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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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 | 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
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