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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import OAI.Computability.PerfectCompleteness.Foundations.CanonicalDirections
import OAI.Computability.PerfectCompleteness.Foundations.QuarterBalanceLemmas
import OAI.Computability.PerfectCompleteness.Foundations.TupleIndexEmitterLemmas
namespace OAI
section
namespace PerfectCompleteness.CanonicalUpperDirectionSplit
noncomputable section
open scoped Classical
open UniqueGamesTheorem.Foundations.Games
variable {n : Nat}
abbrev Complement (rows : Nat → Nat) (upper : Fin n) :=
FiniteProductSplit.Others
(Ω := fun level : Fin n => PrefixTests.LevelDirection rows level) upper
def splitEquiv (rows : Nat → Nat) (upper : Fin n) :
CanonicalDirections.Tuple rows n ≃
PrefixTests.LevelDirection rows upper × Complement rows upper :=
Equiv.piSplitAt upper (fun level => PrefixTests.LevelDirection rows level)
def complement (rows : Nat → Nat) (upper : Fin n)
(directions : CanonicalDirections.Tuple rows n) : Complement rows upper :=
(splitEquiv rows upper directions).2
def assemble (rows : Nat → Nat) (upper : Fin n)
(direction : PrefixTests.LevelDirection rows upper) (rest : Complement rows upper) :
CanonicalDirections.Tuple rows n :=
(splitEquiv rows upper).symm (direction, rest)
@[simp] theorem split_selected (rows : Nat → Nat) (upper : Fin n)
(directions : CanonicalDirections.Tuple rows n) :
(splitEquiv rows upper directions).1 = directions upper := rfl
@[simp] theorem complement_apply (rows : Nat → Nat) (upper : Fin n)
(directions : CanonicalDirections.Tuple rows n) (i : {i : Fin n // i ≠ upper}) :
complement rows upper directions i = directions i.val := rfl
@[simp] theorem assemble_at_upper (rows : Nat → Nat) (upper : Fin n)
(direction : PrefixTests.LevelDirection rows upper) (rest : Complement rows upper) :
assemble rows upper direction rest upper = direction := by
exact congrArg Prod.fst ((splitEquiv rows upper).apply_symm_apply (direction, rest))
@[simp] theorem assemble_at_other (rows : Nat → Nat) (upper : Fin n)
(direction : PrefixTests.LevelDirection rows upper) (rest : Complement rows upper)
(i : Fin n) (hne : i ≠ upper) :
assemble rows upper direction rest i = rest ⟨i, hne⟩ := by
exact congrArg
(fun z : PrefixTests.LevelDirection rows upper × Complement rows upper => z.2 ⟨i, hne⟩)
((splitEquiv rows upper).apply_symm_apply (direction, rest))
theorem assemble_split (rows : Nat → Nat) (upper : Fin n)
(directions : CanonicalDirections.Tuple rows n) :
assemble rows upper (directions upper) (complement rows upper directions) = directions :=
(splitEquiv rows upper).symm_apply_apply directions
theorem lower_preserved (rows : Nat → Nat) (upper lower : Fin n) (hne : lower ≠ upper)
(direction : PrefixTests.LevelDirection rows upper)
(directions : CanonicalDirections.Tuple rows n) :
assemble rows upper direction (complement rows upper directions) lower = directions lower := by
rw [assemble_at_other rows upper direction (complement rows upper directions) lower hne]
rfl
def complementLaw (rows : Nat → Nat) (hrows : ∀ k, 0 < rows (k + 1)) (upper : Fin n) :
FiniteDistribution (Complement rows upper) :=
FiniteProductSplit.otherLaw (CanonicalDirections.levelLaw rows hrows) upper
theorem split_law (rows : Nat → Nat) (hrows : ∀ k, 0 < rows (k + 1)) (upper : Fin n) :
(CanonicalDirections.law rows n hrows).pushforward (splitEquiv rows upper) =
(CanonicalDirections.levelLaw rows hrows upper).product (complementLaw rows hrows upper) :=
FiniteProductSplit.split_law (CanonicalDirections.levelLaw rows hrows) upper
theorem assemble_law (rows : Nat → Nat) (hrows : ∀ k, 0 < rows (k + 1)) (upper : Fin n) :
((CanonicalDirections.levelLaw rows hrows upper).product
(complementLaw rows hrows upper)).pushforward
(fun z => assemble rows upper z.1 z.2) = CanonicalDirections.law rows n hrows := by
have h := congrArg
(fun μ : FiniteDistribution (PrefixTests.LevelDirection rows upper × Complement rows upper) =>
μ.pushforward (splitEquiv rows upper).symm) (split_law rows hrows upper)
rw [FiniteDistribution.pushforward_comp] at h
have hid : (fun directions : CanonicalDirections.Tuple rows n =>
(splitEquiv rows upper).symm (splitEquiv rows upper directions)) = id := by
funext directions
exact (splitEquiv rows upper).symm_apply_apply directions
rw [hid, FiniteDistribution.pushforward_id] at h
exact h.symm
theorem selected_law (rows : Nat → Nat) (hrows : ∀ k, 0 < rows (k + 1)) (upper : Fin n) :
(CanonicalDirections.law rows n hrows).pushforward (fun directions => directions upper) =
CanonicalDirections.levelLaw rows hrows upper :=
FiniteProduct.eval_pushforward (CanonicalDirections.levelLaw rows hrows) upper
theorem probability_split (rows : Nat → Nat) (hrows : ∀ k, 0 < rows (k + 1))
(upper : Fin n) (event : CanonicalDirections.Tuple rows n → Bool) :
(CanonicalDirections.law rows n hrows).probability event =
((CanonicalDirections.levelLaw rows hrows upper).product
(complementLaw rows hrows upper)).probability
(fun z => event (assemble rows upper z.1 z.2)) := by
rw [← assemble_law rows hrows upper, FiniteDistribution.probability_pushforward]
theorem probability_complement_first (rows : Nat → Nat)
(hrows : ∀ k, 0 < rows (k + 1)) (upper : Fin n)
(event : CanonicalDirections.Tuple rows n → Bool) :
(CanonicalDirections.law rows n hrows).probability event =
(complementLaw rows hrows upper).expectation (fun rest =>
(CanonicalDirections.levelLaw rows hrows upper).probability
(fun direction => event (assemble rows upper direction rest))) := by
rw [probability_split rows hrows upper]
exact UsefulAdvice.product_probability_advice_first _ _ _
theorem lower_choice_preserved {branch : Nat → Nat} (rows : Nat → Nat)
(leaf : RecursiveSpaces.Slots branch n) (upper lower : Fin n) (hne : lower ≠ upper)
(direction : PrefixTests.LevelDirection rows upper)
(directions : CanonicalDirections.Tuple rows n) :
PrefixTests.choiceAt rows leaf lower
(assemble rows upper direction (complement rows upper directions) lower) =
PrefixTests.choiceAt rows leaf lower (directions lower) := by
rw [lower_preserved rows upper lower hne direction directions]
end
end PerfectCompleteness.CanonicalUpperDirectionSplit
end
end OAI
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