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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 | import OAI.Computability.PerfectCompleteness.Sampling.SourceChildQuestionLaw
namespace OAI
section
namespace PerfectCompleteness.SourceQuestionPositionSplit
noncomputable section
open scoped Classical
open RecursiveSpaces SourceQuestionReconstruction TreeSourceSpaces
open UniqueGamesTheorem.Foundations.Games
variable {branch : Nat → Nat} {n t m : Nat}
abbrev Questions := PreliminarySampler.Questions branch (n + 1) t m
abbrev Positions := Slots branch (n + 1) → Fin t → Fin 3
abbrev ChildPositions := Slots branch n → Fin t → Fin 3
def splitTree : (Slots branch (n + 1) → SourceTuple m t) ≃
Questions (branch := branch) (n := n) (t := t) (m := m) ×
Positions (branch := branch) (n := n) (t := t) where
toFun tree := (fun leaf k => (tree leaf k).1, fun leaf k => (tree leaf k).2)
invFun parts leaf k := (parts.1 leaf k, parts.2 leaf k)
left_inv _ := rfl
right_inv _ := rfl
def split (designated : Fin (branch n) → Slots branch n) :
SourceChildKernel.Sources (m := m) (t := t) designated ≃
Questions (branch := branch) (n := n) (t := t) (m := m) ×
Positions (branch := branch) (n := n) (t := t) :=
(SourceChildQuestionLaw.assembleEquiv (m := m) (t := t) designated).trans splitTree
def join (designated : Fin (branch n) → Slots branch n)
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(positions : Positions (branch := branch) (n := n) (t := t)) :
SourceChildKernel.Sources (m := m) (t := t) designated :=
(split designated).symm (q, positions)
@[simp] theorem questions_join (designated : Fin (branch n) → Slots branch n)
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(positions : Positions (branch := branch) (n := n) (t := t)) :
SourceChildQuestionLaw.questions designated (join designated q positions) = q :=
congrArg Prod.fst ((split designated).apply_symm_apply (q, positions))
theorem tree_join (designated : Fin (branch n) → Slots branch n)
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(positions : Positions (branch := branch) (n := n) (t := t)) :
SourceChildQuestionLaw.assembleEquiv designated (join designated q positions) =
fun leaf k => (q leaf k, positions leaf k) :=
(SourceChildQuestionLaw.assembleEquiv (m := m) (t := t) designated).apply_symm_apply _
theorem child_tree_join (designated : Fin (branch n) → Slots branch n)
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(positions : Positions (branch := branch) (n := n) (t := t))
(i : Fin (branch n)) (leaf : Slots branch n) (k : Fin t) :
SourceChildKernel.tree designated i (join designated q positions i) leaf k =
(q (i, leaf) k, positions (i, leaf) k) :=
congrFun (congrFun (tree_join designated q positions) (i, leaf)) k
theorem sourceLaw_split [NeZero m] (designated : Fin (branch n) → Slots branch n) :
(SourceChildQuestionLaw.sourceLaw m t designated).pushforward (split designated) =
(PreliminarySampler.questionsLaw (branch := branch) (n := n + 1) (t := t) (m := m)).product
(FiniteDistribution.uniform (Positions (branch := branch) (n := n) (t := t))) := by
rw [SourceChildQuestionLaw.sourceLaw_uniform, FiniteDistribution.pushforward_equiv]
apply FiniteDistribution.eq_of_weight_eq
intro x
change 1 / (Fintype.card (SourceChildKernel.Sources (m := m) (t := t) designated) : ℝ) =
(1 / (Fintype.card (Questions (branch := branch) (n := n) (t := t) (m := m)) : ℝ)) *
(1 / (Fintype.card (Positions (branch := branch) (n := n) (t := t)) : ℝ))
rw [Fintype.card_congr (split (m := m) (t := t) designated), Fintype.card_prod,
Nat.cast_mul, one_div_mul_one_div]
def byChild : Positions (branch := branch) (n := n) (t := t) ≃
(Fin (branch n) → ChildPositions (branch := branch) (n := n) (t := t)) where
toFun positions i leaf k := positions (i, leaf) k
invFun positions leaf k := positions leaf.1 leaf.2 k
left_inv _ := rfl
right_inv _ := rfl
theorem positions_byChild :
(FiniteDistribution.uniform (Positions (branch := branch) (n := n) (t := t))).pushforward
(byChild (branch := branch) (n := n) (t := t)) =
FiniteProduct.law (fun _ : Fin (branch n) =>
FiniteDistribution.uniform (ChildPositions (branch := branch) (n := n) (t := t))) := by
rw [UniformLinearImage.law_uniform, FiniteDistribution.pushforward_equiv]
apply FiniteDistribution.eq_of_weight_eq
intro x
change 1 / (Fintype.card (Positions (branch := branch) (n := n) (t := t)) : ℝ) =
1 / (Fintype.card (Fin (branch n) → ChildPositions (branch := branch) (n := n) (t := t)) : ℝ)
rw [Fintype.card_congr (byChild (branch := branch) (n := n) (t := t))]
theorem parentLeftSlots_join {v : Nat} (clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch n) → Slots branch n)
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(positions : Positions (branch := branch) (n := n) (t := t)) :
SourceChildKernel.parentLeftSlots clauses designated (join designated q positions) =
HierarchicalArrays.sourceSlots clauses (PreliminarySampler.endpoints q) := by
rw [SourceChildQuestionLaw.parentLeftSlots_eq, questions_join]
end
end PerfectCompleteness.SourceQuestionPositionSplit
end
end OAI
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