File size: 4,816 Bytes
d5b7741 | 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 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | 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
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