCC,?_,fun i => hBC (.inl i) rfl,?_⟩ |
| · intro i j hij |
| obtain ⟨v,hv,w,hw,he⟩ := M.adjacent i j hij |
| exact ⟨v,hBC (.inr (.inr i)) hv,w,hBC (.inr (.inr j)) hw,he⟩ |
| · intro q |
| have hmem := hBC (.inr (.inl (e q))) rfl |
| simpa only [owner,Sum.elim_inr,Sum.elim_inl,e.symm_apply_apply] using hmem |
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| end LinearListHadwiger.Recursion |
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| namespace LinearListHadwiger.Recursion |
| open Finset Stage Footprint SmallList Filter |
| open scoped Topology |
| attribute [local instance] Classical.propDecidable |
| variable {V : Type} [Fintype V] [DecidableEq V] |
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| lemma outer_size_cutoff : ∃ T : ℕ, ∀ t : ℕ, T≤t → |
| (t:ℝ)^(53/50:ℝ)+(t:ℝ)≤(t:ℝ)^(11/10:ℝ) := by |
| have he : ∀ᶠ x : ℝ in atTop, 1≤x ∧ 2≤x^(1/25:ℝ) := |
| (eventually_ge_atTop 1).and ((tendsto_rpow_atTop (by norm_num : (0:ℝ)<1/25)).eventually (eventually_ge_atTop 2)) |
| obtain ⟨S,hS⟩ := (eventually_atTop.mp he) |
| refine ⟨⌈S⌉₊,fun t ht => ?_⟩ |
| obtain ⟨h1,h2⟩ := hS t ((Nat.le_ceil S).trans (by exact_mod_cast ht)) |
| have hle : (t:ℝ)≤(t:ℝ)^(53/50:ℝ) := by |
| simpa using Real.rpow_le_rpow_of_exponent_le h1 (by norm_num : (1:ℝ)≤53/50) |
| have heq : (t:ℝ)^(11/10:ℝ)=(t:ℝ)^(53/50:ℝ)*(t:ℝ)^(1/25:ℝ) := by |
| rw [←Real.rpow_add (by positivity : (0:ℝ)<t)] |
| norm_num |
| rw [heq] |
| nlinarith [Real.rpow_nonneg (by positivity : (0:ℝ)≤t) (53/50:ℝ)] |
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| theorem outer_region : ∃ A : ℕ, 1≤A ∧ ∃ T : ℕ, 1≤T ∧ |
| ∀ a : ℕ, 1≤a → T≤64*a → ∀ p : ℕ, p≤3*a → |
| ∀ (V : Type) [Fintype V] [DecidableEq V] (G : SimpleGraph V) |
| (r : Fin a → V) (n : Fin (2*p) → V), |
| Function.Injective r → Function.Injective n → (∀ i j, r i≠n j) → |
| KConnectedOn G univ (30*a) → hadwigerNumber G<64*a → |
| A*a<listChromaticNumber G → |
| (∃ H D : Finset V, ∃ U : Set V, |
| Nonempty (RoleRouting G r n (↑(univ.image r):Set V) (↑H:Set V) D U) ∧ |
| Disjoint H D ∧ KConnectedOn G H (K₀*(64*a)) ∧ |
| KConnectedOn G D (K₀*(64*a)) ∧ |
| listChromaticNumber G≤ell G H+(4*32768+42+5+6*K₀)*(64*a)) ∨ |
| FractionalObstruction G := by |
| obtain ⟨D,hD,Td,hTd⟩ := districts_on K₀ (by norm_num [K₀]) |
| obtain ⟨Tx,hTx⟩ := outer_size_cutoff |
| obtain ⟨Ta,hTa⟩ := additive_deletion_or_obstruction |
| let c : ℕ := 4*32768+42+5 |
| let A := 128*D+10+64*(c+6*K₀) |
| refine ⟨A,by dsimp [A]; omega,max 1 (max Td (max Tx Ta)),by omega,?_⟩ |
| intro a ha ht p hp V _ _ G r n hr hn hrn hc hm hl |
| classical |
| have htd : Td≤64*a := (le_max_left _ _).trans ((le_max_right _ _).trans ht) |
| have htx : Tx≤64*a := (le_max_left _ _).trans ((le_max_right _ _).trans ((le_max_right _ _).trans ht)) |
| have hta : Ta≤64*a := (le_max_right _ _).trans ((le_max_right _ _).trans ((le_max_right _ _).trans ht)) |
| let Z := oldSources r n |
| have hz : Z.card≤7*a := by |
| have h := card_union_le (univ.image r) (univ.image n) |
| have h1 := card_image_le (s:=univ) (f:=r) |
| have h2 := card_image_le (s:=univ) (f:=n) |
| simp only [card_univ,Fintype.card_fin] at h1 h2 |
| change ((univ.image r)∪(univ.image n)).card≤7*a |
| omega |
| have hdel := ell_delete G univ Z |
| rw [ell_univ] at hdel |
| obtain ⟨J,hJS,hJc,hJs⟩ := hTd (64*a) htd V G (univ\Z) hm (by dsimp [A] at hl; nlinarith) |
| have hZJ : Disjoint Z J := disjoint_sdiff_self_right.mono_right hJS |
| have hk : 2*Z.card≤30*a := by omega |
| have hJcard : 2*Z.card≤J.card := by have := hJc.1; dsimp [K₀] at this; omega |
| obtain ⟨R,P,hRU,hRn,hRZ,hRJ,hPR⟩ := first_system univ univ Z J (Subset.refl _) (subset_univ _) (subset_univ _) hZJ |
| (Knitting.kConnected_mono hc hk) hJcard |
| let X := Z∪J |
| let Y := R.used\X |
| have hX : (X.card:ℝ)≤((64*a:ℕ):ℝ)^(11/10:ℝ) := by |
| have hcard := card_union_le Z J |
| have hle : (X.card:ℝ)≤(J.card:ℝ)+(64*a:ℕ) := by exact_mod_cast (by dsimp [X]; omega : X.card≤J.card+64*a) |
| have hcut := hTx (64*a) htx |
| linarith |
| have hY : DegenerateOn G Y (42*(64*a)) := by |
| intro B hBY hB |
| obtain ⟨v,hv,hdeg⟩ := R.degenerate B (hBY.trans sdiff_subset) hB |
| exact ⟨v,hv,hdeg.trans (by rw [hRn]; omega)⟩ |
| have hnY (v) (hv : v∉X∪Y) : (Y.filter (G.Adj v)).card≤42*(64*a) := by |
| have hnot : v∉R.used := by |
| intro hw |
| by_cases hx : v∈X |
| · exact hv (mem_union_left _ hx) |
| · exact hv (mem_union_right _ (mem_sdiff.mpr ⟨hw,hx⟩)) |
| have hd : v∈R.domain := by rw [hRU]; exact mem_univ v |
| exact (card_le_card (filter_subset_filter _ sdiff_subset)).trans |
| ((R.used_neighbors v hd hnot).trans (by rw [hRn]; omega)) |
| rcases hTa (64*a) hta V G X Y 42 hm hX disjoint_sdiff_self_right hY hnY with hloss|hb |
| swap |
| · exact Or.inr hb |
| have hlarge : 6*(K₀*(64*a))<ell G (univ\(X∪Y)) := by |
| change listChromaticNumber G≤ell G (univ\(X∪Y))+c*(64*a) at hloss |
| dsimp [A] at hl |
| nlinarith |
| obtain ⟨H,hHS,hHc,hHl⟩ := extract_on G (univ\(X∪Y)) (by dsimp [K₀]; omega) hlarge |
| have hXH : Disjoint X H := by |
| apply disjoint_left.mpr |
| intro v hv hw |
| exact (mem_sdiff.mp (hHS hw)).2 (mem_union_left _ hv) |
| have hRH : Disjoint R.used H := by |
| apply disjoint_left.mpr |
| intro v hv hw |
| have hout := (mem_sdiff.mp (hHS hw)).2 |
| by_cases hx : v∈X |
| · exact hout (mem_union_left _ hx) |
| · exact hout (mem_union_right _ (mem_sdiff.mpr ⟨hv,hx⟩)) |
| have hZH : Disjoint Z H := hXH.mono_left (subset_union_left) |
| have hJH : Disjoint J H := hXH.mono_left (subset_union_right) |
| have hHsub : H⊆univ\Z := fun v hv => mem_sdiff.mpr ⟨mem_univ _,fun hz => disjoint_left.mp hZH hz hv⟩ |
| have hcc : KConnectedOn G (univ\Z) (2*a) := by |
| exact Knitting.kConnected_mono (Knitting.delete_connected G hc (subset_univ Z) (by omega)) (by omega) |
| obtain ⟨Q⟩ := Second.exists_in_region (univ\Z) (univ\Z) H J (2*a) (Subset.refl _) hHsub hJS hcc |
| (by have := hHc.1; dsimp [K₀] at this; omega) (by have := hJc.1; dsimp [K₀] at this; omega) |
| have hF : univ\(univ.image r\univ.image r)=(univ:Finset V) := by simp |
| have hRu : R.domain=univ\(univ.image r\univ.image r) := by rwa [hF] |
| obtain ⟨routing⟩ := combine_systems (univ.image r) H J r n hr hn hrn |
| (fun i => mem_image.mpr ⟨i,mem_univ _,rfl⟩) hZH hZJ hJH.symm R hRu P hPR hRH Q |
| refine Or.inl ⟨H,J,_,⟨routing⟩,hJH.symm,hHc,hJc,?_⟩ |
| change listChromaticNumber G≤ell G (univ\(X∪Y))+c*(64*a) at hloss |
| dsimp [c] at hloss |
| nlinarith |
|
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| end LinearListHadwiger.Recursion |
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| end OAI |
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