| import OAI.Combinatorics.ListHadwiger.Recursion |
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| namespace OAI |
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| namespace LinearListHadwiger.Recursion |
| open SimpleGraph Finset |
| attribute [local instance] Classical.propDecidable |
| variable {V I : Type} [DecidableEq V] [Fintype I] {G : SimpleGraph V} |
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| theorem reroute_selected (J : Set V) {a b : ℕ} (hJ : WovenOn G J a b) |
| (u : Fin a → V) (hu : Function.Injective u) (huJ : ∀ j, u j∈J) |
| (keep : Fin a → Prop) |
| (x y : I → V) (P : ∀ i, G.Path (x i) (y i)) |
| (hcard : Fintype.card I ≤ b) |
| (hPP : ∀ i j, i≠j → Disjoint {v | v∈(P i).val.support} {v | v∈(P j).val.support}) |
| (hroot : ∀ i j, keep j → u j∈(P i).val.support → u j=x i) : |
| ∃ M : RootedCliqueModel G u, (∀ j, M.branch j⊆J) ∧ |
| ∃ Q : ∀ i, G.Path (x i) (y i), |
| (∀ i j, i≠j → Disjoint {v | v∈(Q i).val.support} {v | v∈(Q j).val.support}) ∧ |
| (∀ i v, v∈(Q i).val.support → v∉J → v∈(P i).val.support) ∧ |
| (∀ i j v, keep j → v∈(Q i).val.support → v∈M.branch j → v=x i ∧ v=u j) := by |
| classical |
| let e : Fin (Fintype.card I) ≃ I := (Fintype.equivFin I).symm |
| obtain ⟨M,hMJ,Q,hQQ,_,hQP,hQM⟩ := reroute_woven_district G J hJ u hu huJ |
| (x ∘ e) (y ∘ e) (fun i => P (e i)) hcard |
| (fun i j hij => hPP _ _ (fun h => hij (e.injective h))) |
| let Q' : ∀ i, G.Path (x i) (y i) := fun i => |
| ⟨(Q (e.symm i)).val.copy (congrArg x (e.apply_symm_apply i)) |
| (congrArg y (e.apply_symm_apply i)),by simp⟩ |
| have hsup (i : I) (v : V) : v∈(Q' i).val.support ↔ v∈(Q (e.symm i)).val.support := by |
| simp [Q'] |
| refine ⟨M,hMJ,Q',?_,?_,?_⟩ |
| · intro i j hij |
| apply Set.disjoint_left.mpr |
| intro v hv hw |
| exact Set.disjoint_left.mp (hQQ _ _ (fun h => hij (e.symm.injective h))) |
| ((hsup i v).mp hv) ((hsup j v).mp hw) |
| · intro i v hv hvJ |
| have he := congrArg (fun j => (P j).val.support) (e.apply_symm_apply i) |
| exact he ▸ hQP _ v ((hsup i v).mp hv) hvJ |
| · intro i j v hjkeep hv hvm |
| obtain ⟨⟨j',rfl⟩,D,hD⟩ := hQM _ j v ((hsup i v).mp hv) hvm |
| have hj : j'=j := by |
| by_contra hne |
| exact Set.disjoint_left.mp (M.disjoint j' j hne) (M.root_mem j') hvm |
| have horig : u j'∈(P i).val.support := by |
| have he := congrArg (fun j => (P j).val.support) (e.apply_symm_apply i) |
| rw [←he] |
| rcases hD with hd | hd |
| · exact hd.symm ▸ D.entry_on |
| · exact hd.symm ▸ D.exit_on |
| exact ⟨hroot i j' (hj ▸ hjkeep) horig,congrArg u hj⟩ |
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| omit [DecidableEq V] [Fintype I] in |
| lemma source_only (x y : I → V) (P : ∀ i, G.Path (x i) (y i)) |
| (hPP : ∀ i j, i≠j → Disjoint {v | v∈(P i).val.support} {v | v∈(P j).val.support}) |
| (i j : I) (h : x j∈(P i).val.support) : x j=x i := by |
| by_cases he : i=j |
| · exact he ▸ rfl |
| · exact (Set.disjoint_left.mp (hPP i j he) h (P j).val.start_mem_support).elim |
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| omit [DecidableEq V] [Fintype I] in |
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| lemma rooted_of_occurrences {O : Type} (G : SimpleGraph V) (r : I → V) |
| (hr : Function.Injective r) (label : O → I) (B : O → Set V) |
| (hB : ∀ o, (G.induce (B o)).Connected) |
| (hBB : ∀ o p, label o≠label p → Disjoint (B o) (B p)) |
| (hroot : ∀ o i, r i∉B o) |
| (hattach : ∀ o, ∃ v∈B o, G.Adj v (r (label o))) |
| (hpairs : ∀ i j, i≠j → ∃ o p, label o=i ∧ label p=j ∧ |
| ∃ v∈B o, ∃ w∈B p, G.Adj v w) : |
| ∃ M : RootedCliqueModel G r, |
| ∀ i, M.branch i={r i}∪{v | ∃ o, label o=i ∧ v∈B o} := by |
| let C (i : I) := {r i}∪{v | ∃ o, label o=i ∧ v∈B o} |
| have hc (i) : (G.induce (C i)).Connected := by |
| have hsingleton : (G.induce ({r i}:Set V)).Connected := by |
| have : Nonempty ({r i}:Set V) := ⟨⟨r i,rfl⟩⟩ |
| exact Connected.of_subsingleton |
| have h := Assembly.connected_glue G {r i} |
| (fun o : {o // label o=i} => B o.val) hsingleton (fun o => hB o.val) (fun o => by |
| obtain ⟨v,hv,he⟩ := hattach o.val |
| exact ⟨v,hv,r i,rfl,by simpa only [o.property] using he⟩) |
| have he : C i={r i}∪{v | ∃ o : {o // label o=i},v∈B o.val} := by |
| ext v |
| simp only [C,Set.mem_union,Set.mem_ofPred_eq,Subtype.exists,exists_prop] |
| exact he ▸ h |
| refine ⟨⟨C,hc,?_,?_,fun _ => Or.inl rfl⟩,fun _ => rfl⟩ |
| · intro i j hij |
| apply Set.disjoint_left.mpr |
| intro v hi hj |
| rcases hi with hi | ⟨o,ho,hi⟩ <;> rcases hj with hj | ⟨p,hp,hj⟩ |
| · exact hij (hr (hi.symm.trans hj)) |
| · exact hroot p i (hi ▸ hj) |
| · exact hroot o j (hj ▸ hi) |
| · exact Set.disjoint_left.mp (hBB o p (by simpa only [ho,hp] using hij)) hi hj |
| · intro i j hij |
| obtain ⟨o,p,ho,hp,v,hv,w,hw,he⟩ := hpairs i j hij |
| exact ⟨v,Or.inr ⟨o,ho,hv⟩,w,Or.inr ⟨p,hp,hw⟩,he⟩ |
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| theorem reroute_disjoint_districts {D : Type} [Fintype D] |
| (J : D → Set V) {a b : ℕ} (hJ : ∀ d, WovenOn G (J d) a b) |
| (hJJ : ∀ d e, d≠e → Disjoint (J d) (J e)) |
| (u : D → Fin a → V) (hu : ∀ d, Function.Injective (u d)) (huJ : ∀ d j, u d j∈J d) |
| (x y : I → V) (P : ∀ i, G.Path (x i) (y i)) |
| (hcard : Fintype.card I≤b) |
| (hPP : ∀ i j, i≠j → Disjoint {v | v∈(P i).val.support} {v | v∈(P j).val.support}) : |
| ∃ M : ∀ d, RootedCliqueModel G (u d), (∀ d j, (M d).branch j⊆J d) ∧ |
| ∃ Q : ∀ i, G.Path (x i) (y i), |
| (∀ i j, i≠j → Disjoint {v | v∈(Q i).val.support} {v | v∈(Q j).val.support}) ∧ |
| (∀ i v, v∈(Q i).val.support → (∀ d, v∉J d) → v∈(P i).val.support) ∧ |
| (∀ d i j v, (∃ k, x k=u d j) → v∈(Q i).val.support → |
| v∈(M d).branch j → v=x i ∧ v=u d j) := by |
| classical |
| have step (S : Finset D) : |
| ∃ M : ∀ d∈S, RootedCliqueModel G (u d), (∀ d hd j, (M d hd).branch j⊆J d) ∧ |
| ∃ Q : ∀ i, G.Path (x i) (y i), |
| (∀ i j, i≠j → Disjoint {v | v∈(Q i).val.support} {v | v∈(Q j).val.support}) ∧ |
| (∀ i v, v∈(Q i).val.support → (∀ d, v∉J d) → v∈(P i).val.support) ∧ |
| (∀ d hd i j v, (∃ k, x k=u d j) → v∈(Q i).val.support → |
| v∈(M d hd).branch j → v=x i ∧ v=u d j) := by |
| induction S using Finset.induction_on with |
| | empty => |
| refine ⟨fun d hd => (notMem_empty d hd).elim,fun d hd => (notMem_empty d hd).elim, |
| P,hPP,fun _ _ hv _ => hv,fun d hd => (notMem_empty d hd).elim⟩ |
| | @insert d S hd ih => |
| obtain ⟨M,hMJ,Q,hQQ,hQP,hQM⟩ := ih |
| obtain ⟨N,hNJ,R,hRR,hRQ,hRN⟩ := reroute_selected (J d) (hJ d) (u d) (hu d) (huJ d) |
| (fun j => ∃ i, x i=u d j) x y Q hcard hQQ (by |
| intro i j hj hv |
| obtain ⟨k,hk⟩ := hj |
| rw [←hk] at hv |
| exact hk.symm.trans (source_only x y Q hQQ i k hv)) |
| let M' : ∀ e∈insert d S, RootedCliqueModel G (u e) := fun e he => |
| if h : e=d then h.symm ▸ N else M e ((mem_insert.mp he).resolve_left h) |
| have hnew (he : d∈insert d S) : M' d he=N := by simp [M'] |
| have hold (e : D) (he : e∈insert d S) (hne : e≠d) : |
| M' e he=M e ((mem_insert.mp he).resolve_left hne) := by simp [M',hne] |
| refine ⟨M',?_,R,hRR,?_,?_⟩ |
| · intro e he j |
| by_cases h : e=d |
| · subst e |
| rw [hnew he] |
| exact hNJ j |
| · rw [hold e he h] |
| exact hMJ e _ j |
| · intro i v hv hvJ |
| exact hQP i v (hRQ i v hv (hvJ d)) hvJ |
| · intro e he i j v hj hv hvm |
| by_cases h : e=d |
| · subst e |
| rw [hnew he] at hvm |
| exact hRN i j v hj hv hvm |
| · rw [hold e he h] at hvm |
| have hvJ : v∉J d := fun hvd => |
| Set.disjoint_left.mp (hJJ e d h) (hMJ e _ j hvm) hvd |
| exact hQM e _ i j v hj (hRQ i v hv hvJ) hvm |
| obtain ⟨M,hMJ,Q,hQQ,hQP,hQM⟩ := step univ |
| exact ⟨fun d => M d (mem_univ d),fun d j => hMJ d _ j,Q,hQQ,hQP, |
| fun d => hQM d (mem_univ d)⟩ |
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| lemma paired_paths_on [Fintype V] [DecidableEq I] |
| (G : SimpleGraph V) (S : Finset V) (z : I × Bool → V) |
| (hz : Function.Injective z) (hzS : ∀ o, z o∈S) |
| (hI : 1≤Fintype.card I) |
| (hc : KConnectedOn G S (65536*(2*Fintype.card I))) : |
| ∃ P : ∀ i, G.Path (z (i,false)) (z (i,true)), |
| (∀ i v, v∈(P i).val.support → v∈S) ∧ |
| ∀ i j, i≠j → Disjoint {v | v∈(P i).val.support} {v | v∈(P j).val.support} := by |
| have hp : 1≤Fintype.card (I × Bool) := by simp only [Fintype.card_prod,Fintype.card_bool]; omega |
| have hc' : KConnectedOn G S (65536*Fintype.card (I×Bool)) := by |
| simpa only [Fintype.card_prod,Fintype.card_bool,Nat.mul_comm (Fintype.card I) 2] using hc |
| obtain ⟨C,hC,hCC,hCS,hzC⟩ := Stage.grouped_ends G S z hz hzS Prod.fst |
| (fun i => ⟨(i,false),rfl⟩) hp hc' |
| have hex (i : I) : ∃ P : G.Path (z (i,false)) (z (i,true)), ∀ v∈P.val.support, v∈C i := |
| Assembly.path_in_connected (hC i) (hzC (i,false)) (hzC (i,true)) |
| choose P hP using hex |
| refine ⟨P,fun i v hv => hCS i (hP i v hv),?_⟩ |
| intro i j hij |
| exact Set.disjoint_left.mpr (fun v hi hj => Set.disjoint_left.mp (hCC i j hij) (hP i v hi) (hP j v hj)) |
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| omit [DecidableEq V] [Fintype I] in |
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| lemma rooted_from_bag_routes {O : Type} (G : SimpleGraph V) |
| (r : I → V) (hr : Function.Injective r) (label : O → I) |
| (x y : O → V) (hx : Function.Injective x) |
| (C : O → Set V) (P : ∀ o, G.Path (x o) (y o)) |
| (hC : ∀ o, (G.induce (C o)).Connected) (hCx : ∀ o, x o∈C o) |
| (hCC : ∀ o p, o≠p → Disjoint (C o) (C p)) |
| (hPP : ∀ o p, o≠p → Disjoint {v | v∈(P o).val.support} {v | v∈(P p).val.support}) |
| (hPC : ∀ o p v, v∈(P o).val.support → v∈C p → v=x o ∧ v=x p) |
| (hRC : ∀ o i, r i∉C o) |
| (hRP : ∀ o i, r i∉(P o).val.support) |
| (hadj : ∀ o, G.Adj (y o) (r (label o))) |
| (hpairs : ∀ i j, i≠j → ∃ o p, label o=i ∧ label p=j ∧ |
| ∃ v∈C o, ∃ w∈C p, G.Adj v w) : |
| ∃ M : RootedCliqueModel G r, ∀ i, |
| M.branch i={r i}∪{v | ∃ o, label o=i ∧ (v∈C o ∨ v∈(P o).val.support)} := by |
| let B (o : O) := C o∪{v | v∈(P o).val.support} |
| have hB (o) : (G.induce (B o)).Connected := |
| Stage.attached_connected C x y P hC hCx o |
| have hBB (o p) (hop : label o≠label p) : Disjoint (B o) (B p) := by |
| have hne : o≠p := fun h => hop (congrArg label h) |
| apply Set.disjoint_left.mpr |
| intro v ho hp |
| rcases ho with ho|ho <;> rcases hp with hp|hp |
| · exact Set.disjoint_left.mp (hCC o p hne) ho hp |
| · have he := hPC p o v hp ho |
| exact hne (hx (he.2.symm.trans he.1)) |
| · have he := hPC o p v ho hp |
| exact hne (hx (he.1.symm.trans he.2)) |
| · exact Set.disjoint_left.mp (hPP o p hne) ho hp |
| apply rooted_of_occurrences G r hr label B hB hBB |
| · intro o i hi |
| exact hi.elim (hRC o i) (hRP o i) |
| · intro o |
| exact ⟨y o,Or.inr (P o).val.end_mem_support,hadj o⟩ |
| · intro i j hij |
| obtain ⟨o,p,ho,hp,v,hv,w,hw,he⟩ := hpairs i j hij |
| exact ⟨o,p,ho,hp,v,Or.inl hv,w,Or.inl hw,he⟩ |
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| end LinearListHadwiger.Recursion |
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| end OAI |
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