openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Machines /CompletionEnumerationProgram.lean
| import OAI.Combinatorics.ContingencyTables.Transport.BoxCompletion | |
| import OAI.Combinatorics.ContingencyTables.Transport.FiniteGridWords | |
| import OAI.Combinatorics.ContingencyTables.Dense.DenseCanonicalProgram | |
| namespace OAI | |
| /-! # Explicit free-box enumeration with binary completion tests -/ | |
| namespace ContingencyTables.DensePrograms | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped ProfilePrograms | |
| open scoped BigOperators Classical | |
| abbrev CompletionMargins := List ℕ×List ℕ | |
| def completionEnumerationData (z : CompletionMargins) : GridEnumerationData := | |
| ((zeroGrid z).map (fun _ => false),z.1.sum+1) | |
| def completionEnumerationDataRealizer : Realizer completionEnumerationData := by | |
| let dims := composition zeroGridRealizer (map (constant false)) | |
| let base : Realizer (fun z : CompletionMargins => z.1.sum+1) := | |
| composition (composition first listSum) successor | |
| exact pair dims base | |
| lemma polynomial_completionEnumerationData : PolynomialTime completionEnumerationDataRealizer := by | |
| unfold completionEnumerationDataRealizer | |
| exact polynomial_pair (polynomial_composition polynomial_zeroGrid | |
| (polynomial_map (polynomial_constant false))) | |
| (polynomial_composition (polynomial_composition polynomial_first polynomial_listSum) | |
| polynomial_successor) | |
| def completionCandidate (z : CompletionMargins×List ℕ) : Option MatrixCode := | |
| acceptCompletion (z.1,unflattenGrid z) | |
| def completionCandidateRealizer : Realizer completionCandidate := | |
| composition (pair first unflattenGridRealizer) acceptCompletionRealizer | |
| lemma polynomial_completionCandidate : PolynomialTime completionCandidateRealizer := | |
| polynomial_composition (polynomial_pair polynomial_first polynomial_unflattenGrid) | |
| polynomial_acceptCompletion | |
| /-- Rejected candidates remain as `none`, allowing later integer sums | |
| to ignore them without adding a second enumeration or an ordering oracle. -/ | |
| def completionCandidates (z : CompletionMargins) : List (Option MatrixCode) := | |
| (gridWords (completionEnumerationData z)).map (fun v => completionCandidate (z,v)) | |
| def completionCandidatesRealizer : Realizer completionCandidates := | |
| composition (pair identity (composition completionEnumerationDataRealizer gridWordsRealizer)) | |
| (mapWith completionCandidateRealizer) | |
| def completionEnumerationMeasure (z : CompletionMargins) : ℕ := | |
| gridStateCount (completionEnumerationData z)+weight z | |
| theorem work_completionCandidates : | |
| WorkBound completionCandidatesRealizer completionEnumerationMeasure := by | |
| have hp : WorkBound completionEnumerationDataRealizer completionEnumerationMeasure := | |
| work_of_polynomial polynomial_completionEnumerationData Polynomial.X (fun z => by | |
| simp only [completionEnumerationMeasure,Polynomial.eval_X]; omega) | |
| obtain ⟨p,hp'⟩ := hp | |
| have hg : WorkBound (composition completionEnumerationDataRealizer gridWordsRealizer) | |
| completionEnumerationMeasure := by | |
| apply work_composition ⟨p,hp'⟩ work_gridWords (Polynomial.X+p) | |
| intro z | |
| have hw := hp' z | |
| simp only [completionEnumerationMeasure,Polynomial.eval_add,Polynomial.eval_X] | |
| change gridStateCount (completionEnumerationData z)+weight (completionEnumerationData z)≤_ | |
| change _≤p.eval (completionEnumerationMeasure z) at hw | |
| unfold completionEnumerationMeasure at hw | |
| omega | |
| unfold completionCandidatesRealizer | |
| apply work_then_polynomial _ (polynomial_mapWith polynomial_completionCandidate) | |
| exact work_pair (work_of_polynomial polynomial_identity Polynomial.X (fun z => by | |
| simp only [completionEnumerationMeasure,Polynomial.eval_X]; omega)) hg Polynomial.X | |
| (fun z => by simp only [completionEnumerationMeasure,Polynomial.eval_X]; omega) | |
| lemma gridWords_finFunctions (xs : List Bool) (N q : ℕ) (hq : xs.length=q) : | |
| gridWords (xs,N+1)=List.ofFn (fun k : Fin ((N+1)^q) => | |
| List.ofFn (fun i : Fin q => ((finFunctionFinEquiv.symm k) i).val)) := by | |
| rw [gridWords_eq_length xs (N+1) q hq] | |
| rfl | |
| theorem completionCandidates_code {m n : ℕ} (R : Option (Fin m) → ℕ) | |
| (P : Option (Fin n) → ℕ) (htotal : ∑ i,R i=∑ j,P j) : | |
| completionCandidates (marginCode R,marginCode P)= | |
| List.ofFn (fun k : Fin (((∑ i,R i)+1)^(m*n)) => | |
| (BoxCompletion.decode R P htotal (∑ i,R i) (finFunctionFinEquiv.symm k)).map tableOutput) := by | |
| have hsum : (marginCode R).sum=∑ i,R i := by | |
| simp only [marginCode,List.sum_cons,List.sum_ofFn,Fintype.sum_option] | |
| have hlen : ((zeroGrid (marginCode R,marginCode P)).map (fun _ => false)).length=m*n := by | |
| simp only [zeroGrid,List.length_map,cellPairs,List.length_range,marginCode, | |
| List.tail_cons,List.length_ofFn] | |
| unfold completionCandidates completionEnumerationData | |
| rw [hsum,gridWords_finFunctions _ (∑ i,R i) (m*n) hlen,List.map_ofFn] | |
| apply congrArg List.ofFn | |
| funext k | |
| dsimp only [Function.comp_apply,completionCandidate] | |
| rw [unflattenGrid_code] | |
| exact BoxCompletion.acceptCompletion_eq R P htotal (finFunctionFinEquiv.symm k) | |
| end ContingencyTables.DensePrograms | |
| end OAI | |