| import OAI.Computability.MatchingCount.RawOddBounds |
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| namespace OAI |
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| section |
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| namespace BinaryMatching.Raw |
| open StackCode RecursiveEval |
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| 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] |
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| 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 [] |
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| 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 |
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| end OAI |
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