File size: 3,459 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 | import OAI.Computability.MatchingCount.RawOddBounds
namespace OAI
section
namespace BinaryMatching.Raw
open StackCode RecursiveEval
def rawTotal (c : Matrix ℕ) : ℕ := c.flatten.sum
lemma rawTotal_rep {n : ℕ} (c : Fin n → Fin n → ℕ) : rawTotal (matrixRep c)=DenseOracle.total c := by
simp only [rawTotal,matrixRep,vectorRep,List.sum_flatten,List.map_ofFn,List.sum_ofFn,DenseOracle.total,Function.comp_def]
def oddFlat (z : ℕ×(Matrix ℕ×Cut)) : Cut :=
let s := ((guardedStep (paramStep empty oddExpand oddCombine))^[2*z.1]
(oddBudget z.1 (rawTotal z.2.1),(z.1,([Command.call (cardinal z.2.2) (z.2.1,z.2.2)],[])))).2
s.2.2.headD []
theorem oddFlat_rep {n : ℕ} (c : Fin n → Fin n → ℕ) (U : Finset (Fin n))
(he : Even U.card) (hU : 2≤U.card) :
oddFlat (n,(matrixRep c,cutRep U))=cutRep (OddMincut.oddCut c U) := by
unfold oddFlat
simp only [rawTotal_rep,cardinal_rep]
rw [guardedStep_eq (paramStep empty oddExpand oddCombine)
(oddBudget n (DenseOracle.total c)) (n,([Command.call U.card (matrixRep c,cutRep U)],[])) (2*n)
(fun j hj => oddOrbit_bound c U _ le_rfl j hj)]
rw [paramStep_iterate]
rw [runs_extra (empty n) (oddExpand n) (oddCombine n) U.card
(matrixRep c,cutRep U) (2*n) (oddCost_bound c U he hU)]
exact oddValue_rep _ c U
namespace Compiler
open StackCode.Poly RecursiveEval.Compile
@[fun_prop] theorem rawTotal_poly : Poly rawTotal := by unfold rawTotal; fun_prop
@[fun_prop] theorem oddBudget_poly : Poly (fun z : ℕ×ℕ => oddBudget z.1 z.2) := by unfold oddBudget; fun_prop
@[fun_prop] theorem oddExpand_poly : Poly (fun z : ℕ×OddArg => oddExpand z.1 z.2) := by
let A (z : ℕ×OddArg) := terminalCut z.1 z.2.1 z.2.2
have hA : Poly A := terminalCut_poly
have h : Poly (fun z : ℕ×OddArg =>
if decide (cardinal (inter z.1 (A z) z.2.2)%2=1) then Branch.leaf (A z) else
if (outside z.1 (A z)).isSome && (outside z.1 (compl z.1 (A z))).isSome then
Branch.fork (z.2.1,(A z,((outside z.1 (A z)).getD 0,(outside z.1 (compl z.1 (A z))).getD 0)))
(contract z.1 z.2.1 (A z) ((outside z.1 (A z)).getD 0),inter z.1 (A z) z.2.2)
(contract z.1 z.2.1 (compl z.1 (A z)) ((outside z.1 (compl z.1 (A z))).getD 0),inter z.1 (compl z.1 (A z)) z.2.2)
else Branch.leaf (empty z.1) : (ℕ×OddArg) → Branch OddArg Cut OddJoin) := by fun_prop
exact h.congr (fun z => by
unfold oddExpand
dsimp only [A]
simp only [decide_eq_true_eq]
by_cases hp : cardinal (inter z.1 (terminalCut z.1 z.2.1 z.2.2) z.2.2)%2=1
· simp only [ite_eq_left hp]
· simp only [ite_eq_right hp]
cases outside z.1 (terminalCut z.1 z.2.1 z.2.2) <;>
cases outside z.1 (compl z.1 (terminalCut z.1 z.2.1 z.2.2)) <;> rfl)
@[fun_prop] theorem oddCombine_poly : Poly (fun z : ℕ×(OddJoin×(Cut×Cut)) => oddCombine z.1 z.2.1 z.2.2.1 z.2.2.2) := by
unfold oddCombine
fun_prop
@[fun_prop] theorem oddFlat_poly : Poly oddFlat := by
have hs := paramStep_poly empty oddExpand oddCombine rawEmpty oddExpand_poly oddCombine_poly
have hg := Poly.guardedIterate (paramStep empty oddExpand oddCombine) hs
have hi := Poly.comp _ (fun z : ℕ×(Matrix ℕ×Cut) =>
(2*z.1,(oddBudget z.1 (rawTotal z.2.1),(z.1,([Command.call (cardinal z.2.2) (z.2.1,z.2.2)],[]))))) hg (by fun_prop)
exact Poly.comp (fun s : ℕ×State OddArg Cut OddJoin => s.2.2.headD []) _ (by fun_prop) hi
end Compiler
end BinaryMatching.Raw
end
end OAI
|