openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /FixedSourceProjectionComparison.lean
| import OAI.Computability.PerfectCompleteness.Decoding.FixedProjectionErrorLemmas | |
| import OAI.Computability.PerfectCompleteness.Decoding.SourceChildProjectionComparison | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedSourceProjectionComparison | |
| open FixedParameters FixedRows RecursiveSpaces | |
| open UniqueGamesTheorem.Foundations.Games | |
| open scoped Classical | |
| noncomputable section | |
| variable {δ : ℚ} {hδ : 0 < δ} (p : Parameters δ hδ) | |
| def projectionFlag (height : Nat) : FiniteDistribution Bool := | |
| ProjectionPosterior.bernoulli (projectionProbability p height : ℝ) | |
| (by exact_mod_cast (projectionProbability_bounds p height).1.le) | |
| (by exact_mod_cast (projectionProbability_bounds p height).2.le) | |
| variable {v m height : Nat} {C : Type*} [Fintype C] | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch p height) → Slots (branch p) height) | |
| abbrev Observation := | |
| SourcePhysicalAssemblyLaw.Observation (C := C) (t := sourceLength p.plan hδ) | |
| (rows p.plan) clauses designated | |
| def mixedLaw [NeZero m] : FiniteDistribution (Observation (C := C) p clauses designated) := | |
| (SourceChildKernel.originalLaw (C := C) (t := sourceLength p.plan hδ) | |
| (rows p.plan) clauses designated (fun _ => projectionFlag p height)).pushforward | |
| (SourcePhysicalAssemblyLaw.observe (rows p.plan) clauses designated) | |
| def uniformLaw [NeZero m] : FiniteDistribution (Observation (C := C) p clauses designated) := | |
| (SourceChildKernel.originalLaw (C := C) (t := sourceLength p.plan hδ) | |
| (rows p.plan) clauses designated (fun _ => SourceChildNativeLaw.falseFlag)).pushforward | |
| (SourcePhysicalAssemblyLaw.observe (rows p.plan) clauses designated) | |
| theorem observed_totalVariation_lt [NeZero m] {Γ : Type*} [Fintype Γ] | |
| (hcalls : Fintype.card C ≤ calls p.plan) | |
| (observe : Observation (C := C) p clauses designated → Γ) : | |
| ((mixedLaw p clauses designated).pushforward observe).totalVariation | |
| ((uniformLaw p clauses designated).pushforward observe) < p.accuracy := by | |
| have hβ : 0 ≤ (projectionProbability p height : ℝ) := by | |
| exact_mod_cast (projectionProbability_bounds p height).1.le | |
| have hβ' : (projectionProbability p height : ℝ) ≤ 1 := by | |
| exact_mod_cast (projectionProbability_bounds p height).2.le | |
| have h := SourceChildProjectionComparison.original_observed_variation | |
| (C := C) (t := sourceLength p.plan hδ) (rows p.plan) clauses designated | |
| (branch_pos p height) (projectionProbability p height : ℝ) hβ hβ' observe | |
| have herror := FixedProjectionError.sparse_error_lt p hcalls height | |
| have hlt := h.trans_lt herror | |
| simpa only [mixedLaw, uniformLaw, projectionFlag, | |
| FiniteDistribution.pushforward_comp] using hlt | |
| theorem event_probability_abs_sub_lt [NeZero m] {Γ : Type*} [Fintype Γ] | |
| (hcalls : Fintype.card C ≤ calls p.plan) | |
| (observe : Observation (C := C) p clauses designated → Γ) (event : Γ → Bool) : | |
| |(mixedLaw p clauses designated).probability (fun record => event (observe record)) - | |
| (uniformLaw p clauses designated).probability (fun record => event (observe record))| < | |
| p.accuracy := by | |
| have h := FiniteDistribution.probability_abs_sub_le_totalVariation | |
| ((mixedLaw p clauses designated).pushforward observe) | |
| ((uniformLaw p clauses designated).pushforward observe) event | |
| have hlt := h.trans_lt (observed_totalVariation_lt p clauses designated hcalls observe) | |
| simpa only [FiniteDistribution.probability_pushforward] using hlt | |
| theorem event_probability_transfer [NeZero m] {Γ : Type*} [Fintype Γ] | |
| (hcalls : Fintype.card C ≤ calls p.plan) | |
| (observe : Observation (C := C) p clauses designated → Γ) (event : Γ → Bool) : | |
| (mixedLaw p clauses designated).probability (fun record => event (observe record)) < | |
| (uniformLaw p clauses designated).probability (fun record => event (observe record)) + | |
| p.accuracy := by | |
| have h := event_probability_abs_sub_lt p clauses designated hcalls observe event | |
| have hsub := lt_of_le_of_lt (le_abs_self | |
| ((mixedLaw p clauses designated).probability (fun record => event (observe record)) - | |
| (uniformLaw p clauses designated).probability (fun record => event (observe record)))) h | |
| exact (sub_lt_iff_lt_add').mp hsub | |
| end | |
| end PerfectCompleteness.FixedSourceProjectionComparison | |
| end | |
| end OAI | |