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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 44 45 46 47 48 49 50 51 52 53 54 55 56 | import OAI.Computability.MatchingCount.RawFlowCompiler
namespace OAI
section
namespace BinaryMatching.Raw
open Finset
def outside (n : ℕ) (A : Cut) : Option ℕ := (List.range n).find? (fun i => !mem A i)
def contractMap (A : Cut) (r i : ℕ) : ℕ := if mem A i then i else r
def contract (n : ℕ) (c : Matrix ℕ) (A : Cut) (r : ℕ) : Matrix ℕ :=
matrix n (fun i j => if i=j then 0 else
(tab n (fun u => (tab n (fun v =>
if contractMap A r u=i ∧ contractMap A r v=j then get c u v else 0)).sum)).sum)
def lift (n : ℕ) (A : Cut) (r : ℕ) (S : Cut) : Cut :=
tab n (fun i => mem S (contractMap A r i))
def cheaper (n : ℕ) (c : Matrix ℕ) (S T : Cut) : Cut :=
if cutCapacity n c S≤cutCapacity n c T then S else T
lemma outside_rep {n : ℕ} (A : Finset (Fin n)) :
outside n (cutRep A)=(OddMincut.outside A).map Fin.val := by
unfold outside OddMincut.outside
rw [range_eq_fin,List.find?_map]
congr 1
congr 1
funext i
simp only [Function.comp_apply,mem_rep,decide_not]
lemma contractMap_rep {n : ℕ} (A : Finset (Fin n)) (r i : Fin n) :
contractMap (cutRep A) r.val i.val=(OddCutParity.contractMap A r i).val := by
simp only [contractMap,mem_rep,decide_eq_true_eq,OddCutParity.contractMap]
split_ifs <;> rfl
lemma contract_rep {n : ℕ} (c : Fin n → Fin n → ℕ) (A : Finset (Fin n)) (r : Fin n) :
contract n (matrixRep c) (cutRep A) r.val=
matrixRep (CutAlgebra.push c (OddCutParity.contractMap A r)) := by
apply matrix_eq_rep
intro i j
simp only [Fin.ext_iff,contractMap_rep,sum_tab,get_rep,CutAlgebra.push]
lemma lift_rep {n : ℕ} (A S : Finset (Fin n)) (r : Fin n) :
lift n (cutRep A) r.val (cutRep S)=cutRep (CutAlgebra.lift (OddCutParity.contractMap A r) S) := by
apply tab_eq_rep
intro i
simp only [contractMap_rep,mem_rep,CutAlgebra.mem_lift]
lemma cheaper_rep {n : ℕ} (c : Fin n → Fin n → ℕ) (S T : Finset (Fin n)) :
cheaper n (matrixRep c) (cutRep S) (cutRep T)=cutRep (OddMincut.cheaper c S T) := by
simp only [cheaper,cutCapacity_rep,OddMincut.cheaper]
split_ifs <;> rfl
end BinaryMatching.Raw
end
end OAI
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