openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionInitializedRestartSemantics.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionInitializedRestartProgram | |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionRestartDataLaw | |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionRestartScheduleSemantics | |
| namespace OAI | |
| /-! # Literal inputs and disjoint slices of initialized restarts -/ | |
| namespace ContingencyTables.CompanionInitializedRestartProgram | |
| open MatchingFPRAS CommonBasesFPRAS FairBits DensePrograms | |
| open CompanionFactorialApplication CompanionComputedInitial CompanionComputedProposal | |
| open CompanionArithmeticProposal CompanionEvaluatedWeightProgram CompanionMultiplierMatrixProgram | |
| open CompanionOracleDensity GridBoundary | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| variable {m n : ℕ} | |
| def data (A : Instance (Fin m) (Fin n)) (C K H T N R L j : ℕ) | |
| (epsilon : TreeFraction.Fraction) (z : View A → ℕ) | |
| (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) | |
| (w : ℕ → LabelIndex A → LabelIndex A → TreeFraction.Fraction) | |
| (estimates : Fin j → TreeFraction.Fraction) (bits : List Bool) : Input := | |
| ((CompanionRestartInitializer.data A C K H T N R L epsilon z v, | |
| List.ofFn (fun i : Fin j => (matrixCode (w (i.val+1)),estimates i))),bits) | |
| lemma stages_data (A : Instance (Fin m) (Fin n)) (C K H T N R L j : ℕ) | |
| (epsilon : TreeFraction.Fraction) (z : View A → ℕ) | |
| (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) | |
| (w : ℕ → LabelIndex A → LabelIndex A → TreeFraction.Fraction) | |
| (estimates : Fin j → TreeFraction.Fraction) (bits : List Bool) : | |
| stages (data A C K H T N R L j epsilon z v w estimates bits)= | |
| CompanionRestartProgram.stagesData A C K j | |
| ([Fintype.card (Labels A),A.W,H,T,N,R,L],epsilon) z v (List.replicate bits.length false) w := by | |
| unfold stages indices data | |
| rw [List.length_ofFn,DirectedBFSCode.range_eq_map] | |
| simp only [List.map_map,Function.comp_def] | |
| rw [←List.ofFn_eq_map] | |
| apply congrArg List.ofFn | |
| funext i | |
| unfold stageAt reserve | |
| simp only [List.getD_eq_getElem?_getD,List.getElem?_ofFn,i.isLt,dite_true,Option.getD_some, | |
| List.map_const'] | |
| rfl | |
| lemma initialWidth_data (A : Instance (Fin m) (Fin n)) (C K H T N R L j : ℕ) | |
| (epsilon : TreeFraction.Fraction) (z : View A → ℕ) | |
| (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) | |
| (w : ℕ → LabelIndex A → LabelIndex A → TreeFraction.Fraction) | |
| (estimates : Fin j → TreeFraction.Fraction) (bits : List Bool) : | |
| initialWidth (data A C K H T N R L j epsilon z v w estimates bits)= | |
| wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H)) := | |
| CompanionRestartInitializer.width_data A C K H T N R L epsilon z v | |
| /-- Exact physical splitting: the initializer sees the first block, | |
| the restart loop sees the remaining block, and all markers are fixed. -/ | |
| theorem output_split (A : Instance (Fin m) (Fin n)) (C K H T N R L j s : ℕ) | |
| (epsilon : TreeFraction.Fraction) (z : View A → ℕ) | |
| (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) | |
| (w : ℕ → LabelIndex A → LabelIndex A → TreeFraction.Fraction) | |
| (estimates : Fin j → TreeFraction.Fraction) | |
| (bits : Fin (wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+s) → Bool) : | |
| output (data A C K H T N R L j epsilon z v w estimates (List.ofFn bits))= | |
| CompanionRestartProgram.output | |
| ((CompanionRestartProgram.stagesData A C K j | |
| ([Fintype.card (Labels A),A.W,H,T,N,R,L],epsilon) z v | |
| (List.replicate (wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+s) false) w, | |
| CompanionRestartInitializer.output | |
| (CompanionRestartInitializer.data A C K H T N R L epsilon z v, | |
| List.ofFn (splitWordEquiv | |
| (wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))) s bits).1)), | |
| List.ofFn (splitWordEquiv | |
| (wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))) s bits).2) := by | |
| unfold output prepare initialCode initialInput | |
| rw [stages_data,initialWidth_data,List.length_ofFn] | |
| dsimp only [data] | |
| rw [ofFn_split_take,ofFn_split_drop] | |
| end | |
| end ContingencyTables.CompanionInitializedRestartProgram | |
| end OAI | |