| import OAI.Combinatorics.ListHadwiger.PairedSources |
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| namespace OAI |
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| namespace LinearListHadwiger.Assembly |
| open SimpleGraph |
| variable {V I O : Type} (G : SimpleGraph V) |
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| lemma connected_glue (C : Set V) (B : O → Set V) |
| (hC : (G.induce C).Connected) |
| (hB : ∀ o, (G.induce (B o)).Connected) |
| (hE : ∀ o, ∃ x ∈ B o, ∃ y ∈ C, G.Adj x y) : |
| (G.induce (C ∪ {v | ∃ o, v ∈ B o})).Connected := by |
| let S := C ∪ {v | ∃ o, v ∈ B o} |
| have hCS : C ⊆ S := Set.subset_union_left |
| obtain ⟨root⟩ := hC.nonempty |
| have hreach : ∀ v : S, (G.induce S).Reachable v ⟨root.val,hCS root.property⟩ := by |
| rintro ⟨v,hv⟩ |
| rcases hv with hv | ⟨o,hv⟩ |
| · exact (hC.preconnected ⟨v,hv⟩ root).map (G.induceHomOfLE hCS).toHom |
| · obtain ⟨x,hx,y,hy,hxy⟩ := hE o |
| have hBS : B o ⊆ S := fun _ hw => Or.inr ⟨o,hw⟩ |
| have h₁ := ((hB o).preconnected ⟨v,hv⟩ ⟨x,hx⟩).map |
| (G.induceHomOfLE hBS).toHom |
| have h₂ : (G.induce S).Adj ⟨x,hBS hx⟩ ⟨y,hCS hy⟩ := hxy |
| have h₃ := (hC.preconnected ⟨y,hy⟩ root).map (G.induceHomOfLE hCS).toHom |
| exact h₁.trans (h₂.reachable.trans h₃) |
| refine { preconnected := ?_, nonempty := ⟨⟨root.val,hCS root.property⟩⟩ } |
| intro v w |
| exact (hreach v).trans (hreach w).symm |
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| theorem clique_of_occurrences {t : ℕ} (label : O → Fin t) |
| (B : O → Set V) (C : Fin t → Set V) |
| (hB : ∀ o, (G.induce (B o)).Connected) |
| (hC : ∀ i, (G.induce (C i)).Connected) |
| (hBB : ∀ o p, label o ≠ label p → Disjoint (B o) (B p)) |
| (hCC : ∀ i j, i ≠ j → Disjoint (C i) (C j)) |
| (hBC : ∀ o i, Disjoint (B o) (C i)) |
| (hattach : ∀ o, ∃ x ∈ B o, ∃ y ∈ C (label o), G.Adj x y) |
| (hpairs : ∀ i j, i ≠ j → ∃ o p, label o = i ∧ label p = j ∧ |
| ∃ x ∈ B o, ∃ y ∈ B p, G.Adj x y) : |
| HasCliqueMinor G t := by |
| let S (i : Fin t) := C i ∪ {v | ∃ o, label o = i ∧ v ∈ B o} |
| refine ⟨S,?_,?_,?_⟩ |
| · intro i |
| have hh := connected_glue G (C i) (fun o : {o // label o = i} => B o.val) |
| (hC i) (fun o => hB o.val) (fun o => by simpa [o.property] using hattach o.val) |
| have he : S i = C i ∪ {v | ∃ o : {o // label o = i}, v ∈ B o.val} := by |
| ext v |
| simp only [S,Set.mem_union,Set.mem_ofPred_eq,Subtype.exists] |
| aesop |
| change (G.induce (S i)).Connected |
| rw [he] |
| exact hh |
| · intro i j hij |
| apply Set.disjoint_left.mpr |
| intro v hvi hvj |
| rcases hvi with hi | ⟨o,ho,hi⟩ <;> rcases hvj with hj | ⟨p,hp,hj⟩ |
| · exact Set.disjoint_left.mp (hCC i j hij) hi hj |
| · exact Set.disjoint_left.mp (hBC p i) hj hi |
| · exact Set.disjoint_left.mp (hBC o j) hi hj |
| · exact Set.disjoint_left.mp (hBB o p (by simpa [ho,hp] using hij)) hi hj |
| · intro i j hij |
| obtain ⟨o,p,ho,hp,x,hx,y,hy,hxy⟩ := hpairs i j hij |
| exact ⟨x,Or.inr ⟨o,ho,hx⟩,y,Or.inr ⟨p,hp,hy⟩,hxy⟩ |
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| lemma attach_ports [Fintype V] [Fintype O] [DecidableEq V] [DecidableRel G.Adj] |
| (port : O → V) (U : Finset V) |
| (hN : ∀ o, Fintype.card O ≤ (U.filter (fun v => G.Adj (port o) v)).card) : |
| ∃ nbr : O → V, Function.Injective nbr ∧ ∀ o, nbr o ∈ U ∧ G.Adj (port o) (nbr o) := by |
| classical |
| let N (o : O) := U.filter (fun v => G.Adj (port o) v) |
| have hall : ∀ S : Finset O, S.card ≤ (S.biUnion N).card := by |
| intro S |
| obtain rfl | hS := S.eq_empty_or_nonempty |
| · simp |
| obtain ⟨o,ho⟩ := hS |
| exact (Finset.card_le_univ S).trans ((hN o).trans |
| (Finset.card_le_card (Finset.subset_biUnion_of_mem N ho))) |
| obtain ⟨nbr,hi,hn⟩ := (Finset.all_card_le_biUnion_card_iff_exists_injective N).mp hall |
| exact ⟨nbr,hi,fun o => Finset.mem_filter.mp (hn o)⟩ |
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| end LinearListHadwiger.Assembly |
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| end OAI |
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