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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 | import OAI.Combinatorics.ContingencyTables.Counting.CompanionBoundedWalkProgram
import OAI.Combinatorics.ContingencyTables.Counting.CompanionMetropolisWordSemantics
import OAI.Combinatorics.ContingencyTables.Machines.FiniteEncodedHistory
namespace OAI
/-! # The bounded physical recorder preserves the complete phase history -/
namespace ContingencyTables.CompanionBoundedWalkProgram
open MatchingFPRAS CompanionFactorialApplication CompanionComputedInitial CompanionComputedProposal
open CompanionArithmeticProposal CompanionOracleDensity CompanionWeightOracle
open CompanionEvaluatedWeightProgram CompanionMultiplierMatrixProgram CommonBasesFPRAS
open CompanionBooleanPhase (State)
open FiniteObservableTransport TwoLiftStates GridBoundary
open scoped Classical
def codeStep (fixed : Fixed) (code : List ℕ) (bits : List Bool) : List ℕ :=
next (bits,(fixed,code))
lemma recordFold_eq (fixed : Fixed) (words : List (List Bool)) (code : List ℕ)
(past : List (List ℕ)) :
words.foldl (fun state bits => recordAdvance (bits,state)) ((fixed,code),past)=
let out := words.foldl (fun state bits =>
let d := codeStep fixed state.1 bits; (d,d::state.2)) (code,past)
((fixed,out.1),out.2) := by
induction words generalizing code past with
| nil => rfl
| cons word words ih =>
simpa only [List.foldl_cons,recordAdvance,advance,codeStep] using
ih (next (word,(fixed,code))) (next (word,(fixed,code))::past)
lemma history_eq_trace (fixed : Fixed) (code : List ℕ) (words : List (List Bool)) :
history ((fixed,code),words)=traceStates (codeStep fixed) code words := by
unfold history recordWalk initialRecord
rw [recordFold_eq]
exact traceStates_fold (codeStep fixed) words code
noncomputable section
variable {m n : ℕ}
lemma cap_encode (A : Instance (Fin m) (Fin n)) (S T : Set (Labels A × Bool)) :
ProfilePrograms.capView (((encodeSet (coordinateOrder A) S).length,1),
encodeSet (coordinateOrder A) T)=encodeSet (coordinateOrder A) T := by
apply ProfilePrograms.capView_eq
· simp only [encodeSet,List.length_ofFn]
· intro a ha
obtain ⟨i,rfl⟩ := List.mem_ofFn.mp ha
unfold ProfilePrograms.setProfile
split_ifs <;> omega
/-- The all-input clipping has no effect on a valid encoded transition;
in particular its full row law agrees with the reference phase step. -/
theorem codeStep_sameLaw (A : Instance (Fin m) (Fin n)) (C K j k : 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)
(xi : ℚ) (hxi : 0 < xi) (z : View A → ℕ)
(v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) (degree depth : List Bool)
(hdegree : degree.length=PowerOfTwoApproximation.rootDegree (roundingTolerance (innerTolerance (xi/2) j)))
(hdepth : depth.length=PowerOfTwoApproximation.depth (roundingTolerance (innerTolerance (xi/2) j)))
(w : LabelIndex A → LabelIndex A → TreeFraction.Fraction) (S₀ S : CompanionBooleanPhase.State A) :
let cfg := RootedLabelWeightProgram.config A C K j xi xi z v degree depth
let fixed : Fixed := (((cfg,matrixCode w),k),encodeSet (coordinateOrder A) S₀.val)
SameLaw (fun bits : Fin (CompanionBooleanPhase.stepBits A C K j k xi) → Bool =>
codeStep fixed (encodeSet (coordinateOrder A) S.val) (List.ofFn bits))
(fun bits => encodeSet (coordinateOrder A)
(RootedBooleanPhase.step A (RootedCompletion.rowColumn A.T) C K j k hK hW xi
(RootedCatalogGrid.gridEquiv A K v) (sourceWeights A w) S bits).val) := by
dsimp only
have h := CompanionMetropolisWordProgram.sameLaw_flat A C K j k hC hj hK hW hB
xi hxi z v degree depth hdegree hdepth w S
have hm := h.map (fun code => ProfilePrograms.capView
(((encodeSet (coordinateOrder A) S₀.val).length,1),code))
intro phi
simpa only [codeStep,next,bound,candidate,transitionInput,CompanionMetropolisWordProgram.packed,
cap_encode] using hm phi
/-- The physical list recorder and the full reference state history have
the same law, before any first-return scan or phase statistic is applied. -/
theorem history_sameLaw (A : Instance (Fin m) (Fin n)) (C K j k : 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)
(xi : ℚ) (hxi : 0 < xi) (z : View A → ℕ)
(v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) (degree depth : List Bool)
(hdegree : degree.length=PowerOfTwoApproximation.rootDegree (roundingTolerance (innerTolerance (xi/2) j)))
(hdepth : depth.length=PowerOfTwoApproximation.depth (roundingTolerance (innerTolerance (xi/2) j)))
(w : LabelIndex A → LabelIndex A → TreeFraction.Fraction) (S : CompanionBooleanPhase.State A) (N : ℕ) :
let cfg := RootedLabelWeightProgram.config A C K j xi xi z v degree depth
let fixed : Fixed := (((cfg,matrixCode w),k),encodeSet (coordinateOrder A) S.val)
SameLaw (fun words : Fin N → (Fin (CompanionBooleanPhase.stepBits A C K j k xi) → Bool) =>
history ((fixed,encodeSet (coordinateOrder A) S.val),List.ofFn (fun i => List.ofFn (words i))))
(fun words : Fin N → (Fin (CompanionBooleanPhase.stepBits A C K j k xi) → Bool) => (traceStates
(RootedBooleanPhase.step A (RootedCompletion.rowColumn A.T) C K j k hK hW xi
(RootedCatalogGrid.gridEquiv A K v) (sourceWeights A w)) S (List.ofFn words)).map
(fun T => encodeSet (coordinateOrder A) T.val)) := by
dsimp only
let cfg := RootedLabelWeightProgram.config A C K j xi xi z v degree depth
let fixed : Fixed := (((cfg,matrixCode w),k),encodeSet (coordinateOrder A) S.val)
have ht := traceStates_sameLaw
(fun code bits => codeStep fixed code (List.ofFn bits))
(RootedBooleanPhase.step A (RootedCompletion.rowColumn A.T) C K j k hK hW xi
(RootedCatalogGrid.gridEquiv A K v) (sourceWeights A w))
(fun T : CompanionBooleanPhase.State A => encodeSet (coordinateOrder A) T.val)
(codeStep_sameLaw A C K j k hC hj hK hW hB xi hxi z v degree depth hdegree hdepth w S) S N
intro phi
rw [show (fun words : Fin N → (Fin (CompanionBooleanPhase.stepBits A C K j k xi) → Bool) =>
phi (history ((fixed,encodeSet (coordinateOrder A) S.val),List.ofFn (fun i => List.ofFn (words i)))))=
(fun words => phi (traceStates (fun code bits => codeStep fixed code (List.ofFn bits))
(encodeSet (coordinateOrder A) S.val) (List.ofFn words))) by
funext words
rw [history_eq_trace]
have hm : List.ofFn (fun i => List.ofFn (words i))=(List.ofFn words).map List.ofFn := by
rw [List.map_ofFn]
rfl
rw [hm,traceStates_map]]
exact ht phi
end
end ContingencyTables.CompanionBoundedWalkProgram
end OAI
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