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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 | import OAI.Computability.PerfectCompleteness.Algebra.FixedRankContradiction
import OAI.Computability.PerfectCompleteness.Decoding.FixedDecoderCleanRateLemmas
namespace OAI
section
namespace PerfectCompleteness.DecoderRankSplit
noncomputable section
open scoped BigOperators Classical
open TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
abbrev F2 := ZMod 2
variable {branch rows repeats : Nat → Nat} {n h t v m : Nat} [NeZero m]
{upper : Nodes branch n} {d : HierarchicalFrozenTables.LowerNodes upper (h + 1)}
(S : CleanDecoderRate.Setup branch rows repeats n h t v m upper d)
def meeting (x : CleanDecoderRate.Sample S) (answer : CleanDecoderPairLaw.Pair S x) : Bool :=
decide (answer.2 = DecoderSourcePullback.upperTarget
(CleanDecoderOddLists.fullProjection (CleanDecoderPairLaw.context S x)
(CleanDecoderPairLaw.occurrences S x)) upper answer.1)
variable (hbranch : ∀ k < n, 0 < branch k)
def nativeRank (x : CleanDecoderRate.Sample S)
(answer : CleanDecoderPairLaw.Pair S x) : Nat :=
OddListExtraction.formRank
(H (CleanDecoderOddLists.leftNative (CleanDecoderPairLaw.context S x)
(CleanDecoderPairLaw.leftOwn S x)))
(CleanLeftDecoder.form
(CleanDecoderContext.leftExposed (CleanDecoderPairLaw.context S x)) upper (h + 1)
hbranch d (CleanDecoderPairLaw.leftOwn S x) answer.1)
def low (s : Nat) (x : CleanDecoderRate.Sample S)
(answer : CleanDecoderPairLaw.Pair S x) : Bool :=
decide (nativeRank S hbranch x answer ≤ s)
def highAgrees (s : Nat) (x : CleanDecoderRate.Sample S)
(answer : CleanDecoderPairLaw.Pair S x) : Bool :=
meeting S x answer && !(low S hbranch s x answer)
@[simp] theorem highAgrees_eq_true (s : Nat) (x : CleanDecoderRate.Sample S)
(answer : CleanDecoderPairLaw.Pair S x) :
highAgrees S hbranch s x answer = true ↔
s < nativeRank S hbranch x answer ∧ meeting S x answer = true := by
simp [highAgrees, low, and_comm]
private theorem decide_and_comm (P Q : Prop) [Decidable P] [Decidable Q] :
(decide P && decide Q) = decide (Q ∧ P) := by
by_cases hP : P <;> by_cases hQ : Q <;> simp [hP, hQ]
theorem meeting_low (s : Nat) (x : CleanDecoderRate.Sample S)
(answer : CleanDecoderPairLaw.Pair S x) :
(meeting S x answer && low S hbranch s x answer) =
CleanDecoderPairLaw.pairAgrees S hbranch s x answer :=
decide_and_comm
(answer.2 = DecoderSourcePullback.upperTarget
(CleanDecoderOddLists.fullProjection (CleanDecoderPairLaw.context S x)
(CleanDecoderPairLaw.occurrences S x)) upper answer.1)
(nativeRank S hbranch x answer ≤ s)
theorem probability_split
(μ : FiniteDistribution (Σ x : CleanDecoderRate.Sample S, CleanDecoderPairLaw.Pair S x))
(s : Nat) :
μ.probability (fun z => meeting S z.1 z.2) =
μ.probability (fun z => CleanDecoderPairLaw.pairAgrees S hbranch s z.1 z.2) +
μ.probability (fun z => highAgrees S hbranch s z.1 z.2) := by
have hsplit := FixedRankContradiction.probability_split μ
(fun z => meeting S z.1 z.2) (fun z => low S hbranch s z.1 z.2)
simpa only [meeting_low, highAgrees] using hsplit
variable (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t))
(useful : (record : CleanDecoderRate.Record S) →
CleanLeftDecoder.Useful (CleanDecoderContext.leftExposed (CleanDecoderRate.context S record))
upper (h + 1))
(r : Nat) (ρ : ℝ) (A : ManyGoodRows.RowMap (Block rows upper) r)
(W : Submodule F2 (OwnInputReference.UpperVector rows upper))
(a : OwnInputReference.LowerVector rows (HierarchicalLeftDecoder.LowerNode upper (h + 1) d))
(threshold : ℝ)
theorem pair_probability_split (s cube : Nat) (hcube : 0 < cube) :
(CleanDecoderPairLaw.pairLaw S σ useful r ρ A W a threshold cube hcube).probability
(fun z => meeting S z.1 z.2) =
(CleanDecoderPairLaw.pairLaw S σ useful r ρ A W a threshold cube hcube).probability
(fun z => CleanDecoderPairLaw.pairAgrees S hbranch s z.1 z.2) +
(CleanDecoderPairLaw.pairLaw S σ useful r ρ A W a threshold cube hcube).probability
(fun z => highAgrees S hbranch s z.1 z.2) :=
probability_split S hbranch _ s
end
end PerfectCompleteness.DecoderRankSplit
end
end OAI
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