openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /CompletedSignedLaw.lean
| import OAI.Computability.PerfectCompleteness.Reduction.FinitePreliminaryCompletion | |
| import OAI.Computability.PerfectCompleteness.Reduction.SignedCompletionSchedule | |
| import OAI.Computability.PerfectCompleteness.Sampling.MetadataFreeLaw | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CompletedSignedLaw | |
| open MetadataFreeSampler MetadataFreeLaw | |
| open UniqueGamesTheorem.Foundations.Games | |
| open CompletionSoundness CompletionSoundness.LegalProjectionGame | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n t v m : Nat} | |
| (clauses : Fin m → SourceClause.NormalizedClause v) (rows repeats : Nat → Nat) | |
| local notation "locationCount" => TreeCanonical.locationCount branch n t | |
| local notation "hq" δ => CanonicalLocalCompletion.alphabet_positive locationCount δ | |
| local notation "large" δ => CanonicalLocalCompletion.partitionWidth_le_alphabet locationCount δ | |
| local notation "F" => FinitePreliminaryCompletion.blockFamily clauses branch n t rows repeats | |
| local instance leftVertex_fintype : Fintype (CanonicalGame.LeftVertex F) := Fintype.ofFinite _ | |
| local instance rightVertex_fintype : Fintype (CanonicalGame.RightVertex F) := Fintype.ofFinite _ | |
| local instance leftLabel_fintype (x : CanonicalGame.LeftVertex F) : | |
| Fintype (CanonicalGame.LeftLabel F x) := Fintype.ofFinite _ | |
| local instance rightLabel_fintype (y : CanonicalGame.RightVertex F) : | |
| Fintype (CanonicalGame.RightLabel F y) := Fintype.ofFinite _ | |
| abbrev SignedCompleted (δ : ℚ) := | |
| Σ source : PreliminarySampler.Questions branch n t m, | |
| SignedCompletionSchedule.Completed (rows := rows) (repeats := repeats) | |
| (hq δ) (large δ) (sourceSigns clauses source) | |
| variable [NeZero m] (hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k) | |
| (hrows : ∀ k, 0 < rows (k + 1)) (δ : ℚ) | |
| abbrev Completed := | |
| Σ raw : PreliminarySampler.Raw clauses branch n t rows repeats, | |
| Fin ((FinitePreliminaryCompletion.actualFamily (t := t) | |
| clauses rows repeats hn hbranch hrows δ).size raw) | |
| theorem size_eq_active (e : PreliminarySampler.Raw clauses branch n t rows repeats) : | |
| (FinitePreliminaryCompletion.actualFamily (t := t) | |
| clauses rows repeats hn hbranch hrows δ).size e = | |
| SignedCompletionSchedule.active (hq δ) (large δ) | |
| (sourceSigns clauses e.1.1) (encodeOutcome clauses rows repeats e) := rfl | |
| def completedEquiv : | |
| Completed (t := t) clauses rows repeats hn hbranch hrows δ ≃ | |
| SignedCompleted (branch := branch) (n := n) (t := t) clauses rows repeats δ := | |
| (Equiv.sigmaCongrLeft | |
| (β := fun y : (Σ source : PreliminarySampler.Questions branch n t m, | |
| SignedOutcome (sourceSigns clauses source) rows repeats) => | |
| Fin (SignedCompletionSchedule.active (hq δ) (large δ) | |
| (sourceSigns clauses y.1) y.2)) | |
| (rawEquiv clauses rows repeats)).trans | |
| (Equiv.sigmaAssoc (fun source outcome => | |
| Fin (SignedCompletionSchedule.active (hq δ) (large δ) | |
| (sourceSigns clauses source) outcome))) | |
| @[simp] theorem completedEquiv_apply | |
| (e : PreliminarySampler.Raw clauses branch n t rows repeats) | |
| (seed : Fin ((FinitePreliminaryCompletion.actualFamily (t := t) | |
| clauses rows repeats hn hbranch hrows δ).size e)) : | |
| completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ ⟨e, seed⟩ = | |
| ⟨e.1.1, encodeOutcome clauses rows repeats e, seed⟩ := rfl | |
| @[simp] theorem completedEquiv_source | |
| (e : Completed (t := t) clauses rows repeats hn hbranch hrows δ) : | |
| (completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ e).1 = e.1.1.1 := rfl | |
| theorem completedEquiv_symm_source | |
| (e : SignedCompleted (branch := branch) (n := n) (t := t) clauses rows repeats δ) : | |
| ((completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ).symm e).1.1.1 = e.1 := by | |
| have h := congrArg Sigma.fst | |
| ((completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ).apply_symm_apply e) | |
| exact h | |
| theorem descriptorTable_eq_map | |
| (e : Completed (t := t) clauses rows repeats hn hbranch hrows δ) | |
| (a : Fin (2 * FinitePreliminaryCompletion.alphabet branch n t δ)) : | |
| SignedCompletionSchedule.descriptorTable (hq δ) (large δ) | |
| ⟨sourceSigns clauses (completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ e).1, | |
| (completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ e).2⟩ a = | |
| (FinitePreliminaryCompletion.actualFamily (t := t) | |
| clauses rows repeats hn hbranch hrows δ).map e.1 e.2 a := rfl | |
| def law : FiniteDistribution | |
| (SignedCompleted (branch := branch) (n := n) (t := t) clauses rows repeats δ) := | |
| sigmaLaw PreliminarySampler.questionsLaw (fun source => | |
| SignedCompletionSchedule.law (rows := rows) (repeats := repeats) | |
| (hq δ) (large δ) hn hbranch hrows (sourceSigns clauses source)) | |
| theorem completed_weight (e : Completed (t := t) clauses rows repeats hn hbranch hrows δ) : | |
| (FinitePreliminaryCompletion.actualFamily (t := t) | |
| clauses rows repeats hn hbranch hrows δ).game.occurrences.weight e = | |
| (law (t := t) clauses rows repeats hn hbranch hrows δ).weight | |
| (completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ e) := by | |
| rcases e with ⟨raw, seed⟩ | |
| change (PreliminarySampler.law (t := t) clauses rows repeats hn hbranch hrows).weight raw * | |
| (1 / (Fintype.card (Fin ((FinitePreliminaryCompletion.actualFamily (t := t) | |
| clauses rows repeats hn hbranch hrows δ).size raw)) : ℝ)) = | |
| (1 / (Fintype.card (PreliminarySampler.Questions branch n t m) : ℝ)) * | |
| (SignedCompletionSchedule.law (rows := rows) (repeats := repeats) | |
| (hq δ) (large δ) hn hbranch hrows | |
| (sourceSigns clauses raw.1.1)).weight | |
| ⟨encodeOutcome clauses rows repeats raw, seed⟩ | |
| rw [Fintype.card_fin, raw_weight_eq, SignedCompletionSchedule.law_weight] | |
| rw [size_eq_active (t := t) clauses rows repeats hn hbranch hrows δ raw] | |
| simp only [div_eq_mul_inv, one_mul, mul_assoc] | |
| theorem actualcompletedLaw_transport : | |
| (FinitePreliminaryCompletion.actualFamily (t := t) | |
| clauses rows repeats hn hbranch hrows δ).game.occurrences.transport | |
| (completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ) = | |
| sigmaLaw PreliminarySampler.questionsLaw (fun source => | |
| SignedCompletionSchedule.law (rows := rows) (repeats := repeats) | |
| (hq δ) (large δ) hn hbranch hrows | |
| (sourceSigns clauses source)) := by | |
| apply FiniteDistribution.eq_of_weight_eq | |
| intro e | |
| change (FinitePreliminaryCompletion.actualFamily (t := t) | |
| clauses rows repeats hn hbranch hrows δ).game.occurrences.weight | |
| ((completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ).symm e) = | |
| (law (t := t) clauses rows repeats hn hbranch hrows δ).weight e | |
| simpa only [Equiv.apply_symm_apply] using | |
| completed_weight (t := t) clauses rows repeats hn hbranch hrows δ | |
| ((completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ).symm e) | |
| theorem probability_reindex | |
| (event : SignedCompleted (branch := branch) (n := n) (t := t) clauses rows repeats δ → Bool) : | |
| (FinitePreliminaryCompletion.actualFamily (t := t) | |
| clauses rows repeats hn hbranch hrows δ).game.occurrences.probability | |
| (fun e => event (completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ e)) = | |
| (law (t := t) clauses rows repeats hn hbranch hrows δ).probability event := by | |
| rw [← FiniteDistribution.probability_transport] | |
| rw [actualcompletedLaw_transport] | |
| rfl | |
| theorem probability_eq | |
| (event : Completed (t := t) clauses rows repeats hn hbranch hrows δ → Bool) : | |
| (FinitePreliminaryCompletion.actualFamily (t := t) | |
| clauses rows repeats hn hbranch hrows δ).game.occurrences.probability event = | |
| (law (t := t) clauses rows repeats hn hbranch hrows δ).probability | |
| (fun e => event ((completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ).symm e)) := by | |
| simpa only [Equiv.symm_apply_apply] using | |
| probability_reindex (t := t) clauses rows repeats hn hbranch hrows δ | |
| (fun e => event ((completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ).symm e)) | |
| theorem probability_iterated | |
| (event : Completed (t := t) clauses rows repeats hn hbranch hrows δ → Bool) : | |
| (FinitePreliminaryCompletion.actualFamily (t := t) | |
| clauses rows repeats hn hbranch hrows δ).game.occurrences.probability event = | |
| PreliminarySampler.questionsLaw.expectation (fun source => | |
| (SignedCompletionSchedule.law (rows := rows) (repeats := repeats) | |
| (hq δ) (large δ) hn hbranch hrows | |
| (sourceSigns clauses source)).probability (fun outcome => | |
| event ((completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ).symm | |
| ⟨source, outcome⟩))) := by | |
| rw [probability_eq] | |
| exact sigmaLaw_probability _ _ _ | |
| theorem probability_schedule | |
| (event : Completed (t := t) clauses rows repeats hn hbranch hrows δ → Bool) : | |
| (FinitePreliminaryCompletion.actualFamily (t := t) | |
| clauses rows repeats hn hbranch hrows δ).game.occurrences.probability event = | |
| PreliminarySampler.questionsLaw.expectation (fun source => | |
| (((SignedCompletionSchedule.schedule (rows := rows) (repeats := repeats) | |
| (hq δ) (large δ) hn hbranch hrows | |
| (sourceSigns clauses source)).filter (fun outcome => | |
| event ((completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ).symm | |
| ⟨source, outcome⟩))).length : ℝ) / | |
| (SignedCompletionSchedule.schedule (rows := rows) (repeats := repeats) | |
| (hq δ) (large δ) hn hbranch hrows | |
| (sourceSigns clauses source)).length) := by | |
| rw [probability_iterated] | |
| congr 1 | |
| funext source | |
| exact (SignedCompletionSchedule.schedule_acceptance (hq δ) (large δ) hn hbranch hrows | |
| (sourceSigns clauses source) (fun outcome => | |
| event ((completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ).symm | |
| ⟨source, outcome⟩))).symm | |
| end | |
| end PerfectCompleteness.CompletedSignedLaw | |
| end | |
| end OAI | |