openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Machines /CachedCompletionWeights.lean
| import OAI.Combinatorics.ContingencyTables.Transport.CanonicalIntegerLaw | |
| namespace OAI | |
| /-! # Explicit table weights from the computed dense-law cache -/ | |
| namespace ContingencyTables.DensePrograms | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped GridBoundary | |
| open scoped BigOperators Classical | |
| abbrev CompletionCache := CanonicalParameters×List ℕ | |
| def cachedCompletionWeight (z : CompletionCache×Option MatrixCode) : ℕ := | |
| z.2.elim 0 (fun X => z.1.2.getD (canonicalTableIndex (z.1.1,X)) 0) | |
| def cachedCompletionWeightRealizer : Realizer cachedCompletionWeight := by | |
| let context : Realizer (fun z : CompletionCache×MatrixCode => z.1.1) := | |
| composition first first | |
| let cache : Realizer (fun z : CompletionCache×MatrixCode => z.1.2) := | |
| composition first second | |
| let index := composition (pair context second) canonicalTableIndexRealizer | |
| let entry := composition (pair cache index) (getDFast 0) | |
| exact optionCases (constant 0) entry | |
| lemma polynomial_cachedCompletionWeight : PolynomialTime cachedCompletionWeightRealizer := by | |
| unfold cachedCompletionWeightRealizer | |
| exact polynomial_optionCases (polynomial_constant 0) | |
| (polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_first polynomial_second) | |
| (polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_first polynomial_first) polynomial_second) | |
| polynomial_canonicalTableIndex)) (polynomial_getDFast 0)) | |
| def completionWeightEntry (z : CompletionCache×Option MatrixCode) : Option MatrixCode×ℕ := | |
| (z.2,cachedCompletionWeight z) | |
| def completionWeightEntryRealizer : Realizer completionWeightEntry := | |
| pair second cachedCompletionWeightRealizer | |
| lemma polynomial_completionWeightEntry : PolynomialTime completionWeightEntryRealizer := | |
| polynomial_pair polynomial_second polynomial_cachedCompletionWeight | |
| def completionWeightInput (z : CanonicalParameters) : CompletionCache×List (Option MatrixCode) := | |
| ((z,canonicalDrawCounts z),completionCandidates z.1) | |
| def completionWeightInputRealizer : Realizer completionWeightInput := | |
| pair (pair identity canonicalDrawCountsRealizer) | |
| (composition first completionCandidatesRealizer) | |
| def completionWeightMeasure (z : CanonicalParameters) : ℕ := | |
| canonicalTabulationMeasure z+completionEnumerationMeasure z.1 | |
| private lemma weight_le_completionMeasure (z : CanonicalParameters) : | |
| weight z≤completionWeightMeasure z := by | |
| unfold completionWeightMeasure canonicalTabulationMeasure | |
| omega | |
| lemma work_completionWeightInput : | |
| WorkBound completionWeightInputRealizer completionWeightMeasure := by | |
| have hw : WorkBound canonicalDrawCountsRealizer completionWeightMeasure := | |
| work_remeasure work_canonicalDrawCounts Polynomial.X (fun z => by | |
| simp only [Polynomial.eval_X,completionWeightMeasure]; omega) | |
| have he : WorkBound (composition first completionCandidatesRealizer) completionWeightMeasure := by | |
| apply work_composition (work_of_polynomial polynomial_first Polynomial.X | |
| (fun z => by simpa using weight_le_completionMeasure z)) work_completionCandidates Polynomial.X | |
| intro z | |
| simp only [Polynomial.eval_X,completionWeightMeasure] | |
| omega | |
| have hi : WorkBound (identity (A := CanonicalParameters)) completionWeightMeasure := | |
| work_of_polynomial polynomial_identity Polynomial.X | |
| (fun z => by simpa using weight_le_completionMeasure z) | |
| unfold completionWeightInputRealizer | |
| apply work_pair _ he Polynomial.X (fun z => by simpa using weight_le_completionMeasure z) | |
| exact work_pair hi hw Polynomial.X (fun z => by simpa using weight_le_completionMeasure z) | |
| def completionWeights (z : CanonicalParameters) : List (Option MatrixCode×ℕ) := | |
| (completionCandidates z.1).map | |
| (fun o => (o,cachedCompletionWeight ((z,canonicalDrawCounts z),o))) | |
| def completionWeightsRealizer : Realizer completionWeights := | |
| composition completionWeightInputRealizer (mapWith completionWeightEntryRealizer) | |
| section | |
| attribute [local irreducible] completionWeightInputRealizer | |
| theorem work_completionWeights : WorkBound completionWeightsRealizer completionWeightMeasure := by | |
| unfold completionWeightsRealizer | |
| exact work_then_polynomial work_completionWeightInput | |
| (polynomial_mapWith polynomial_completionWeightEntry) | |
| end | |
| attribute [local irreducible] canonicalDrawCounts canonicalCounts | |
| lemma cachedCompletionWeight_actual {m n d : ℕ} (R : Option (Fin m) → ℕ) | |
| (P : Option (Fin n) → ℕ) (hd : 14≤d) (hq : m*n≤d) | |
| (hR : ∀ i,d^12≤R i) (hP : ∀ j,d^12≤P j) (htotal : ∑ i,R i=∑ j,P j) | |
| (h : ℕ) (o : Option (Table R P)) : | |
| cachedCompletionWeight ((((marginCode R,marginCode P),(d,h)), | |
| canonicalDrawCounts ((marginCode R,marginCode P),(d,h))),o.map tableOutput)= | |
| o.elim 0 (fun X => (canonicalCounts R P hd hR hP htotal h).count X) := by | |
| cases o with | |
| | none => rfl | |
| | some X => exact canonicalCounts_computed R P hd hR hP htotal h hq X | |
| /-- The explicit list contains every feasible table once. Each rejected | |
| free vector has zero weight; the surviving weights are the actual draw counts. -/ | |
| theorem completionWeights_actual {m n d : ℕ} (R : Option (Fin m) → ℕ) | |
| (P : Option (Fin n) → ℕ) (hd : 14≤d) (hq : m*n≤d) | |
| (hR : ∀ i,d^12≤R i) (hP : ∀ j,d^12≤P j) (htotal : ∑ i,R i=∑ j,P j) (h : ℕ) : | |
| completionWeights ((marginCode R,marginCode P),(d,h))= | |
| List.ofFn (fun k : Fin (((∑ i,R i)+1)^(m*n)) => | |
| let o := BoxCompletion.decode R P htotal (∑ i,R i) (finFunctionFinEquiv.symm k) | |
| (o.map tableOutput,o.elim 0 (fun X => (canonicalCounts R P hd hR hP htotal h).count X))) := by | |
| unfold completionWeights | |
| rw [completionCandidates_code R P htotal,List.map_ofFn] | |
| apply congrArg List.ofFn | |
| funext k | |
| dsimp only [Function.comp_apply] | |
| congr 1 | |
| exact cachedCompletionWeight_actual R P hd hq hR hP htotal h _ | |
| end ContingencyTables.DensePrograms | |
| end OAI | |