openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /AllSamplingSemantics.lean
| import OAI.Combinatorics.ContingencyTables.Sampling.AllSamplingProgram | |
| import OAI.Combinatorics.ContingencyTables.Sampling.UnitTrialSemantics | |
| import OAI.Combinatorics.ContingencyTables.Sampling.ProfilePublicSemantics | |
| namespace OAI | |
| /-! # One executable program chooses the correct rectangle branch -/ | |
| namespace ContingencyTables.SmallGraphProfiles | |
| open ProfilePrograms DensePrograms CompletionCounts FirstPaperPhysicalMarginal | |
| open FirstSuccess FairBits SmallContextCoordinates | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| attribute [local irreducible] unitProfileRetry unitCodeOuterDraw | |
| variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ) | |
| local notation "d" => dimensionAllowance (I := Fin m) (J := Fin n) | |
| local notation "b" => smallBinaryLength r c | |
| local notation "s" => smallProposalBits (I := Fin m) (J := Fin n) | |
| lemma hasLargeRectangle_spec (k : ℕ) : | |
| hasLargeRectangle ((List.ofFn r,List.ofFn c),k)=true ↔ | |
| 0<(largeIndices (d^20,List.ofFn r)).length ∧ | |
| 0<(largeIndices (d^20,List.ofFn c)).length := by | |
| simp only [hasLargeRectangle,parameterLargeIndices,marginThreshold_ofFn, | |
| ite_true,Bool.false_eq_true,ite_false,Bool.and_eq_true,decide_eq_true_eq] | |
| lemma allSmall_of_no_large (k : ℕ) | |
| (h : hasLargeRectangle ((List.ofFn r,List.ofFn c),k)≠true) : | |
| ∀ i j,paperSmall r c i j=true := by | |
| intro i j | |
| apply Bool.eq_true_of_ne_false | |
| intro hij | |
| obtain ⟨hi,hj⟩ := (firstPaperSmall_false_iff r c (d^20) i j).mp hij | |
| have a := (largeIndexEquiv r (d^20)).symm ⟨i,hi⟩ | |
| have b' := (largeIndexEquiv c (d^20)).symm ⟨j,hj⟩ | |
| apply h | |
| apply (hasLargeRectangle_spec r c k).mpr | |
| exact ⟨Nat.zero_lt_of_lt a.isLt,Nat.zero_lt_of_lt b'.isLt⟩ | |
| lemma parameterUnitWords_ofFn (k : ℕ) : | |
| parameterUnitWords ((List.ofFn r,List.ofFn c),k)=outerTrials d k*(outerSteps d b k*s) := by | |
| rw [parameterUnitWords,parameterTrials_ofFn,parameterSteps_ofFn,parameterProposalWidth_ofFn] | |
| lemma unitCodeBits_le (k : ℕ) : outerTrials d k*(outerSteps d b k*s)≤finiteSamplerBits r c k := by | |
| apply Nat.mul_le_mul_left | |
| exact (Nat.mul_le_mul_left (outerSteps d b k) | |
| (Nat.le_add_right s (25*(d+b+outerAccuracy d b k+1)^62))).trans (Nat.le_add_right _ _) | |
| lemma scheduled_steps_pos (k : ℕ) : 0<outerSteps d b k := by | |
| have hd : 0<d := by unfold dimensionAllowance; omega | |
| have hb := (smallBinaryLength_parameters r c).2.1 | |
| unfold outerSteps | |
| exact Nat.mul_pos (pow_pos hd _) (pow_pos (by omega) _) | |
| /-- The all-small branch uses a prefix of the same public fixed-length word. -/ | |
| lemma unitSamplingMatrix_code (hsmall : ∀ i j,paperSmall r c i j=true) | |
| (htotal : (∑ i,r i)=∑ j,c j) (k : ℕ) (bits : Fin (finiteSamplerBits r c k) → Bool) : | |
| unitSamplingMatrix (((List.ofFn r,List.ofFn c),k),List.ofFn bits)= | |
| matrixCode (unitCodeOuterDraw r c (computedCellEquiv r c) (allSmallFibre r c hsmall) | |
| (tableState r c (GreedyFeasibleTable.table r c htotal)) (outerSteps d b k) (outerTrials d k) | |
| (GreedyFeasibleTable.table r c htotal) (prefixWord (unitCodeBits_le r c k) bits)).val := by | |
| have hdraw := unitProfileRetry_code r c (computedCellEquiv r c) hsmall htotal | |
| (outerAccuracy d b k) (tableState r c (GreedyFeasibleTable.table r c htotal)) | |
| (outerSteps d b k) (outerTrials d k) (25*(d+b+outerAccuracy d b k+1)^62) | |
| (scheduled_steps_pos r c k) (GreedyFeasibleTable.table r c htotal) | |
| (prefixWord (unitCodeBits_le r c k) bits) | |
| unfold unitSamplingMatrix unitSamplingInput unitSamplingBits samplingRetryParameters | |
| rw [parameterUnitWords_ofFn,take_ofFn_prefix (unitCodeBits_le r c k), | |
| initialConfiguration_code r c htotal,samplingSchedule_ofFn,GreedyFeasibleTable.listMatrix_ofFn] | |
| have hcode : matrixCode (GreedyFeasibleTable.table r c htotal).val= | |
| List.ofFn (fun i => List.ofFn (GreedyFeasibleTable.matrix r c i)) := rfl | |
| rw [hcode] at hdraw | |
| simpa only [Table] using hdraw | |
| end | |
| end ContingencyTables.SmallGraphProfiles | |
| end OAI | |