openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Reduction /SignedCompletionSchedule.lean
| import OAI.Computability.PerfectCompleteness.Foundations.ExactConditionalOutput | |
| import OAI.Computability.PerfectCompleteness.Foundations.RawOccurrenceOrderLemmas | |
| import OAI.Computability.PerfectCompleteness.Machines.FiniteConstraintMachine | |
| namespace OAI | |
| namespace PerfectCompleteness.SignedCompletionSchedule | |
| open MetadataFreeSampler MetadataFreeLaw | |
| open UniqueGamesTheorem.Foundations.Games | |
| open Turing UniqueGamesTheorem.Foundations.Complexity | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n t q : Nat} {rows repeats : Nat → Nat} | |
| (hq : 0 < q) | |
| (large : CanonicalKeyEncoding.partitionWidth (TreeCanonical.locationCount branch n t) ≤ q) | |
| def result (signs : SignTuple branch n t) (e : SignedOutcome signs rows repeats) : | |
| LocalCompletionFamily.Result q := FiniteConstraintMachine.result hq large ⟨signs, e⟩ | |
| def active (signs : SignTuple branch n t) (e : SignedOutcome signs rows repeats) : Nat := | |
| LocalCompletionFamily.activeCount (result (rows := rows) (repeats := repeats) hq large signs e) | |
| theorem active_positive (signs : SignTuple branch n t) (e : SignedOutcome signs rows repeats) : | |
| 0 < active (rows := rows) (repeats := repeats) hq large signs e := LocalCompletionFamily.activeCount_positive _ | |
| theorem active_le (signs : SignTuple branch n t) (e : SignedOutcome signs rows repeats) : | |
| active (rows := rows) (repeats := repeats) hq large signs e ≤ 2 * q := LocalCompletionFamily.activeCount_le _ | |
| abbrev Completed (signs : SignTuple branch n t) := | |
| Σ e : SignedOutcome signs rows repeats, Fin (active (rows := rows) (repeats := repeats) hq large signs e) | |
| abbrev Descriptor := Σ signs : SignTuple branch n t, Completed (rows := rows) (repeats := repeats) hq large signs | |
| def descriptorTable (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) : Fin (2 * q) → Fin q := | |
| LocalCompletionFamily.table (result (rows := rows) (repeats := repeats) hq large descriptor.1 descriptor.2.1) descriptor.2.2 | |
| theorem descriptorTable_eq_parsed (signs : SignTuple branch n t) | |
| (e : Completed (rows := rows) (repeats := repeats) hq large signs) (a : Fin (2 * q)) : | |
| descriptorTable (rows := rows) (repeats := repeats) hq large ⟨signs, e⟩ a = | |
| (LocalCompletionSerialization.projection | |
| (FiniteConstraintMachine.result_correct (rows := rows) (repeats := repeats) hq large ⟨signs, e.1⟩) e.2).images[a] := | |
| (FiniteConstraintMachine.parsed_projection (rows := rows) (repeats := repeats) hq large ⟨signs, e.1⟩ e.2 a).symm | |
| def seedLaw (signs : SignTuple branch n t) (e : SignedOutcome signs rows repeats) : | |
| FiniteDistribution (Fin (active (rows := rows) (repeats := repeats) hq large signs e)) := by | |
| letI : Nonempty (Fin (active (rows := rows) (repeats := repeats) hq large signs e)) := ⟨⟨0, active_positive (rows := rows) (repeats := repeats) hq large signs e⟩⟩ | |
| exact FiniteDistribution.uniform _ | |
| variable (hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k) | |
| (hrows : ∀ k, 0 < rows (k + 1)) | |
| def law (signs : SignTuple branch n t) : FiniteDistribution (Completed (rows := rows) (repeats := repeats) hq large signs) := | |
| CompletionSoundness.sigmaLaw (signedLaw signs rows repeats hn hbranch hrows) | |
| (seedLaw (rows := rows) (repeats := repeats) hq large signs) | |
| theorem law_weight (signs : SignTuple branch n t) (e : Completed (rows := rows) (repeats := repeats) hq large signs) : | |
| (law (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).weight e = | |
| (signedLaw signs rows repeats hn hbranch hrows).weight e.1 / | |
| active (rows := rows) (repeats := repeats) hq large signs e.1 := by | |
| change (signedLaw signs rows repeats hn hbranch hrows).weight e.1 * | |
| (1 / (Fintype.card (Fin (active (rows := rows) (repeats := repeats) hq large signs e.1)) : ℝ)) = _ | |
| simp only [Fintype.card_fin, div_eq_mul_inv, one_mul] | |
| def count (signs : SignTuple branch n t) (e : Completed (rows := rows) (repeats := repeats) hq large signs) : Nat := | |
| ExactSeedMultiplicity.perSeed (2 * q) (active (rows := rows) (repeats := repeats) hq large signs e.1) | |
| (SignedMultiplicity.baseCopies branch n t rows repeats hn hbranch hrows signs e.1) | |
| theorem count_positive (signs : SignTuple branch n t) (e : Completed (rows := rows) (repeats := repeats) hq large signs) : | |
| 0 < count (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs e := | |
| ExactSeedMultiplicity.perSeed_positive (active_positive (rows := rows) (repeats := repeats) hq large signs e.1) | |
| (active_le (rows := rows) (repeats := repeats) hq large signs e.1) | |
| (SignedMultiplicity.baseCopies_positive branch n t rows repeats hn hbranch hrows signs e.1) | |
| theorem count_total (signs : SignTuple branch n t) : | |
| (∑ e, count (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs e) = | |
| SignedMultiplicity.denominator branch n t rows repeats hn hbranch hrows q := | |
| ExactSeedMultiplicity.sum_multiplicity | |
| (SignedMultiplicity.baseCopies branch n t rows repeats hn hbranch hrows signs) | |
| (active (rows := rows) (repeats := repeats) hq large signs) (active_positive (rows := rows) (repeats := repeats) hq large signs) (active_le (rows := rows) (repeats := repeats) hq large signs) | |
| (SignedMultiplicity.baseCopies_total branch n t rows repeats hn hbranch hrows signs) | |
| theorem count_ratio (signs : SignTuple branch n t) (e : Completed (rows := rows) (repeats := repeats) hq large signs) : | |
| (count (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs e : ℚ) / | |
| SignedMultiplicity.denominator branch n t rows repeats hn hbranch hrows q = | |
| SignedMultiplicity.weights branch n t rows repeats hn hbranch hrows signs e.1 / | |
| active (rows := rows) (repeats := repeats) hq large signs e.1 := by | |
| change (ExactSeedMultiplicity.perSeed (2 * q) (active (rows := rows) (repeats := repeats) hq large signs e.1) | |
| (SignedMultiplicity.baseCopies branch n t rows repeats hn hbranch hrows signs e.1) : ℚ) / | |
| ExactSeedMultiplicity.denominator | |
| (SignedMultiplicity.baseDenominator branch n t rows repeats hn hbranch hrows) (2 * q) = _ | |
| rw [ExactSeedMultiplicity.perSeed_ratio | |
| (SignedMultiplicity.baseDenominator_positive branch n t rows repeats hn hbranch hrows) | |
| (active_positive (rows := rows) (repeats := repeats) hq large signs e.1) (active_le (rows := rows) (repeats := repeats) hq large signs e.1), | |
| SignedMultiplicity.baseCopies_ratio] | |
| theorem count_mass (signs : SignTuple branch n t) (e : Completed (rows := rows) (repeats := repeats) hq large signs) : | |
| (count (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs e : ℝ) / | |
| SignedMultiplicity.denominator branch n t rows repeats hn hbranch hrows q = | |
| (law (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).weight e := by | |
| have h := congrArg (fun z : ℚ => (z : ℝ)) (count_ratio (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs e) | |
| simp only [Rat.cast_div, Rat.cast_natCast] at h | |
| rw [law_weight, (signedRationalLaw signs rows repeats hn hbranch hrows).cast_weight] | |
| exact h | |
| def schedule (signs : SignTuple branch n t) : List (Completed (rows := rows) (repeats := repeats) hq large signs) := | |
| ExactConditionalOutput.copies (count (rows := rows) (repeats := repeats) hq large hn hbranch hrows) signs | |
| theorem schedule_length (signs : SignTuple branch n t) : | |
| (schedule (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).length = | |
| SignedMultiplicity.denominator branch n t rows repeats hn hbranch hrows q := by | |
| rw [schedule, ExactConditionalOutput.copies_length, count_total] | |
| theorem schedule_nonempty (signs : SignTuple branch n t) : | |
| schedule (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs ≠ [] := by | |
| intro hempty | |
| have h := schedule_length (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs | |
| rw [hempty, List.length_nil] at h | |
| exact (Nat.ne_of_gt | |
| (SignedMultiplicity.denominator_positive branch n t rows repeats hn hbranch hrows q)) h.symm | |
| theorem schedule_acceptance (signs : SignTuple branch n t) | |
| (accepts : Completed (rows := rows) (repeats := repeats) hq large signs → Bool) : | |
| (((schedule (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).filter accepts).length : ℝ) / | |
| (schedule (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).length = | |
| (law (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).probability accepts := by | |
| rw [schedule_length] | |
| change (((ExactConditionalOutput.copies (count (rows := rows) (repeats := repeats) hq large hn hbranch hrows) signs).filter | |
| accepts).length : ℝ) / _ = _ | |
| rw [ExactConditionalOutput.copies_filter_length, Nat.cast_sum, Finset.sum_div, | |
| FiniteDistribution.probability] | |
| apply Finset.sum_congr rfl | |
| intro e _ | |
| cases accepts e <;> simp [count_mass] | |
| def descriptorEnum : Descriptor (rows := rows) (repeats := repeats) hq large ≃ Fin (Fintype.card (Descriptor (rows := rows) (repeats := repeats) hq large)) := | |
| Fintype.equivFin _ | |
| def descriptorBits (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) : List Bool := | |
| FiniteBlockMachine.word (descriptorEnum (rows := rows) (repeats := repeats) hq large) descriptor | |
| @[simp] theorem descriptorBits_length (descriptor : Descriptor (rows := rows) (repeats := repeats) hq large) : | |
| (descriptorBits (rows := rows) (repeats := repeats) hq large descriptor).length = Fintype.card (Descriptor (rows := rows) (repeats := repeats) hq large) := | |
| FiniteBlockMachine.word_length _ _ | |
| def outputBits (signs : SignTuple branch n t) : List Bool := | |
| (schedule (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).flatMap | |
| (fun e => descriptorBits (rows := rows) (repeats := repeats) hq large ⟨signs, e⟩) | |
| def signsEnum (branch : Nat → Nat) (n t : Nat) : | |
| SignTuple branch n t ≃ Fin (Fintype.card (SignTuple branch n t)) := Fintype.equivFin _ | |
| def inputBits (signs : SignTuple branch n t) : List Bool := | |
| FiniteBlockMachine.word (signsEnum branch n t) signs | |
| def computation : | |
| TM2ComputableInPolyTime (inputBits (branch := branch) (n := n) (t := t)) | |
| id (outputBits (rows := rows) (repeats := repeats) hq large hn hbranch hrows) := | |
| FiniteWordMachine.computation (signsEnum branch n t) (fun _ _ => (false, false, false)) | |
| (outputBits (rows := rows) (repeats := repeats) hq large hn hbranch hrows) | |
| def machine_output (signs : SignTuple branch n t) : | |
| TM2OutputsInTime (computation (rows := rows) (repeats := repeats) hq large hn hbranch hrows).tm (inputBits signs) | |
| (some (outputBits (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs)) 1 := by | |
| have h := (computation (rows := rows) (repeats := repeats) hq large hn hbranch hrows).outputsFun signs | |
| change TM2OutputsInTime (computation (rows := rows) (repeats := repeats) hq large hn hbranch hrows).tm | |
| ((inputBits signs).map id) (some ((outputBits (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs).map id)) | |
| ((1 : Polynomial Nat).eval _) at h | |
| rw [Polynomial.eval_one] at h | |
| exact Eq.mp (congrArg₂ | |
| (fun (i o : List Bool) => | |
| TM2OutputsInTime (computation (rows := rows) (repeats := repeats) hq large hn hbranch hrows).tm i (some o) 1) | |
| (List.map_id (inputBits signs)) | |
| (List.map_id (outputBits (rows := rows) (repeats := repeats) hq large hn hbranch hrows signs))) h | |
| theorem workAlphabet_finite (tape : (computation (rows := rows) (repeats := repeats) hq large hn hbranch hrows).tm.K) : | |
| Finite ((computation (rows := rows) (repeats := repeats) hq large hn hbranch hrows).tm.Γ tape) := by | |
| change Finite Bool | |
| infer_instance | |
| end | |
| end PerfectCompleteness.SignedCompletionSchedule | |
| end OAI | |