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97c3405 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | import OAI.Combinatorics.ContingencyTables.Counting.CompanionInitializedRestartLaw
import OAI.Combinatorics.ContingencyTables.Machines.FiniteWordJointPadding
namespace OAI
/-! # Joint restart law with the unused trajectory and query suffix -/
namespace ContingencyTables.CompanionInitializedRestartProgram
open MatchingFPRAS CommonBasesFPRAS FirstSuccess FairBits FiniteObservableTransport
open CompanionFactorialApplication CompanionComputedInitial CompanionComputedProposal
open CompanionArithmeticProposal CompanionEvaluatedWeightProgram
open CompanionOracleDensity TwoLiftStates GridBoundary
open scoped BigOperators Classical
noncomputable section
variable {m n : ℕ}
def jointModel (A : Instance (Fin m) (Fin n)) (C K H T N R L j padding : ℕ)
(epsilon : TreeFraction.Fraction) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W)
(v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K)
(w : ℕ → LabelIndex A → LabelIndex A → TreeFraction.Fraction)
(bits : Fin (CompanionInitialDraw.initialBits A
(CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
(CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) → Bool) :
PlainIndex (pairs : List (CompanionRationalUpdates.Pair A))×(Fin padding → Bool) :=
let B := CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H)
let widths := CompanionRestartProgram.sourceWidth A C K (parameters A H T N R L epsilon)
let blocks := splitWordEquiv (CompanionInitialDraw.initialBits A B)
(CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding) bits
let tail := splitWordEquiv (CompanionRestartProgram.sumBits A C K j
(parameters A H T N R L epsilon)) padding blocks.2
(varyingWalk widths
(CompanionRestartProgram.sourceStep A C K (parameters A H T N R L epsilon) hK hW v w)
(CompanionInitialDraw.draw A B blocks.1) j (varyingWordEquiv widths j tail.1),tail.2)
theorem output_joint_sameLaw (A : Instance (Fin m) (Fin n)) (C K H T N R L j padding : ℕ)
(hC : 0 < C) (hj : j ≤ rationalHeight A C) (hK : 1 ≤ K)
(hW : 100*A.d*K ≤ A.W) (hB : (K:ℝ)=CompanionOracleDensity.binWidth A C)
(epsilon : TreeFraction.Fraction)
(hxi : 0 < TreeFraction.val (CountingPhaseParametersProgram.noise (parameters A H T N R L epsilon)))
(z : View A → ℕ) (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K)
(w : ℕ → LabelIndex A → LabelIndex A → TreeFraction.Fraction)
(estimates : Fin j → TreeFraction.Fraction)
(hdegree : ∀ i : Fin j,RootedOracleConfigProgram.rootDegree
(CompanionRestartStageProgram.oracleData A C K (i.val+1) (parameters A H T N R L epsilon) z v
(List.replicate (wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
(CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) false)) ≤
wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
(CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding))
(hdepth : ∀ i : Fin j,RootedOracleConfigProgram.depth
(CompanionRestartStageProgram.oracleData A C K (i.val+1) (parameters A H T N R L epsilon) z v
(List.replicate (wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
(CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) false)) ≤
wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
(CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) :
SameLaw (fun bits : Fin (wordLength A
(CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
(CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding)) → Bool =>
(output (data A C K H T N R L j epsilon z v w estimates (List.ofFn bits)),
(splitWordEquiv (CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon))
padding (splitWordEquiv
(wordLength A (CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H)))
(CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)+padding) bits).2).2))
(fun bits =>
(encodeSet (coordinateOrder A) (jointModel A C K H T N R L j padding epsilon hK hW v w bits).1.val.val,
(jointModel A C K H T N R L j padding epsilon hK hW v w bits).2)) := by
let B := CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H)
let p := parameters A H T N R L epsilon
let n := CompanionRestartProgram.sumBits A C K j p
let reserve := List.replicate (wordLength A B+(n+padding)) false
let stageList := CompanionRestartProgram.stagesData A C K j p z v reserve w
let init := fun bits : Fin (wordLength A B) → Bool => CompanionRestartInitializer.output
(CompanionRestartInitializer.data A C K H T N R L epsilon z v,List.ofFn bits)
let realInit := CompanionInitialDraw.draw A B
let encode := fun S : PlainIndex (pairs : List (CompanionRationalUpdates.Pair A)) =>
encodeSet (coordinateOrder A) S.val.val
let run := fun code (bits : Fin (n+padding) → Bool) =>
(CompanionRestartProgram.output ((stageList,code),List.ofFn bits),(splitWordEquiv n padding bits).2)
let sourceRun := fun S (bits : Fin n → Bool) => encode (varyingWalk
(CompanionRestartProgram.sourceWidth A C K p)
(CompanionRestartProgram.sourceStep A C K p hK hW v w) S j
(varyingWordEquiv (CompanionRestartProgram.sourceWidth A C K p) j bits))
let realRun := fun S (bits : Fin (n+padding) → Bool) =>
(sourceRun S (splitWordEquiv n padding bits).1,(splitWordEquiv n padding bits).2)
have hi : SameLaw init (fun bits => encode (realInit bits)) :=
CompanionRestartInitializer.output_sameLaw A C K H T N R L epsilon z v
have hs : ∀ S,SameLaw (run (encode S)) (realRun S) := by
intro S
have hn := CompanionRestartProgram.widthSum_data A C K j p z v reserve w S.val.val
have hh := CompanionRestartProgram.output_native_sameLaw A C K j hC hj hK hW hB p hxi z v reserve
(fun i => by simpa only [reserve,List.length_replicate] using hdegree i)
(fun i => by simpa only [reserve,List.length_replicate] using hdepth i) w S
exact restart_append_joint_sameLaw (CompanionRestartProgram.baseData A C K j p z v reserve w S.val.val)
n n hn (sourceRun S) hh padding
have hf := flat_encoded_bind (wordLength A B) (n+padding) (CompanionInitialDraw.initialBits A B) (n+padding)
init realInit encode run realRun hi hs
intro phi
have h := hf phi
have he : (fun bits : Fin (wordLength A B+(n+padding)) → Bool =>
(output (data A C K H T N R L j epsilon z v w estimates (List.ofFn bits)),
(splitWordEquiv n padding (splitWordEquiv (wordLength A B) (n+padding) bits).2).2))=
(fun bits => run (init (splitWordEquiv (wordLength A B) (n+padding) bits).1)
(splitWordEquiv (wordLength A B) (n+padding) bits).2) := by
funext bits
apply Prod.ext
· exact output_split A C K H T N R L j (n+padding) epsilon z v w estimates bits
· rfl
rw [he]
exact h
theorem jointModel_sameLaw (A : Instance (Fin m) (Fin n)) (C K H T N R L j padding : ℕ)
(epsilon : TreeFraction.Fraction) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W)
(v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K)
(w : ℕ → LabelIndex A → LabelIndex A → TreeFraction.Fraction) :
SameLaw (fun bits =>
(encodeSet (coordinateOrder A) (jointModel A C K H T N R L j padding epsilon hK hW v w bits).1.val.val,
(jointModel A C K H T N R L j padding epsilon hK hW v w bits).2))
(fun bits :
(Fin (CompanionInitialDraw.initialBits A
(CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H))+
CompanionRestartProgram.sumBits A C K j (parameters A H T N R L epsilon)) → Bool)×
(Fin padding → Bool) =>
(encodeSet (coordinateOrder A) (model A C K H T N R L j epsilon hK hW v w bits.1).val.val,bits.2)) := by
let B := CompanionInitialDraw.precision A (CountingPhaseSchedule.initialPrecision H)
let p := parameters A H T N R L epsilon
let widths := CompanionRestartProgram.sourceWidth A C K p
exact nested_word_joint (CompanionInitialDraw.initialBits A B)
(CompanionRestartProgram.sumBits A C K j p) padding
(fun initBits stageBits => encodeSet (coordinateOrder A)
(varyingWalk widths (CompanionRestartProgram.sourceStep A C K p hK hW v w)
(CompanionInitialDraw.draw A B initBits) j (varyingWordEquiv widths j stageBits)).val.val)
end
end ContingencyTables.CompanionInitializedRestartProgram
end OAI
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