File size: 8,603 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 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 | import OAI.Computability.PerfectCompleteness.Machines.MetadataFreeTapeLemmas
import OAI.Computability.PerfectCompleteness.Sampling.SamplerCardinalityLemmas
namespace OAI
section
namespace PerfectCompleteness.MetadataFreeLaw
open RecursiveSpaces DescendantSpaces TreeSourceSpaces
open MetadataFreeSampler MetadataFreeTape
open UniqueGamesTheorem.Foundations.Games
open scoped BigOperators Classical
noncomputable section
variable {branch : Nat → Nat} {n d t v m : Nat}
theorem tape_weight_eq (clauses : Fin m → SourceClause.NormalizedClause v)
(rows repeats : Nat → Nat) (p : Path branch n d) :
∀ (q : PreliminarySampler.Questions branch n t m)
(ω : WholeArraySampler.Tape rows repeats p (sourceSlots clauses q)),
(WholeArraySampler.tapeLaw rows repeats p (sourceSlots clauses q)).weight ω =
(WholeArraySampler.tapeLaw rows repeats p (dummyTreeSlots (sourceSigns clauses q))).weight
(tapeEquiv clauses rows repeats p q ω) := by
induction p with
| refl n =>
intro q ω
rfl
| @step n d i p ih =>
intro q ω
change (∏ k : WholeArraySampler.StepIndex i,
(WholeArraySampler.componentLaw rows repeats i p (sourceSlots clauses q) k).weight (ω k)) =
∏ k : WholeArraySampler.StepIndex i,
(WholeArraySampler.componentLaw rows repeats i p
(dummyTreeSlots (sourceSigns clauses q)) k).weight
(tapeEquiv clauses rows repeats (.step i p) q ω k)
apply Finset.prod_congr rfl
intro k _
rcases k with u | (u | j)
· rfl
· cases u
exact ih (fun leaf k => q (i, leaf) k) (ω (.inr (.inl ())))
· rfl
theorem tapeLaw_transport (clauses : Fin m → SourceClause.NormalizedClause v)
(rows repeats : Nat → Nat) (p : Path branch n d)
(q : PreliminarySampler.Questions branch n t m) :
(WholeArraySampler.tapeLaw rows repeats p (sourceSlots clauses q)).transport
(tapeEquiv clauses rows repeats p q) =
WholeArraySampler.tapeLaw rows repeats p (dummyTreeSlots (sourceSigns clauses q)) := by
apply FiniteDistribution.eq_of_weight_eq
intro ω
change (WholeArraySampler.tapeLaw rows repeats p (sourceSlots clauses q)).weight
((tapeEquiv clauses rows repeats p q).symm ω) = _
simpa only [Equiv.apply_symm_apply] using
tape_weight_eq clauses rows repeats p q ((tapeEquiv clauses rows repeats p q).symm ω)
abbrev SignedOutcome (signs : SignTuple branch n t) (rows repeats : Nat → Nat) :=
Σ leaf : Slots branch n,
WholeArraySampler.Tape rows repeats (GeometricPath.leafPath leaf) (dummyTreeSlots signs) ×
PreliminarySampler.Choice rows leaf
def signedLaw (signs : SignTuple branch n t) (rows repeats : Nat → Nat)
(hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k)
(hrows : ∀ k, 0 < rows (k + 1)) : FiniteDistribution (SignedOutcome signs rows repeats) :=
CompletionSoundness.sigmaLaw (GeometricPath.law n hbranch) fun leaf =>
(WholeArraySampler.tapeLaw rows repeats (GeometricPath.leafPath leaf) (dummyTreeSlots signs)).product
(PreliminarySampler.choiceLaw leaf hn hrows)
def signedRationalLaw (signs : SignTuple branch n t) (rows repeats : Nat → Nat)
(hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1)) :
RationalFiniteLaw.RationalLaw (signedLaw signs rows repeats hn hbranch hrows) :=
RationalFiniteLaw.sigma (SamplerRationality.pathLaw hbranch) fun leaf =>
RationalFiniteLaw.product
(SamplerRationality.wholeArrayTapeLaw rows repeats (GeometricPath.leafPath leaf)
(dummyTreeSlots signs))
(SamplerRationality.choiceLaw leaf hn hrows)
theorem signedLaw_positive (signs : SignTuple branch n t) (rows repeats : Nat → Nat)
(hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1))
(e : SignedOutcome signs rows repeats) :
0 < (signedLaw signs rows repeats hn hbranch hrows).weight e :=
RationalFiniteLaw.real_positive (signedRationalLaw signs rows repeats hn hbranch hrows) e
def encodeOutcome (clauses : Fin m → SourceClause.NormalizedClause v) (rows repeats : Nat → Nat)
(e : PreliminarySampler.Raw clauses branch n t rows repeats) :
SignedOutcome (sourceSigns clauses e.1.1) rows repeats :=
⟨e.1.2, tapeEquiv clauses rows repeats (GeometricPath.leafPath e.1.2) e.1.1 e.2.1, e.2.2⟩
theorem raw_weight_eq [NeZero m] (clauses : Fin m → SourceClause.NormalizedClause v)
(rows repeats : Nat → Nat) (hn : 0 < n)
(hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1))
(e : PreliminarySampler.Raw clauses branch n t rows repeats) :
(PreliminarySampler.law clauses rows repeats hn hbranch hrows).weight e =
(1 / (Fintype.card (PreliminarySampler.Questions branch n t m) : ℝ)) *
(signedLaw (sourceSigns clauses e.1.1) rows repeats hn hbranch hrows).weight
(encodeOutcome clauses rows repeats e) := by
change ((1 / (Fintype.card (PreliminarySampler.Questions branch n t m) : ℝ)) *
(GeometricPath.law n hbranch).weight e.1.2) *
((WholeArraySampler.tapeLaw rows repeats (GeometricPath.leafPath e.1.2)
(sourceSlots clauses e.1.1)).weight e.2.1 *
(PreliminarySampler.choiceLaw e.1.2 hn hrows).weight e.2.2) =
(1 / (Fintype.card (PreliminarySampler.Questions branch n t m) : ℝ)) *
((GeometricPath.law n hbranch).weight e.1.2 *
((WholeArraySampler.tapeLaw rows repeats (GeometricPath.leafPath e.1.2)
(dummyTreeSlots (sourceSigns clauses e.1.1))).weight
(tapeEquiv clauses rows repeats (GeometricPath.leafPath e.1.2) e.1.1 e.2.1) *
(PreliminarySampler.choiceLaw e.1.2 hn hrows).weight e.2.2))
rw [tape_weight_eq clauses rows repeats (GeometricPath.leafPath e.1.2) e.1.1 e.2.1]
exact mul_assoc _ _ _
theorem raw_weight_rational [NeZero m] (clauses : Fin m → SourceClause.NormalizedClause v)
(rows repeats : Nat → Nat) (hn : 0 < n)
(hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1))
(e : PreliminarySampler.Raw clauses branch n t rows repeats) :
(PreliminarySampler.law clauses rows repeats hn hbranch hrows).weight e =
(1 / (Fintype.card (PreliminarySampler.Questions branch n t m) : ℝ)) *
((signedRationalLaw (sourceSigns clauses e.1.1) rows repeats hn hbranch hrows).weights
(encodeOutcome clauses rows repeats e) : ℝ) := by
rw [raw_weight_eq, (signedRationalLaw (sourceSigns clauses e.1.1)
rows repeats hn hbranch hrows).cast_weight]
def rawEquiv (clauses : Fin m → SourceClause.NormalizedClause v) (rows repeats : Nat → Nat) :
PreliminarySampler.Raw clauses branch n t rows repeats ≃
(Σ q : PreliminarySampler.Questions branch n t m,
SignedOutcome (sourceSigns clauses q) rows repeats) where
toFun e := ⟨e.1.1, encodeOutcome clauses rows repeats e⟩
invFun y := ⟨(y.1, y.2.1),
(tapeEquiv clauses rows repeats (GeometricPath.leafPath y.2.1) y.1).symm y.2.2.1,
y.2.2.2⟩
left_inv := by
rintro ⟨⟨q, leaf⟩, ω, choice⟩
simp only [encodeOutcome]
exact congrArg
(fun tape : WholeArraySampler.Tape rows repeats (GeometricPath.leafPath leaf)
(sourceSlots clauses q) =>
(⟨(q, leaf), tape, choice⟩ : PreliminarySampler.Raw clauses branch n t rows repeats))
((tapeEquiv clauses rows repeats (GeometricPath.leafPath leaf) q).symm_apply_apply ω)
right_inv := by
rintro ⟨q, leaf, ω, choice⟩
simp only [encodeOutcome, Equiv.apply_symm_apply]
theorem rawLaw_transport [NeZero m] (clauses : Fin m → SourceClause.NormalizedClause v)
(rows repeats : Nat → Nat) (hn : 0 < n)
(hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1)) :
(PreliminarySampler.law (t := t) clauses rows repeats hn hbranch hrows).transport
(rawEquiv clauses rows repeats) =
CompletionSoundness.sigmaLaw PreliminarySampler.questionsLaw
(fun q => signedLaw (sourceSigns clauses q) rows repeats hn hbranch hrows) := by
apply FiniteDistribution.eq_of_weight_eq
rintro ⟨q, leaf, ω, choice⟩
change (PreliminarySampler.law clauses rows repeats hn hbranch hrows).weight
⟨(q, leaf), (tapeEquiv clauses rows repeats (GeometricPath.leafPath leaf) q).symm ω,
choice⟩ =
(1 / (Fintype.card (PreliminarySampler.Questions branch n t m) : ℝ)) *
(signedLaw (sourceSigns clauses q) rows repeats hn hbranch hrows).weight ⟨leaf, ω, choice⟩
rw [raw_weight_eq]
simp only [encodeOutcome, Equiv.apply_symm_apply]
end
end PerfectCompleteness.MetadataFreeLaw
end
end OAI
|