File size: 5,763 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 | import Mathlib.Logic.Equiv.Prod
import OAI.Computability.PerfectCompleteness.Construction.HiddenBucketBias
import OAI.Computability.PerfectCompleteness.Foundations.TerminalCalls
import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerRefinement
import OAI.Computability.UniqueGames.Foundations.SamplingLemmas
namespace OAI
section
namespace PerfectCompleteness.CutTerminalSplit
open PointwiseSpaces RecursiveSpaces DescendantSpaces RecursiveSampler TerminalCalls
open UniqueGamesTheorem.Foundations.Games
open scoped Classical
noncomputable section
universe u w
variable {branch : Nat → Nat} {n m : Nat}
theorem refinedSpace_eq_terminal (𝕜 : Type w) [Field 𝕜]
(repeats : Nat → Nat) (p : Path branch n m) :
∀ (A : Slots branch n → Type u) (terminal : TerminalIndex repeats p),
CutSamplerRefinement.RefinedSpace 𝕜 repeats p A (drawIndex repeats p terminal) =
CutSamplerRefinement.RefinedSpace 𝕜 repeats (.refl m) (p.family A) () := by
induction p with
| refl n =>
intro A terminal
cases terminal
rfl
| step i p ih =>
intro A terminal
exact ih (childFamily A i) terminal.2
variable (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat)
(p : Path branch n (m + 1)) (A : Slots branch n → Type u)
abbrev TerminalFactor :=
(i : Fin (branch m)) → squareSpace (space 𝕜 branch m (childFamily (p.family A) i))
abbrev TerminalTape := TerminalIndex repeats p → TerminalFactor 𝕜 p A
abbrev ExteriorIndex := {j : DrawIndex repeats p // ¬ IsTerminal repeats p j}
abbrev ExteriorTape :=
(j : ExteriorIndex repeats p) → CutSamplerRefinement.RefinedSpace 𝕜 repeats p A j.val
def terminalFactorEquiv (terminal : TerminalIndex repeats p) :
CutSamplerRefinement.RefinedSpace 𝕜 repeats p A (drawIndex repeats p terminal) ≃
TerminalFactor 𝕜 p A :=
Equiv.cast (refinedSpace_eq_terminal 𝕜 repeats p A terminal)
def terminalTapeEquiv :
((j : {j : DrawIndex repeats p // IsTerminal repeats p j}) →
CutSamplerRefinement.RefinedSpace 𝕜 repeats p A j.val) ≃
TerminalTape 𝕜 repeats p A :=
((Equiv.piCongrLeft
(fun j : {j : DrawIndex repeats p // IsTerminal repeats p j} =>
CutSamplerRefinement.RefinedSpace 𝕜 repeats p A j.val)
(terminalSubtypeEquiv repeats p)).symm).trans
(Equiv.piCongrRight (terminalFactorEquiv 𝕜 repeats p A))
def splitTape :
CutSamplerRefinement.Tape 𝕜 repeats p A ≃
TerminalTape 𝕜 repeats p A × ExteriorTape 𝕜 repeats p A :=
(Equiv.piEquivPiSubtypeProd (IsTerminal repeats p)
(CutSamplerRefinement.RefinedSpace 𝕜 repeats p A)).trans
(Equiv.prodCongr (terminalTapeEquiv 𝕜 repeats p A) (Equiv.refl _))
@[simp] theorem splitTape_terminal_apply
(x : CutSamplerRefinement.Tape 𝕜 repeats p A) (terminal : TerminalIndex repeats p) :
(splitTape 𝕜 repeats p A x).1 terminal =
terminalFactorEquiv 𝕜 repeats p A terminal (x (drawIndex repeats p terminal)) := rfl
@[simp] theorem splitTape_exterior_apply
(x : CutSamplerRefinement.Tape 𝕜 repeats p A) (j : ExteriorIndex repeats p) :
(splitTape 𝕜 repeats p A x).2 j = x j.val := rfl
variable [Finite 𝕜] [∀ s, Finite (A s)]
instance childSquareFintype (i : Fin (branch m)) :
Fintype (squareSpace (space 𝕜 branch m (childFamily (p.family A) i))) :=
Fintype.ofFinite _
def terminalLaw : FiniteDistribution (TerminalTape 𝕜 repeats p A) :=
FiniteProduct.law (fun _ : TerminalIndex repeats p =>
FiniteProduct.law (fun i : Fin (branch m) =>
FiniteDistribution.uniform
(squareSpace (space 𝕜 branch m (childFamily (p.family A) i)))))
def exteriorLaw : FiniteDistribution (ExteriorTape 𝕜 repeats p A) :=
FiniteProduct.law (fun j : ExteriorIndex repeats p =>
FiniteDistribution.uniform (CutSamplerRefinement.RefinedSpace 𝕜 repeats p A j.val))
theorem terminalLaw_uniform :
terminalLaw 𝕜 repeats p A = FiniteDistribution.uniform (TerminalTape 𝕜 repeats p A) := by
simp only [terminalLaw, UniformLinearImage.law_uniform]
theorem exteriorLaw_uniform :
exteriorLaw 𝕜 repeats p A = FiniteDistribution.uniform (ExteriorTape 𝕜 repeats p A) :=
UniformLinearImage.law_uniform
theorem split_tapeLaw :
(CutSamplerRefinement.tapeLaw 𝕜 repeats p A).pushforward (splitTape 𝕜 repeats p A) =
(terminalLaw 𝕜 repeats p A).product (exteriorLaw 𝕜 repeats p A) := by
unfold CutSamplerRefinement.tapeLaw
rw [UniformLinearImage.law_uniform, FiniteDistribution.pushforward_equiv,
terminalLaw_uniform, exteriorLaw_uniform]
apply FiniteDistribution.eq_of_weight_eq
intro x
change 1 / (Fintype.card (CutSamplerRefinement.Tape 𝕜 repeats p A) : ℝ) =
(1 / (Fintype.card (TerminalTape 𝕜 repeats p A) : ℝ)) *
(1 / (Fintype.card (ExteriorTape 𝕜 repeats p A) : ℝ))
rw [Fintype.card_congr (splitTape 𝕜 repeats p A), Fintype.card_prod]
simp only [Nat.cast_mul, one_div, mul_inv]
theorem terminal_marginal :
(CutSamplerRefinement.tapeLaw 𝕜 repeats p A).pushforward
(fun x => (splitTape 𝕜 repeats p A x).1) = terminalLaw 𝕜 repeats p A := by
rw [← FiniteDistribution.pushforward_comp (CutSamplerRefinement.tapeLaw 𝕜 repeats p A)
(splitTape 𝕜 repeats p A) Prod.fst, split_tapeLaw]
exact HiddenBucketBias.product_fst
(terminalLaw 𝕜 repeats p A) (exteriorLaw 𝕜 repeats p A)
theorem terminal_marginal_uniform :
(CutSamplerRefinement.tapeLaw 𝕜 repeats p A).pushforward
(fun x => (splitTape 𝕜 repeats p A x).1) =
FiniteDistribution.uniform (TerminalTape 𝕜 repeats p A) := by
rw [terminal_marginal, terminalLaw_uniform]
end
end PerfectCompleteness.CutTerminalSplit
end
end OAI
|