File size: 7,974 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 | import OAI.Computability.PerfectCompleteness.Construction.ChildCallNumbering
import OAI.Computability.PerfectCompleteness.Foundations.DependentPredictionDifferenceLemmas
import OAI.Computability.PerfectCompleteness.Foundations.MultiplicationFormInjectiveLemmas
import OAI.Computability.PerfectCompleteness.Sampling.RationalFiniteLawLemmas
namespace OAI
section
namespace PerfectCompleteness.ChildCallNumberingLaws
noncomputable section
open scoped BigOperators Classical
open RecursiveSpaces TreeSourceSpaces
open UniqueGamesTheorem.Foundations.Games
variable {branch : Nat → Nat} {n t : Nat} {C : Type*} [Fintype C]
def liftedLaw (rows : Nat → Nat)
{slots projected : Slots branch n → Fin t → MixedSupport.Slot}
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) :
FiniteDistribution (ChildCallNumbering.Raw (C := C) rows slots) :=
(FiniteDistribution.uniform (ChildCallNumbering.Raw (C := C) rows projected)).pushforward
(ChildCallNumbering.pullback rows p)
theorem liftedLaw_numbered (rows : Nat → Nat)
{slots projected : Slots branch n → Fin t → MixedSupport.Slot}
(p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) :
(liftedLaw (C := C) rows p).pushforward (ChildCallNumbering.equiv rows slots) =
ChildBlockProjection.liftedLaw (Fintype.card C) rows p := by
unfold liftedLaw ChildBlockProjection.liftedLaw
rw [← ChildCallNumbering.uniform_equiv (C := C) rows projected]
rw [FiniteDistribution.pushforward_comp, FiniteDistribution.pushforward_comp]
congr 1
def mixtureLaw {Z : Type*} [Fintype Z] (rows : Nat → Nat)
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(projected : Z → Slots branch n → Fin t → MixedSupport.Slot)
(p : ∀ z s k, MixedSupport.Projection (slots s k) (projected z s k))
(choiceLaw : FiniteDistribution Z) :
FiniteDistribution (ChildCallNumbering.Raw (C := C) rows slots) :=
choiceLaw.mixture (fun z => liftedLaw rows (p z))
theorem mixtureLaw_numbered {Z : Type*} [Fintype Z] (rows : Nat → Nat)
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(projected : Z → Slots branch n → Fin t → MixedSupport.Slot)
(p : ∀ z s k, MixedSupport.Projection (slots s k) (projected z s k))
(choiceLaw : FiniteDistribution Z) :
(mixtureLaw (C := C) rows slots projected p choiceLaw).pushforward
(ChildCallNumbering.equiv rows slots) =
ChildBlockProjection.mixtureLaw (Fintype.card C) rows slots projected p choiceLaw := by
simp only [mixtureLaw, FiniteDistribution.pushforward_mixture,
liftedLaw_numbered, ChildBlockProjection.mixtureLaw]
variable {I : Type*} [Fintype I] [DecidableEq I]
{Z : I → Type*} [∀ i, Fintype (Z i)]
def replacementLaws (rows : Nat → Nat)
(slots : I → Slots branch n → Fin t → MixedSupport.Slot)
(projected : (i : I) → Z i → Slots branch n → Fin t → MixedSupport.Slot)
(p : ∀ i z s k, MixedSupport.Projection (slots i s k) (projected i z s k))
(choiceLaw : (i : I) → FiniteDistribution (Z i)) :
(i : I) → FiniteDistribution (ChildCallNumbering.Raw (C := C) rows (slots i)) :=
fun i => mixtureLaw rows (slots i) (projected i) (p i) (choiceLaw i)
def number (rows : Nat → Nat)
(slots : I → Slots branch n → Fin t → MixedSupport.Slot)
(x : (i : I) → ChildCallNumbering.Raw (C := C) rows (slots i)) :
(i : I) → ChildBlockCardinality.Raw (Fintype.card C) rows (slots i) :=
fun i => ChildCallNumbering.equiv rows (slots i) (x i)
def unnumber (rows : Nat → Nat)
(slots : I → Slots branch n → Fin t → MixedSupport.Slot)
(x : (i : I) → ChildBlockCardinality.Raw (Fintype.card C) rows (slots i)) :
(i : I) → ChildCallNumbering.Raw (C := C) rows (slots i) :=
fun i => (ChildCallNumbering.equiv rows (slots i)).symm (x i)
omit [Fintype I] [DecidableEq I] in
@[simp] theorem unnumber_number (rows : Nat → Nat)
(slots : I → Slots branch n → Fin t → MixedSupport.Slot)
(x : (i : I) → ChildCallNumbering.Raw (C := C) rows (slots i)) :
unnumber rows slots (number rows slots x) = x := by
funext i
exact (ChildCallNumbering.equiv rows (slots i)).symm_apply_apply (x i)
omit [Fintype I] [DecidableEq I] in
@[simp] theorem number_unnumber (rows : Nat → Nat)
(slots : I → Slots branch n → Fin t → MixedSupport.Slot)
(x : (i : I) → ChildBlockCardinality.Raw (Fintype.card C) rows (slots i)) :
number rows slots (unnumber (C := C) rows slots x) = x := by
funext i
exact (ChildCallNumbering.equiv rows (slots i)).apply_symm_apply (x i)
theorem uniform_product_numbered (rows : Nat → Nat)
(slots : I → Slots branch n → Fin t → MixedSupport.Slot) :
(FiniteProduct.law (fun i => FiniteDistribution.uniform
(ChildCallNumbering.Raw (C := C) rows (slots i)))).pushforward (number rows slots) =
FiniteProduct.law (fun i => FiniteDistribution.uniform
(ChildBlockCardinality.Raw (Fintype.card C) rows (slots i))) := by
unfold number
rw [FiniteProduct.pushforward_map]
congr 1
funext i
exact ChildCallNumbering.uniform_equiv rows (slots i)
theorem conditional_product_numbered (rows : Nat → Nat)
(slots : I → Slots branch n → Fin t → MixedSupport.Slot)
(projected : (i : I) → Z i → Slots branch n → Fin t → MixedSupport.Slot)
(p : ∀ i z s k, MixedSupport.Projection (slots i s k) (projected i z s k))
(choiceLaw : (i : I) → FiniteDistribution (Z i)) (mask : I → Bool) :
(FiniteProduct.law (fun i => if mask i then
replacementLaws (C := C) rows slots projected p choiceLaw i else
FiniteDistribution.uniform (ChildCallNumbering.Raw (C := C) rows (slots i)))).pushforward
(number rows slots) =
FiniteProduct.law (fun i => if mask i then
ChildBlockProjection.replacementLaws (Fintype.card C) rows slots projected p choiceLaw i else
FiniteDistribution.uniform (ChildBlockCardinality.Raw (Fintype.card C) rows (slots i))) := by
unfold number
rw [FiniteProduct.pushforward_map]
congr 1
funext i
cases mask i with
| false =>
simp only [Bool.false_eq_true, ite_false]
exact ChildCallNumbering.uniform_equiv rows (slots i)
| true =>
simp only [ite_true]
exact mixtureLaw_numbered rows (slots i) (projected i) (p i) (choiceLaw i)
theorem markedLaw_numbered (rows : Nat → Nat)
(slots : I → Slots branch n → Fin t → MixedSupport.Slot)
(projected : (i : I) → Z i → Slots branch n → Fin t → MixedSupport.Slot)
(p : ∀ i z s k, MixedSupport.Projection (slots i s k) (projected i z s k))
(choiceLaw : (i : I) → FiniteDistribution (Z i))
(β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1) :
(SparseReplacement.markedLaw
(fun i => FiniteDistribution.uniform (ChildCallNumbering.Raw (C := C) rows (slots i)))
(replacementLaws rows slots projected p choiceLaw) β hβ hβ').pushforward
(fun z => (z.1, number rows slots z.2)) =
SparseReplacement.markedLaw
(fun i => FiniteDistribution.uniform
(ChildBlockCardinality.Raw (Fintype.card C) rows (slots i)))
(ChildBlockProjection.replacementLaws (Fintype.card C) rows slots projected p choiceLaw)
β hβ hβ' := by
apply SigmaObservation.eq_of_probability_eq
intro event
rw [FiniteDistribution.probability_pushforward]
unfold SparseReplacement.markedLaw
rw [CleanConditioning.probability_kernelJoint, CleanConditioning.probability_kernelJoint]
apply Finset.sum_congr rfl
intro mask _
apply congrArg (fun x : ℝ =>
(FiniteProduct.law (fun _ : I => ProjectionPosterior.bernoulli β hβ hβ')).weight mask * x)
have h := congrArg (fun μ => μ.probability (fun x => event (mask, x)))
(conditional_product_numbered (C := C) rows slots projected p choiceLaw mask)
simpa only [FiniteDistribution.probability_pushforward] using h
end
end PerfectCompleteness.ChildCallNumberingLaws
end
end OAI
|