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namespace OAI
namespace LinearListHadwiger.Recursion
open SimpleGraph Finset
attribute [local instance] Classical.propDecidable
variable {V I : Type} [DecidableEq V] [Fintype I] {G : SimpleGraph V}
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⟩
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
omit [DecidableEq V] [Fintype I] in
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⟩
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)⟩
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))
omit [DecidableEq V] [Fintype I] in
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⟩
end LinearListHadwiger.Recursion
end OAI
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