File size: 2,547 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 | import OAI.Computability.PerfectCompleteness.Foundations.WholeCutGrouping
namespace OAI
section
namespace PerfectCompleteness.CutCallExtension
open scoped Classical
open TreeSourceSpaces HierarchicalArrays PointwiseSpaces
open UniqueGamesTheorem.Foundations.Games
noncomputable section
variable {branch : Nat → Nat} {n t : Nat} {C : Type*} [Fintype C]
(slots : RecursiveSpaces.Slots branch (n + 1) → Fin t → MixedSupport.Slot)
(rows : Nat → Nat)
abbrev Extended := CutChildGrouping.Raw (C := Option C) slots rows
def original (x : Extended (C := C) slots rows) :
CutChildGrouping.Raw (C := C) slots rows :=
fun i => (fun c => (x i).1 (some c), (x i).2)
def fresh (x : Extended (C := C) slots rows) : H slots :=
UniformChildSum.recursiveSum (LeafDomain slots) (fun i => (x i).1 none)
def observe (x : Extended (C := C) slots rows) :
CutChildGrouping.Assembled (C := C) slots rows × H slots :=
(CutChildGrouping.assemble slots rows (original slots rows x), fresh slots rows x)
def splitAssembled : CutChildGrouping.Assembled (C := Option C) slots rows →ₗ[F2]
CutChildGrouping.Assembled (C := C) slots rows × H slots where
toFun x := ((fun c => x.1 (some c), x.2), x.1 none)
map_add' _ _ := rfl
map_smul' _ _ := rfl
omit [Fintype C] in
theorem splitAssembled_surjective : Function.Surjective
(splitAssembled (C := C) slots rows) := by
rintro ⟨⟨old, arrays⟩, extra⟩
refine ⟨(fun c => match c with | none => extra | some c => old c, arrays), ?_⟩
rfl
omit [Fintype C] in
@[simp] theorem split_assemble (x : Extended (C := C) slots rows) :
splitAssembled slots rows (CutChildGrouping.assemble slots rows x) =
observe slots rows x := rfl
theorem observe_law :
(CutChildGrouping.rawLaw (C := Option C) slots rows).pushforward
(observe slots rows) =
(FiniteDistribution.uniform (CutChildGrouping.Assembled (C := C) slots rows)).product
(FiniteDistribution.uniform (H slots)) := by
change (CutChildGrouping.rawLaw (C := Option C) slots rows).pushforward
(fun x => splitAssembled slots rows (CutChildGrouping.assemble slots rows x)) = _
rw [← FiniteDistribution.pushforward_comp, CutChildGrouping.assemble_law,
UniformLinearImage.uniform_pushforward_linearMap
(splitAssembled slots rows) (splitAssembled_surjective slots rows)]
exact WholeCutSampler.uniform_product.symm
theorem call_count : Fintype.card (Option C) = Fintype.card C + 1 :=
Fintype.card_option
end
end PerfectCompleteness.CutCallExtension
end
end OAI
|