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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 | import OAI.Computability.PerfectCompleteness.Construction.SourceQuestionOrderLemmas
import OAI.Computability.PerfectCompleteness.Sampling.CompletedSignedLaw
namespace OAI
section
namespace PerfectCompleteness.CompletedOccurrenceOrder
open MetadataFreeSampler
open UniqueGamesTheorem.Foundations.Games
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)
[NeZero m] (hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k)
(hrows : ∀ k, 0 < rows (k + 1)) (δ : ℚ)
local notation "locationCount" => TreeCanonical.locationCount branch n t
local notation "hq" => CanonicalLocalCompletion.alphabet_positive locationCount δ
local notation "large" => CanonicalLocalCompletion.partitionWidth_le_alphabet locationCount δ
local notation "D" => SignedMultiplicity.denominator branch n t rows repeats hn hbranch hrows
(FinitePreliminaryCompletion.alphabet branch n t 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 _
local notation "C" =>
FinitePreliminaryCompletion.actualFamily (t := t) clauses rows repeats hn hbranch hrows δ
local notation "Completed" => CompletedSignedLaw.Completed (t := t) clauses rows repeats hn hbranch hrows δ
local notation "transport" =>
Equiv.symm (CompletedSignedLaw.completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ)
def localOrder (source : PreliminarySampler.Questions branch n t m) : List Completed :=
(SignedCompletionSchedule.schedule (branch := branch) (n := n) (t := t)
(rows := rows) (repeats := repeats) hq large hn hbranch hrows
(sourceSigns clauses source)).map (fun outcome => transport ⟨source, outcome⟩)
def completedOrder : List Completed :=
(SourceQuestionOrder.sourceOrder branch n t m).flatMap
(localOrder (t := t) clauses rows repeats hn hbranch hrows δ)
theorem localOrder_length (source : PreliminarySampler.Questions branch n t m) :
(localOrder (t := t) clauses rows repeats hn hbranch hrows δ source).length = D := by
rw [localOrder, List.length_map, SignedCompletionSchedule.schedule_length]
theorem completedOrder_length :
(completedOrder (t := t) clauses rows repeats hn hbranch hrows δ).length =
D * m ^ locationCount := by
change (ExactConditionalOutput.output
(SignedCompletionSchedule.count (branch := branch) (n := n) (t := t)
(rows := rows) (repeats := repeats) hq large hn hbranch hrows) (sourceSigns clauses)
(fun source outcome => transport ⟨source, outcome⟩)
(SourceQuestionOrder.sourceOrder branch n t m)).length = _
rw [ExactConditionalOutput.output_length _ _ _ D
(SignedCompletionSchedule.count_total (branch := branch) (n := n) (t := t)
(rows := rows) (repeats := repeats) hq large hn hbranch hrows),
SourceQuestionOrder.sourceOrder_length]
theorem completedOrder_nonempty :
completedOrder (t := t) clauses rows repeats hn hbranch hrows δ ≠ [] := by
intro hempty
have h := completedOrder_length (t := t) clauses rows repeats hn hbranch hrows δ
rw [hempty, List.length_nil] at h
exact (Nat.ne_of_gt (Nat.mul_pos
(SignedMultiplicity.denominator_positive branch n t rows repeats hn hbranch hrows _)
(Nat.pow_pos (NeZero.pos m)))) h.symm
private theorem mem_copies {S : Type*} {E : S → Type*}
[Fintype S] [∀ s, Fintype (E s)] (count : ∀ s, E s → Nat)
(s : S) (e : E s) (positive : 0 < count s e) :
e ∈ ExactConditionalOutput.copies count s := by
unfold ExactConditionalOutput.copies
refine List.mem_flatten.mpr ⟨List.replicate (count s e) e, ?_, ?_⟩
· refine List.mem_ofFn.mpr ⟨ExactConditionalOutput.enumeration s e, ?_⟩
simp only [Equiv.symm_apply_apply]
· exact List.mem_replicate.mpr ⟨Nat.ne_of_gt positive, rfl⟩
theorem mem_completedOrder (e : Completed) :
e ∈ completedOrder (t := t) clauses rows repeats hn hbranch hrows δ := by
let descriptor := CompletedSignedLaw.completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ e
apply List.mem_flatMap.mpr
refine ⟨descriptor.1, SourceQuestionOrder.mem_sourceOrder descriptor.1, ?_⟩
apply List.mem_map.mpr
refine ⟨descriptor.2, ?_, ?_⟩
· exact mem_copies (SignedCompletionSchedule.count (branch := branch) (n := n) (t := t)
(rows := rows) (repeats := repeats) hq large hn hbranch hrows)
(sourceSigns clauses descriptor.1) descriptor.2
(SignedCompletionSchedule.count_positive (branch := branch) (n := n) (t := t)
(rows := rows) (repeats := repeats) hq large hn hbranch hrows _ _)
· exact (CompletedSignedLaw.completedEquiv (t := t) clauses rows repeats hn hbranch hrows δ).symm_apply_apply e
def rawOrder : List (PreliminarySampler.Raw clauses branch n t rows repeats) :=
(completedOrder (t := t) clauses rows repeats hn hbranch hrows δ).map Sigma.fst
theorem mem_rawOrder (e : PreliminarySampler.Raw clauses branch n t rows repeats) :
e ∈ rawOrder (t := t) clauses rows repeats hn hbranch hrows δ := by
apply List.mem_map.mpr
refine ⟨⟨e, ⟨0, (C).positive e⟩⟩,
mem_completedOrder (t := t) clauses rows repeats hn hbranch hrows δ _, rfl⟩
theorem rawOrder_length :
(rawOrder (t := t) clauses rows repeats hn hbranch hrows δ).length = D * m ^ locationCount := by
rw [rawOrder, List.length_map,
completedOrder_length (t := t) clauses rows repeats hn hbranch hrows δ]
theorem completedOrder_filter_length (event : Completed → Bool) :
((completedOrder (t := t) clauses rows repeats hn hbranch hrows δ).filter event).length =
∑ source : PreliminarySampler.Questions branch n t m,
((SignedCompletionSchedule.schedule (branch := branch) (n := n) (t := t)
(rows := rows) (repeats := repeats) hq large hn hbranch hrows
(sourceSigns clauses source)).filter
(fun outcome => event (transport ⟨source, outcome⟩))).length := by
rw [completedOrder, List.filter_flatMap, List.length_flatMap]
simp only [localOrder, List.filter_map, List.length_map, Function.comp_def]
exact SourceQuestionOrder.sum_map_sourceOrder _
theorem completedOrder_probability (event : Completed → Bool) :
(((completedOrder (t := t) clauses rows repeats hn hbranch hrows δ).filter event).length : ℝ) /
(completedOrder (t := t) clauses rows repeats hn hbranch hrows δ).length =
(C).game.occurrences.probability event := by
have hcard : Fintype.card (PreliminarySampler.Questions branch n t m) =
m ^ locationCount := by
rw [Fintype.card_congr (SourceQuestionOrder.questionTupleEquiv branch n t m)]
simp only [Fintype.card_fun, Fintype.card_fin]
rw [completedOrder_filter_length (t := t) clauses rows repeats hn hbranch hrows δ event,
Nat.cast_sum, completedOrder_length (t := t) clauses rows repeats hn hbranch hrows δ,
Nat.cast_mul,
CompletedSignedLaw.probability_schedule (t := t) clauses rows repeats hn hbranch hrows δ event]
simp only [FiniteDistribution.expectation, PreliminarySampler.questionsLaw,
FiniteDistribution.uniform, SignedCompletionSchedule.schedule_length]
rw [← Finset.mul_sum, ← Finset.sum_div, hcard]
simp only [div_eq_mul_inv, mul_inv_rev]
ring
end
end PerfectCompleteness.CompletedOccurrenceOrder
end
end OAI
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