File size: 4,109 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.Foundations.SourcePhysicalSourceSwapLemmas
import OAI.Computability.PerfectCompleteness.Reduction.FixedCallBudgetLemmas
namespace OAI
section
namespace PerfectCompleteness.FixedPhysicalStoppedComparison
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces FixedParameters FixedRows
open UniqueGamesTheorem.Foundations.Games
variable {δ : ℚ} {hδ : 0 < δ} (params : Parameters δ hδ)
{root h v m : Nat} (p : Path (branch params) root (h + 1))
(outside : Slots (branch params) root → Fin (sourceLength params.plan hδ) → MixedSupport.Slot)
(placeholder : Slots (branch params) (h + 1) → Fin (sourceLength params.plan hδ) → MixedSupport.Slot)
(clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch params h) → Slots (branch params) h)
theorem beta_nonneg : 0 ≤ (projectionProbability params h : ℝ) := by
exact_mod_cast (projectionProbability_bounds params h).1.le
theorem beta_le_one : (projectionProbability params h : ℝ) ≤ 1 := by
exact_mod_cast (projectionProbability_bounds params h).2.le
theorem calls_le (hroot : root ≤ params.plan.depth) :
Fintype.card (WholeCutCalls.Index (rows params.plan) (repeats params.plan) p) ≤
calls params.plan := by
have hactual := FixedCallBudget.modified_calls_le (rows params.plan)
(repeats params.plan) p hroot
apply hactual.trans
unfold calls
omega
variable (source : FiniteDistribution
(SourceChildKernel.Sources (m := m) (t := sourceLength params.plan hδ) designated))
abbrev modifiedLaw :=
SourcePhysicalStoppedComparison.sourceModifiedLaw (rows params.plan) (repeats params.plan)
p outside placeholder clauses designated source (projectionProbability params h : ℝ)
(beta_nonneg params) (beta_le_one params)
abbrev stoppedLaw :=
SourcePhysicalStoppedComparison.sourceStoppedLaw (rows params.plan) (repeats params.plan)
p outside clauses designated source
theorem variation_lt_accuracy (hroot : root ≤ params.plan.depth) :
(modifiedLaw params p outside placeholder clauses designated source).totalVariation
(stoppedLaw params p outside clauses designated source) < params.accuracy := by
have htv := SourcePhysicalStoppedComparison.source_variation
(rows params.plan) (repeats params.plan) p outside placeholder clauses designated
source (branch_pos params h) (projectionProbability params h : ℝ)
(beta_nonneg params) (beta_le_one params)
apply htv.trans_lt
simpa only [SourceChildProjectionComparison.error] using
FixedProjectionError.sparse_error_lt params (calls_le params p hroot) h
theorem probability_gt_sub_accuracy (hroot : root ≤ params.plan.depth)
(event : SourcePhysicalStoppedComparison.Observation (rows params.plan) p outside clauses designated → Bool) :
(stoppedLaw params p outside clauses designated source).probability event - params.accuracy <
(modifiedLaw params p outside placeholder clauses designated source).probability event := by
have htv := variation_lt_accuracy params p outside placeholder clauses designated source hroot
have hp := FiniteDistribution.probability_le_add_totalVariation
(stoppedLaw params p outside clauses designated source)
(modifiedLaw params p outside placeholder clauses designated source) event
rw [FiniteDistribution.totalVariation_comm] at hp
linarith
theorem probability_gt_of_reserve (hroot : root ≤ params.plan.depth)
(event : SourcePhysicalStoppedComparison.Observation (rows params.plan) p outside clauses designated → Bool)
(c : ℝ)
(hreserve : c + params.accuracy ≤
(stoppedLaw params p outside clauses designated source).probability event) :
c < (modifiedLaw params p outside placeholder clauses designated source).probability event := by
have h := probability_gt_sub_accuracy params p outside placeholder clauses designated source hroot event
linarith
end
end PerfectCompleteness.FixedPhysicalStoppedComparison
end
end OAI
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