File size: 3,479 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 | import OAI.Computability.PerfectCompleteness.Decoding.LowerCutPairLemmas
namespace OAI
section
namespace PerfectCompleteness.CutCallOriginalLaw
open RecursiveSpaces TreeSourceSpaces HierarchicalArrays WholeCutGrouping
open UniqueGamesTheorem.Foundations.Games
open scoped Classical
noncomputable section
section Original
variable {branch : Nat → Nat} {n t : Nat} {C : Type*} [Fintype C]
(slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot)
(rows : Nat → Nat)
def originalLinear : CutCallExtension.Extended (C := C) slots rows →ₗ[ZMod 2]
CutChildGrouping.Raw (C := C) slots rows where
toFun := CutCallExtension.original slots rows
map_add' _ _ := rfl
map_smul' _ _ := rfl
omit [Fintype C] in
theorem originalLinear_surjective : Function.Surjective
(originalLinear (C := C) slots rows) := by
intro old
refine ⟨fun i => (fun c => match c with
| none => 0
| some c => (old i).1 c, (old i).2), ?_⟩
rfl
theorem original_law :
(CutChildGrouping.rawLaw (C := Option C) slots rows).pushforward
(CutCallExtension.original slots rows) =
CutChildGrouping.rawLaw (C := C) slots rows := by
unfold CutChildGrouping.rawLaw
rw [UniformLinearImage.law_uniform, UniformLinearImage.law_uniform]
exact UniformLinearImage.uniform_pushforward_linearMap
(originalLinear slots rows) (originalLinear_surjective slots rows)
theorem original_probability (event : CutChildGrouping.Raw (C := C) slots rows → Bool) :
(CutChildGrouping.rawLaw (C := Option C) slots rows).probability
(fun raw => event (CutCallExtension.original slots rows raw)) =
(CutChildGrouping.rawLaw (C := C) slots rows).probability event := by
rw [← FiniteDistribution.probability_pushforward, original_law]
end Original
section First
variable {branch : Nat → Nat} {n m t : Nat}
(rows repeats : Nat → Nat) (p : DescendantSpaces.Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot)
theorem first_law (exterior : Exterior rows repeats p slots) :
(CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p))
(cutSlots p slots) rows).pushforward
(LowerCutPair.first rows repeats p slots exterior) =
(CutChildGrouping.rawLaw (C := LowerCutPair.Calls rows repeats p)
(cutSlots p slots) rows).pushforward
(LowerCutRows.reconstruct rows repeats p slots exterior) := by
change (CutChildGrouping.rawLaw
(C := Option (LowerCutPair.Calls rows repeats p)) (cutSlots p slots) rows).pushforward
(fun raw => LowerCutRows.reconstruct rows repeats p slots exterior
(CutCallExtension.original (cutSlots p slots) rows raw)) = _
rw [← FiniteDistribution.pushforward_comp, original_law]
theorem first_probability (exterior : Exterior rows repeats p slots)
(event : Arrays slots rows → Bool) :
(CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p))
(cutSlots p slots) rows).probability
(fun raw => event (LowerCutPair.first rows repeats p slots exterior raw)) =
(CutChildGrouping.rawLaw (C := LowerCutPair.Calls rows repeats p)
(cutSlots p slots) rows).probability
(fun raw => event (LowerCutRows.reconstruct rows repeats p slots exterior raw)) := by
exact original_probability (cutSlots p slots) rows
(fun raw => event (LowerCutRows.reconstruct rows repeats p slots exterior raw))
end First
end
end PerfectCompleteness.CutCallOriginalLaw
end
end OAI
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