File size: 4,341 Bytes
2f887db | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | 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
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