openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionComputedProposal.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionComputedInitial | |
| import OAI.Combinatorics.ContingencyTables.Machines.PairedCatalogProgram | |
| namespace OAI | |
| /-! # The companion proposal with computed metadata | |
| The executable proposal receives only a binary profile and fresh bits. | |
| It computes the consecutive-pair catalog itself and scans coordinate | |
| exchanges. Its semantic law is the same ordered proposal law. | |
| -/ | |
| namespace ContingencyTables.CompanionComputedProposal | |
| open MatchingFPRAS.TreeTyped CompanionFactorialApplication CompanionComputedInitial | |
| open CompanionArithmeticProposal CompanionOrderedProposal CompanionNeighborProgram | |
| open CompanionBooleanPhase | |
| open scoped Classical | |
| def proposalProgram (z : List ℕ×List Bool) : Option (List ℕ) := | |
| codeProposal ((PairedCatalogProgram.catalog z.1,z.1),z.2) | |
| def catalogInputRealizer : Realizer (fun z : List ℕ×List Bool => PairedCatalogProgram.catalog z.1) := | |
| composition (f := fun z : List ℕ×List Bool => z.1) (g := PairedCatalogProgram.catalog) | |
| first PairedCatalogProgram.catalogRealizer | |
| def proposalProgramRealizer : Realizer proposalProgram := | |
| composition (f := fun z : List ℕ×List Bool => ((PairedCatalogProgram.catalog z.1,z.1),z.2)) | |
| (g := codeProposal) (pair (pair catalogInputRealizer first) second) codeProposalRealizer | |
| theorem polynomial_proposalProgram : PolynomialTime proposalProgramRealizer := | |
| polynomial_composition (polynomial_pair (polynomial_pair | |
| (polynomial_composition polynomial_first PairedCatalogProgram.polynomial_catalog) | |
| polynomial_first) polynomial_second) polynomial_codeProposal | |
| noncomputable section | |
| variable {m n : ℕ} (A : Instance (Fin m) (Fin n)) | |
| def pairLabels : Fin (CanonicalInitialEncoding.copyOrder (capacities A)).length ≃ Labels A := | |
| (PairedCatalogProgram.labelOrder (capacities A)).trans (copyOrder A).symm | |
| lemma pair_coordinate (i : Fin (CanonicalInitialEncoding.copyOrder (capacities A)).length) | |
| (side : Bool) : | |
| coordinateOrder A (pairLabels A i,side)= | |
| CanonicalInitialEncoding.coordinateEquiv (capacities A) | |
| (PairedCatalogProgram.labelOrder (capacities A) i,side) := by | |
| simp [coordinateOrder,pairOrder,pairLabels] | |
| lemma generated_catalog (S : Set (Labels A×Bool)) : | |
| PairedCatalogProgram.catalog (encodeSet (coordinateOrder A) S)= | |
| orderedCatalog A (pairLabels A) (coordinateOrder A) := by | |
| have he : encodeSet (coordinateOrder A) S= | |
| encodeSet (CanonicalInitialEncoding.coordinateEquiv (capacities A)) | |
| ((pairOrder A).symm ⁻¹' S) := rfl | |
| rw [he,PairedCatalogProgram.catalog_encode] | |
| unfold orderedCatalog | |
| apply congrArg List.ofFn | |
| funext i | |
| rw [pair_coordinate,pair_coordinate] | |
| /-- Every proposed code is the encoding of the actual ordered proposal | |
| state, and unused binary indices give precisely the same `none` result. -/ | |
| theorem proposalProgram_eq (S : State A) (bits : Fin (proposalBits A) → Bool) : | |
| proposalProgram (encodeSet (coordinateOrder A) S.val,List.ofFn bits)= | |
| (orderedProposal A (coordinateOrder A) S bits).map | |
| (fun T => encodeSet (coordinateOrder A) T.val) := by | |
| unfold proposalProgram | |
| rw [generated_catalog] | |
| exact codeProposal_ordered A (pairLabels A) (coordinateOrder A) S bits | |
| end | |
| end ContingencyTables.CompanionComputedProposal | |
| end OAI | |