File size: 3,141 Bytes
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 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import OAI.Computability.MatchingCount.OracleBounds
namespace OAI
section
namespace BinaryMatching.Raw
open RecursiveEval
abbrev OddArg := Matrix ℕ×Cut
abbrev OddJoin := Matrix ℕ×(Cut×(ℕ×ℕ))
def oddExpand (n : ℕ) (a : OddArg) : Branch OddArg Cut OddJoin :=
let A := terminalCut n a.1 a.2
if cardinal (inter n A a.2)%2=1 then .leaf A else
match outside n A,outside n (compl n A) with
| some r,some q => .fork (a.1,(A,(r,q)))
(contract n a.1 A r,inter n A a.2)
(contract n a.1 (compl n A) q,inter n (compl n A) a.2)
| _,_ => .leaf (empty n)
def oddCombine (n : ℕ) (c : OddJoin) (X Y : Cut) : Cut :=
cheaper n c.1 (lift n c.2.1 c.2.2.1 X) (lift n (compl n c.2.1) c.2.2.2 Y)
def oddValue (n k : ℕ) (a : OddArg) : Cut :=
value (empty n) (oddExpand n) (oddCombine n) k a
theorem oddValue_rep {n : ℕ} (k : ℕ) (c : Fin n → Fin n → ℕ) (U : Finset (Fin n)) :
oddValue n k (matrixRep c,cutRep U)=cutRep (OddMincut.recurse k c U) := by
induction k generalizing c U with
| zero => exact empty_rep _
| succ k ih =>
simp only [oddValue,value,oddExpand,terminalCut_rep,inter_rep,cardinal_rep,compl_rep,outside_rep,
OddMincut.recurse,OddCutParity.OddSide,Nat.odd_iff]
split_ifs with hodd
· rfl
· cases hr : OddMincut.outside (TerminalMincut.terminalCut c U) <;>
cases hq : OddMincut.outside (TerminalMincut.terminalCut c U)ᶜ <;>
simp only [Option.map_none,Option.map_some,contract_rep]
· exact empty_rep _
· exact empty_rep _
· exact empty_rep _
· rename_i r q
change oddCombine n (matrixRep c,(cutRep (TerminalMincut.terminalCut c U),(r.val,q.val)))
(oddValue n k (matrixRep _,cutRep _)) (oddValue n k (matrixRep _,cutRep _))=_
rw [ih,ih]
simp only [oddCombine,compl_rep,lift_rep,cheaper_rep]
theorem oddCalls_rep {n : ℕ} (k : ℕ) (c : Fin n → Fin n → ℕ) (U : Finset (Fin n)) :
calls (oddExpand n) k (matrixRep c,cutRep U)=
DenseOracle.calls k (DenseOracle.make c) U := by
induction k generalizing c U with
| zero => rfl
| succ k ih =>
simp only [calls,oddExpand,terminalCut_rep,inter_rep,cardinal_rep,compl_rep,outside_rep,
DenseOracle.calls,DenseOracle.terminalCut_eq,DenseOracle.view_make,OddCutParity.OddSide,Nat.odd_iff]
split_ifs with hodd
· rfl
· cases hr : OddMincut.outside (TerminalMincut.terminalCut c U) <;>
cases hq : OddMincut.outside (TerminalMincut.terminalCut c U)ᶜ <;>
simp only [Option.map_none,Option.map_some,contract_rep]
all_goals simp only [ih,DenseOracle.contract,DenseOracle.view_make]
theorem oddCost_bound {n : ℕ} (c : Fin n → Fin n → ℕ) (U : Finset (Fin n))
(he : Even U.card) (hU : 2≤U.card) :
cost (oddExpand n) U.card (matrixRep c,cutRep U)≤2*n := by
have hb := DenseOracle.calls_bound U.card (DenseOracle.make c) U he hU le_rfl
rw [←oddCalls_rep] at hb
have hc := cost_le_calls (oddExpand n) U.card (matrixRep c,cutRep U)
have hu := Finset.card_le_univ U
simp only [Fintype.card_fin] at hu
omega
end BinaryMatching.Raw
end
end OAI
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