openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Machines /FiniteGraphCutProgram.lean
| import OAI.Combinatorics.ContingencyTables.Transport.FiniteGraphRootSemantics | |
| import OAI.Combinatorics.ContingencyTables.Rooted.RootedCut | |
| namespace OAI | |
| /-! # Executing the selected root-away cut | |
| Deletion filters the stored adjacency rows. The endpoint and cut side | |
| are then computed by the bounded reachability program. | |
| -/ | |
| namespace ContingencyTables.FiniteGraphCutProgram | |
| open MatchingFPRAS.TreeTyped Polynomial FiniteGraphRootProgram | |
| abbrev EdgeInput := GraphCode×(ℕ×ℕ) | |
| abbrev KeepInput := ((ℕ×ℕ)×ℕ)×ℕ | |
| def keep (z : KeepInput) : Bool := | |
| if z.1.2=z.1.1.1 then decide (z.2≠z.1.1.2) | |
| else if z.1.2=z.1.1.2 then decide (z.2≠z.1.1.1) else true | |
| def keepRealizer : Realizer keep := by | |
| let I : Realizer (fun z : KeepInput => z.1.2) := composition first second | |
| let U : Realizer (fun z : KeepInput => z.1.1.1) := composition (composition first first) first | |
| let V : Realizer (fun z : KeepInput => z.1.1.2) := composition (composition first first) second | |
| let neqU := composition (composition (pair second U) natEqual) boolNot | |
| let neqV := composition (composition (pair second V) natEqual) boolNot | |
| exact (choose (composition (pair I U) natEqual) neqV | |
| (choose (composition (pair I V) natEqual) neqU (constant true))).congr (by | |
| intro z | |
| simp [keep]) | |
| lemma polynomial_keep : PolynomialTime keepRealizer := by | |
| unfold keepRealizer | |
| have hi : PolynomialTime (composition (first (A := (ℕ×ℕ)×ℕ) (B := ℕ)) second) := | |
| polynomial_composition polynomial_first polynomial_second | |
| have hu : PolynomialTime (composition | |
| (composition (first (A := (ℕ×ℕ)×ℕ) (B := ℕ)) first) first) := | |
| polynomial_composition (polynomial_composition polynomial_first polynomial_first) polynomial_first | |
| have hv : PolynomialTime (composition | |
| (composition (first (A := (ℕ×ℕ)×ℕ) (B := ℕ)) first) second) := | |
| polynomial_composition (polynomial_composition polynomial_first polynomial_first) polynomial_second | |
| exact polynomial_congr _ (polynomial_choose | |
| (polynomial_composition (polynomial_pair hi hu) polynomial_natEqual) | |
| (polynomial_composition (polynomial_composition (polynomial_pair polynomial_second hv) | |
| polynomial_natEqual) polynomial_boolNot) | |
| (polynomial_choose (polynomial_composition (polynomial_pair hi hv) polynomial_natEqual) | |
| (polynomial_composition (polynomial_composition (polynomial_pair polynomial_second hu) | |
| polynomial_natEqual) polynomial_boolNot) (polynomial_constant true))) | |
| def deleteRow (z : EdgeInput×ℕ) : List ℕ := | |
| (z.1.1.getD z.2 []).filter (fun a => keep ((z.1.2,z.2),a)) | |
| def deleteRowRealizer : Realizer deleteRow := | |
| composition (f := fun z : EdgeInput×ℕ => | |
| ((z.1.2,z.2),z.1.1.getD z.2 [])) | |
| (g := fun z : ((ℕ×ℕ)×ℕ)×List ℕ => z.2.filter (fun a => keep (z.1,a))) | |
| (pair (pair (composition first second) second) | |
| (composition (pair (composition first first) second) (getDFast []))) | |
| (filterWith keepRealizer) | |
| lemma polynomial_deleteRow : PolynomialTime deleteRowRealizer := | |
| polynomial_composition (polynomial_pair | |
| (polynomial_pair (polynomial_composition polynomial_first polynomial_second) polynomial_second) | |
| (polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_first polynomial_first) polynomial_second) | |
| (polynomial_getDFast []))) (polynomial_filterWith polynomial_keep) | |
| def deleteGraph (z : EdgeInput) : GraphCode := | |
| (indices z.1).map (fun i => deleteRow (z,i)) | |
| def deleteGraphRealizer : Realizer deleteGraph := | |
| composition (pair identity (composition first indicesRealizer)) (mapWith deleteRowRealizer) | |
| theorem polynomial_deleteGraph : PolynomialTime deleteGraphRealizer := | |
| polynomial_composition (polynomial_pair polynomial_identity | |
| (polynomial_composition polynomial_first polynomial_indices)) | |
| (polynomial_mapWith polynomial_deleteRow) | |
| def reachableCode (z : Input×ℕ) : Bool := decide (z.2∈reached z.1) | |
| def reachableCodeRealizer : Realizer reachableCode := | |
| composition (pair (composition first reachedRealizer) second) membership | |
| lemma polynomial_reachableCode : PolynomialTime reachableCodeRealizer := | |
| polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_first polynomial_reached) polynomial_second) | |
| polynomial_membership | |
| abbrev CutInput := EdgeInput×ℕ | |
| def awayCode (z : CutInput) : ℕ := | |
| if reachableCode ((deleteGraph z.1,z.1.2.1),z.2) then z.1.2.2 else z.1.2.1 | |
| def awayTestRealizer : Realizer (fun z : CutInput => | |
| reachableCode ((deleteGraph z.1,z.1.2.1),z.2)) := | |
| composition (pair (pair (composition first deleteGraphRealizer) | |
| (composition (composition first second) first)) second) reachableCodeRealizer | |
| lemma polynomial_awayTest : PolynomialTime awayTestRealizer := | |
| polynomial_composition (polynomial_pair | |
| (polynomial_pair (polynomial_composition polynomial_first polynomial_deleteGraph) | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_second) | |
| polynomial_first)) polynomial_second) polynomial_reachableCode | |
| def awayCodeRealizer : Realizer awayCode := | |
| choose awayTestRealizer (composition (composition first second) second) | |
| (composition (composition first second) first) | |
| lemma polynomial_awayCode : PolynomialTime awayCodeRealizer := | |
| polynomial_choose polynomial_awayTest | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_second) | |
| polynomial_second) | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_second) | |
| polynomial_first) | |
| def cutCode (z : CutInput×ℕ) : Bool := | |
| reachableCode ((deleteGraph z.1.1,awayCode z.1),z.2) | |
| def cutCodeRealizer : Realizer cutCode := | |
| composition (pair (pair | |
| (composition (composition first first) deleteGraphRealizer) | |
| (composition first awayCodeRealizer)) second) reachableCodeRealizer | |
| theorem polynomial_cutCode : PolynomialTime cutCodeRealizer := | |
| polynomial_composition (polynomial_pair (polynomial_pair | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_first) | |
| polynomial_deleteGraph) | |
| (polynomial_composition polynomial_first polynomial_awayCode)) polynomial_second) | |
| polynomial_reachableCode | |
| def canonicalCutCode (z : EdgeInput×ℕ) : Bool := | |
| cutCode ((z.1,rootCode (z.1.1,z.1.2.1)),z.2) | |
| def canonicalCutCodeRealizer : Realizer canonicalCutCode := | |
| composition (pair (pair first | |
| (composition (pair (composition first first) | |
| (composition (composition first second) first)) rootCodeRealizer)) second) | |
| cutCodeRealizer | |
| theorem polynomial_canonicalCutCode : PolynomialTime canonicalCutCodeRealizer := | |
| polynomial_composition (polynomial_pair (polynomial_pair polynomial_first | |
| (polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_first polynomial_first) | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_second) | |
| polynomial_first)) polynomial_rootCode)) polynomial_second) polynomial_cutCode | |
| end ContingencyTables.FiniteGraphCutProgram | |
| end OAI | |