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ea76aac | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 | import OAI.Computability.MatchingCount.OddCutParity
namespace OAI
section
namespace BinaryMatching.OddCutUncross
open Finset UnitFlow CutAlgebra TerminalMincut OddCutParity
variable {n : ℕ}
lemma separates_compl (U A : Finset (Fin n)) (hA : Separates U A) : Separates U Aᶜ := by
obtain ⟨a,ha,z,hz,haa,hza⟩ := hA
exact ⟨z,hz,a,ha,mem_compl.mpr hza,by simpa using haa⟩
lemma uncross_on_side (c : Fin n → Fin n → ℕ) (hc : ∀ i j, c i j=c j i)
(U A S : Finset (Fin n)) (hsep : Separates U A) (heven : Even (A∩U).card)
(hmin : ∀ T, Separates U T → cutCapacity c A ≤ cutCapacity c T)
(hinside : OddSide (A∩U) S) :
∃ T, T⊆A ∧ OddSide U T ∧ cutCapacity c T ≤ cutCapacity c S := by
obtain ⟨a,ha,b,hb,haa,hba⟩ := hsep
have horient : ∃ S', OddSide (A∩U) S' ∧ cutCapacity c S'=cutCapacity c S ∧
∃ z∈U, z∉A ∧ z∉S' := by
by_cases hex : ∃ z∈U, z∉A ∧ z∉S
· exact ⟨S,hinside,rfl,hex⟩
· have hbS : b∈S := by
by_contra hh
exact hex ⟨b,hb,hba,hh⟩
exact ⟨Sᶜ,odd_other (A∩U) S heven hinside,cut_compl c hc S,
b,hb,hba,by simpa using hbS⟩
obtain ⟨S',hodd,hcap,z,hz,hza,hzs⟩ := horient
have hunion : Separates U (S'∪A) :=
⟨a,ha,z,hz,mem_union_right _ haa,by simpa only [mem_union,not_or] using And.intro hzs hza⟩
have hlow := hmin (S'∪A) hunion
have hsub := cut_submodular c S' A
refine ⟨S'∩A,inter_subset_right,?_,?_⟩
· simpa only [OddSide,inter_assoc] using hodd
· omega
end BinaryMatching.OddCutUncross
end
end OAI
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