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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 | import OAI.Computability.PerfectCompleteness.Decoding.CanonicalUpperDirectionSplit
import OAI.Computability.PerfectCompleteness.Foundations.StoppedProjectedExperiment
namespace OAI
section
namespace PerfectCompleteness.StoppedProjectedBucketIndex
noncomputable section
open scoped Classical
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
variable {branch : Nat → Nat} {n i j t v m : Nat}
abbrev Base := GeometricCutSplit.Prefix branch (j + 1) (i + 1) ×
SourceProjectedTag.Tag (branch := branch) (n := i) (t := t)
def baseOf (rows : Nat → Nat)
(k : StoppedProjectedExperiment.Inner (branch := branch) (n := n)
(i := i) (j := j) (t := t) rows) : Base (branch := branch) (i := i) (j := j) (t := t) :=
(k.1, k.2.1)
def withDirections (rows : Nat → Nat)
(base : Base (branch := branch) (i := i) (j := j) (t := t))
(directions : CanonicalDirections.Tuple rows n) :
StoppedProjectedExperiment.Inner (branch := branch) (n := n)
(i := i) (j := j) (t := t) rows :=
(base.1, (base.2, directions))
@[simp] theorem baseOf_withDirections (rows : Nat → Nat)
(base : Base (branch := branch) (i := i) (j := j) (t := t))
(directions : CanonicalDirections.Tuple rows n) :
baseOf rows (withDirections rows base directions) = base := rfl
@[simp] theorem withDirections_baseOf (rows : Nat → Nat)
(k : StoppedProjectedExperiment.Inner (branch := branch) (n := n)
(i := i) (j := j) (t := t) rows) :
withDirections rows (baseOf rows k) k.2.2 = k := rfl
def defaultDirections (rows : Nat → Nat) (hrows : ∀ k, 0 < rows (k + 1)) :
CanonicalDirections.Tuple rows n :=
fun level => Classical.choice
(PreliminarySampler.directionNonempty (rows (level.val + 1)) (hrows level.val))
def baseLaw (hij : i < j) (hbranch : ∀ k < j + 1, 0 < branch k)
(flag : Fin (branch i) → FiniteDistribution Bool) :
FiniteDistribution (Base (branch := branch) (i := i) (j := j) (t := t)) :=
(GeometricCutSplit.prefixLaw (Nat.succ_le_succ hij.le) hbranch).product
(SourceProjectedTag.law (t := t) flag)
variable (clauses : Fin m → SourceClause.NormalizedClause v)
(rows repeats : Nat → Nat) (hupper : j + 1 ≤ n) (hij : i < j)
(designated : Fin (branch i) → Slots branch i)
(hrows : ∀ k, 0 < rows (k + 1))
(o : StoppedProjectedExperiment.Outer
(branch := branch) (n := n) (j := j) (t := t) (m := m))
abbrev key (base : Base (branch := branch) (i := i) (j := j) (t := t)) :=
withDirections (n := n) rows base (defaultDirections rows hrows)
abbrev Tape (base : Base (branch := branch) (i := i) (j := j) (t := t)) :=
WholeArraySampler.Tape rows repeats
(StoppedProjectedExperiment.stoppedPath rows hupper hij o (key rows hrows base))
(StoppedProjectedExperiment.projectedSlots clauses rows hupper hij designated o
(key rows hrows base))
def tapeLaw (base : Base (branch := branch) (i := i) (j := j) (t := t)) :
FiniteDistribution (Tape clauses rows repeats hupper hij designated hrows o base) :=
WholeArraySampler.tapeLaw rows repeats
(StoppedProjectedExperiment.stoppedPath rows hupper hij o (key rows hrows base))
(StoppedProjectedExperiment.projectedSlots clauses rows hupper hij designated o
(key rows hrows base))
theorem projectedSlots_eq
(k : StoppedProjectedExperiment.Inner (branch := branch) (n := n)
(i := i) (j := j) (t := t) rows) :
StoppedProjectedExperiment.projectedSlots clauses rows hupper hij designated o k =
StoppedProjectedExperiment.projectedSlots clauses rows hupper hij designated o
(key rows hrows (baseOf rows k)) := rfl
theorem expectation_coordinate
(hbranch : ∀ k < j + 1, 0 < branch k)
(flag : Fin (branch i) → FiniteDistribution Bool) (level : Fin n)
(f : (base : Base (branch := branch) (i := i) (j := j) (t := t)) →
PrefixTests.LevelDirection rows level →
Tape clauses rows repeats hupper hij designated hrows o base → ℝ) :
(StoppedProjectedExperiment.tapeLaw clauses rows repeats hupper hij designated
hbranch hrows flag o).expectation
(fun sample => f (baseOf rows sample.1) (sample.1.2.2 level) sample.2) =
(baseLaw (t := t) hij hbranch flag).expectation (fun base =>
(CanonicalDirections.levelLaw rows hrows level).expectation (fun direction =>
(tapeLaw clauses rows repeats hupper hij designated hrows o base).expectation
(f base direction))) := by
rw [StoppedProjectedExperiment.tapeLaw, CandidateCoupling.expectation_sigmaLaw]
simp only [StoppedProjectedExperiment.innerLaw, baseLaw,
FiniteDistribution.expectation_product]
apply FiniteDistribution.expectation_congr
intro lowerPrefix
apply FiniteDistribution.expectation_congr
intro tag
exact FiniteProduct.expectation_eval (CanonicalDirections.levelLaw rows hrows) level
(fun direction =>
(tapeLaw clauses rows repeats hupper hij designated hrows o (lowerPrefix, tag)).expectation
(f (lowerPrefix, tag) direction))
end
end PerfectCompleteness.StoppedProjectedBucketIndex
end
end OAI
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