| import OAI.Combinatorics.ContingencyTables.Rooted.ComputedForestCode |
| import OAI.Combinatorics.ContingencyTables.Transport.ComputedCompanionInstance |
| import OAI.Combinatorics.ContingencyTables.Rooted.RootedDemandProgram |
|
|
| namespace OAI |
|
|
| /-! # Rooted-oracle metadata from the executed original-input preprocessing -/ |
|
|
| namespace ContingencyTables.ComputedRootedInput |
| open MatchingFPRAS MatchingFPRAS.TreeTyped |
| open DensePrograms NetworkVertexCode |
|
|
| abbrev Input := NetworkVertexCode.Input×ℕ |
|
|
| def forestFrame (z : Input) : LargeEdgeCatalogProgram.Input := |
| (TightenedInputProgram.compute z.1,z.2) |
| def forestFrameRealizer : Realizer forestFrame := |
| pair (composition (f := fun z : Input => z.1) (g := TightenedInputProgram.compute) |
| first TightenedInputProgram.computeRealizer) second |
| lemma polynomial_forestFrame : PolynomialTime forestFrameRealizer := |
| polynomial_pair (polynomial_composition polynomial_first TightenedInputProgram.polynomial_compute) |
| polynomial_second |
|
|
| def catalogData (z : Input) : RootedForestCatalog.Data := |
| (((ForestGreedyProgram.forest (forestFrame z),z.2),(TightenedInputProgram.compute z.1).1), |
| (TightenedInputProgram.compute z.1).2) |
| def catalogDataRealizer : Realizer catalogData := |
| pair (pair (pair |
| (composition (f := forestFrame) (g := ForestGreedyProgram.forest) |
| forestFrameRealizer ForestGreedyProgram.forestRealizer) second) |
| (composition (composition (f := fun z : Input => z.1) (g := TightenedInputProgram.compute) |
| first TightenedInputProgram.computeRealizer) first)) |
| (composition (composition (f := fun z : Input => z.1) (g := TightenedInputProgram.compute) |
| first TightenedInputProgram.computeRealizer) second) |
| lemma polynomial_catalogData : PolynomialTime catalogDataRealizer := |
| polynomial_pair (polynomial_pair (polynomial_pair |
| (polynomial_composition polynomial_forestFrame ForestGreedyProgram.polynomial_forest) polynomial_second) |
| (polynomial_composition (polynomial_composition polynomial_first TightenedInputProgram.polynomial_compute) |
| polynomial_first)) |
| (polynomial_composition (polynomial_composition polynomial_first TightenedInputProgram.polynomial_compute) |
| polynomial_second) |
|
|
| def prepared (z : Input) : RootedForestCatalog.Data×MatrixCode := |
| (catalogData z,TightenedInputProgram.baseline z.1) |
| def preparedRealizer : Realizer prepared := |
| pair catalogDataRealizer (composition (f := fun z : Input => z.1) |
| (g := TightenedInputProgram.baseline) first TightenedInputProgram.baselineRealizer) |
| lemma polynomial_prepared : PolynomialTime preparedRealizer := |
| polynomial_pair polynomial_catalogData |
| (polynomial_composition polynomial_first TightenedInputProgram.polynomial_baseline) |
|
|
| open scoped Classical |
| variable {m n : ℕ} |
|
|
| lemma catalogData_instance (hEK : PublishedEdmondsKarp.EdmondsKarp1972Theorem1) |
| (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) |
| (hne : ∃ X,TableTightening.Feasible r c b X) (W : ℕ) (hW : 2 ≤ W) : |
| catalogData (data r c b,W)=RootedForestCatalog.data |
| (ComputedCompanionInstance.instanceData hEK r c b hne W hW) := by |
| unfold catalogData forestFrame |
| rw [TightenedInputProgram.compute_code] |
| unfold RootedForestCatalog.data ComputedCompanionInstance.instanceData |
| rw [ComputedLargeForest.computed_code _ _ _ W (by omega)] |
| simp only [data,Int.toNat_natCast,ResidualGraphProgram.natCode,matrixCode] |
|
|
| lemma baseline_instance (hEK : PublishedEdmondsKarp.EdmondsKarp1972Theorem1) |
| (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) |
| (hne : ∃ X,TableTightening.Feasible r c b X) (W : ℕ) (hW : 2 ≤ W) : |
| TightenedInputProgram.baseline (data r c b)=matrixCode |
| (fun i j => ((ComputedCompanionInstance.instanceData hEK r c b hne W hW).X i j).toNat) := by |
| obtain ⟨Y,hY,_,_⟩ := TightenedInputProgram.baseline_correct hEK r c b hne |
| rw [hY] |
| unfold matrixCode |
| apply congrArg List.ofFn |
| funext i |
| apply congrArg List.ofFn |
| funext j |
| simp only [ComputedCompanionInstance.instanceData,ComputedCompanionInstance.baseline, |
| hY,ResidualGraphProgram.natCode,List.getD_eq_getElem?_getD,List.getElem?_ofFn, |
| i.isLt,j.isLt,dite_true,Option.getD_some,Fin.eta,Int.toNat_natCast] |
|
|
| lemma prepared_instance (hEK : PublishedEdmondsKarp.EdmondsKarp1972Theorem1) |
| (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) |
| (hne : ∃ X,TableTightening.Feasible r c b X) (W : ℕ) (hW : 2 ≤ W) : |
| prepared (data r c b,W)= |
| (RootedForestCatalog.data (ComputedCompanionInstance.instanceData hEK r c b hne W hW), |
| matrixCode (fun i j => |
| ((ComputedCompanionInstance.instanceData hEK r c b hne W hW).X i j).toNat)) := by |
| exact Prod.ext (catalogData_instance hEK r c b hne W hW) |
| (baseline_instance hEK r c b hne W hW) |
|
|
| end ContingencyTables.ComputedRootedInput |
|
|
| end OAI |
|
|
