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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 | import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerReplayTransportLemmas
import OAI.Computability.PerfectCompleteness.Sampling.SamplerComposition
namespace OAI
section
namespace PerfectCompleteness.SamplerComposition
open PointwiseSpaces RecursiveSpaces DescendantSpaces RecursiveSampler
open UniqueGamesTheorem.Foundations.Games
noncomputable section
universe u w
variable {branch : Nat → Nat} {n m k : Nat}
variable (𝕜 : Type w) [Field 𝕜]
instance expansionSpaceNonempty (repeats : Nat → Nat) (p : Path branch n m)
(q : Path branch m k) (A : Slots branch n → Type u) (j : DrawIndex repeats p) :
Nonempty (ExpansionSpace 𝕜 repeats p q A j) :=
⟨zeroFactor 𝕜 repeats p q A j⟩
instance expandedTapeNonempty (repeats : Nat → Nat) (p : Path branch n m)
(q : Path branch m k) (A : Slots branch n → Type u) :
Nonempty (ExpandedTape 𝕜 repeats p q A) :=
⟨zeroExpandedTape 𝕜 repeats p q A⟩
variable [Finite 𝕜]
theorem expansionSpace_finite (repeats : Nat → Nat) (p : Path branch n m) :
∀ (q : Path branch m k) (A : Slots branch n → Type u),
(∀ s, Finite (A s)) → ∀ j : DrawIndex repeats p,
Finite (ExpansionSpace 𝕜 repeats p q A j) := by
induction p with
| refl n =>
intro q A hA j
let : ∀ s, Finite (A s) := hA
let : ∀ d : DrawIndex repeats q, Fintype (DrawSpace 𝕜 repeats q A d) :=
fun d => drawSpaceFintype 𝕜 repeats q A d
change Finite (RecursiveSampler.Tape 𝕜 repeats q A)
infer_instance
| @step n m i p ih =>
intro q A hA j
cases j with
| inl j =>
let : ∀ s, Finite (childFamily A j.val s) := fun s => hA (j.val, s)
change Finite (squareSpace (space 𝕜 branch n (childFamily A j.val)))
infer_instance
| inr j =>
exact ih q (childFamily A i) (fun s => hA (i, s)) j.2
instance expansionSpaceFinite (repeats : Nat → Nat) (p : Path branch n m)
(q : Path branch m k) (A : Slots branch n → Type u) [∀ s, Finite (A s)]
(j : DrawIndex repeats p) : Finite (ExpansionSpace 𝕜 repeats p q A j) :=
expansionSpace_finite 𝕜 repeats p q A (fun _ => inferInstance) j
instance expansionSpaceFintype (repeats : Nat → Nat) (p : Path branch n m)
(q : Path branch m k) (A : Slots branch n → Type u) [∀ s, Finite (A s)]
(j : DrawIndex repeats p) : Fintype (ExpansionSpace 𝕜 repeats p q A j) :=
Fintype.ofFinite _
instance expandedTapeFintype (repeats : Nat → Nat) (p : Path branch n m)
(q : Path branch m k) (A : Slots branch n → Type u) [∀ s, Finite (A s)] :
Fintype (ExpandedTape 𝕜 repeats p q A) :=
@Pi.instFintype (DrawIndex repeats p) (ExpansionSpace 𝕜 repeats p q A)
(drawIndexDecidableEq repeats p) (drawIndexFintype repeats p)
(fun j => expansionSpaceFintype 𝕜 repeats p q A j)
def expandedTapeLaw (repeats : Nat → Nat) (p : Path branch n m)
(q : Path branch m k) (A : Slots branch n → Type u) [∀ s, Finite (A s)] :
FiniteDistribution (ExpandedTape 𝕜 repeats p q A) := by
letI : ∀ j : DrawIndex repeats p, Fintype (ExpansionSpace 𝕜 repeats p q A j) :=
fun j => expansionSpaceFintype 𝕜 repeats p q A j
letI : ∀ j : DrawIndex repeats p, Nonempty (ExpansionSpace 𝕜 repeats p q A j) :=
fun j => expansionSpaceNonempty 𝕜 repeats p q A j
exact FiniteProduct.law (fun j => FiniteDistribution.uniform (ExpansionSpace 𝕜 repeats p q A j))
theorem expandedTapeLaw_eq_uniform (repeats : Nat → Nat) (p : Path branch n m)
(q : Path branch m k) (A : Slots branch n → Type u) [∀ s, Finite (A s)] :
expandedTapeLaw 𝕜 repeats p q A = FiniteDistribution.uniform (ExpandedTape 𝕜 repeats p q A) := by
let : ∀ j : DrawIndex repeats p, Fintype (ExpansionSpace 𝕜 repeats p q A j) :=
fun j => expansionSpaceFintype 𝕜 repeats p q A j
let : ∀ j : DrawIndex repeats p, Nonempty (ExpansionSpace 𝕜 repeats p q A j) :=
fun j => expansionSpaceNonempty 𝕜 repeats p q A j
exact UniformLinearImage.law_uniform
theorem expand_tapeLaw (repeats : Nat → Nat) (p : Path branch n m)
(q : Path branch m k) (A : Slots branch n → Type u) [∀ s, Finite (A s)] :
(tapeLaw 𝕜 repeats (p.append q) A).pushforward (expandEquiv 𝕜 repeats p q A) =
expandedTapeLaw 𝕜 repeats p q A := by
rw [tapeLaw_eq_uniform, expandedTapeLaw_eq_uniform,
← FiniteDistribution.transport_eq_pushforward]
exact UniformConditioning.uniform_transport (expandEquiv 𝕜 repeats p q A)
def factorLaw (repeats : Nat → Nat) (p : Path branch n m)
(q : Path branch m k) (A : Slots branch n → Type u) [∀ s, Finite (A s)]
(j : DrawIndex repeats p) : FiniteDistribution (DrawSpace 𝕜 repeats p A j) :=
(FiniteDistribution.uniform (ExpansionSpace 𝕜 repeats p q A j)).pushforward
(collapseFactor 𝕜 repeats p q A j)
theorem collapseExpanded_expandedTapeLaw (repeats : Nat → Nat) (p : Path branch n m)
(q : Path branch m k) (A : Slots branch n → Type u) [∀ s, Finite (A s)] :
(expandedTapeLaw 𝕜 repeats p q A).pushforward (collapseExpanded 𝕜 repeats p q A) =
FiniteProduct.law (factorLaw 𝕜 repeats p q A) := by
let : ∀ j : DrawIndex repeats p, Fintype (ExpansionSpace 𝕜 repeats p q A j) :=
fun j => expansionSpaceFintype 𝕜 repeats p q A j
let : ∀ j : DrawIndex repeats p, Nonempty (ExpansionSpace 𝕜 repeats p q A j) :=
fun j => expansionSpaceNonempty 𝕜 repeats p q A j
exact FiniteProduct.pushforward_map _ (collapseFactor 𝕜 repeats p q A)
theorem collapseSuffix_tapeLaw (repeats : Nat → Nat) (p : Path branch n m)
(q : Path branch m k) (A : Slots branch n → Type u) [∀ s, Finite (A s)] :
(tapeLaw 𝕜 repeats (p.append q) A).pushforward (collapseSuffix 𝕜 repeats p q A) =
FiniteProduct.law (factorLaw 𝕜 repeats p q A) := by
calc
_ = ((tapeLaw 𝕜 repeats (p.append q) A).pushforward
(expandEquiv 𝕜 repeats p q A)).pushforward (collapseExpanded 𝕜 repeats p q A) :=
(FiniteDistribution.pushforward_comp _ _ _).symm
_ = _ := by rw [expand_tapeLaw, collapseExpanded_expandedTapeLaw]
theorem law_append (repeats : Nat → Nat) (p : Path branch n m)
(q : Path branch m k) (A : Slots branch n → Type u) [∀ s, Finite (A s)]
[Fintype (space 𝕜 branch n A)] :
RecursiveSampler.law 𝕜 repeats (p.append q) A =
(FiniteProduct.law (factorLaw 𝕜 repeats p q A)).pushforward
(RecursiveSampler.evaluate 𝕜 repeats p A) := by
calc
_ = ((tapeLaw 𝕜 repeats (p.append q) A).pushforward
(collapseSuffix 𝕜 repeats p q A)).pushforward
(RecursiveSampler.evaluate 𝕜 repeats p A) := by
rw [FiniteDistribution.pushforward_comp]
unfold RecursiveSampler.law
congr 1
funext t
exact (evaluate_collapseSuffix 𝕜 repeats p q A t).symm
_ = _ := by rw [collapseSuffix_tapeLaw]
end
end PerfectCompleteness.SamplerComposition
end
end OAI
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