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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | import OAI.Combinatorics.ContingencyTables.Counting.CompanionArithmeticProposal
import OAI.Combinatorics.ContingencyTables.Counting.CompanionPhaseTrajectory
namespace OAI
/-! # Coordinate arithmetic for the companion proposal
The arithmetic and abstract proposals consume equal fixed word lengths.
Oracle and acceptance words are disjoint, and their full one-step laws
agree, giving the same trajectory error bound.
-/
namespace ContingencyTables.CompanionArithmeticStep
open CommonBasesFPRAS FirstSuccess ResidualMixture FairBits GridBoundary
open CompanionFactorialApplication CompanionOracleDensity CompanionWeightOracle
open CompanionBooleanPhase RealKernelApproximation
open scoped BigOperators Classical
noncomputable section
private def regroup (L P Q : Type*) : (L×(P×Q)) ≃ P×(L×Q) where
toFun b := (b.2.1,(b.1,b.2.2))
invFun b := (b.2.1,(b.1,b.2.2))
left_inv _ := rfl
right_inv _ := rfl
private lemma word_finish_law {B P Q X Z : Type*}
[Fintype B] [Fintype P] [Fintype Q] [Fintype X] [Fintype Z]
[Nonempty B] [Nonempty P] [Nonempty Q] [DecidableEq X] [DecidableEq Z]
(E : B ≃ P×Q) (draw : P → X) (finish : X → Q → Z) :
mapLaw (uniformLaw (α := B)) (fun b => finish (draw (E b).1) (E b).2)=
bindLaw (mapLaw (uniformLaw (α := P)) draw)
(fun x => mapLaw (uniformLaw (α := Q)) (finish x)) := by
have he := map_equiv E (uniformLaw (α := B)) (fun b : P×Q => finish (draw b.1) b.2)
rw [uniform_equiv,←uniform_product] at he
exact he.symm.trans (mapLaw_product_finish _ _ draw finish)
variable {I J : Type} [Fintype I] [Fintype J]
abbrev RestWord (A : Instance I J) (C K j k : ℕ) (xi : ℚ) :=
(Fin 1 → Bool)×((WeightWord A C K j xi×WeightWord A C K j xi)×(Fin k → Bool))
def wordSplit (A : Instance I J) (C K j k : ℕ) (xi : ℚ) :
(Fin (stepBits A C K j k xi) → Bool) ≃
((Fin (proposalBits A) → Bool)×RestWord A C K j k xi) :=
(stepWordEquiv A C K j k xi).trans (regroup _ _ _)
def finish (A : Instance I J) (C K j k : ℕ)
(hK : 1≤K) (hW : 100*A.d*K≤A.W) (xi : ℚ)
(v : Grid (I := Fin (dimension A)) K) (w : CompanionRationalUpdates.Pair A → CompanionRationalUpdates.Pair A → ℚ)
(S : State A) (U : Option (State A)) (rest : RestWord A C K j k xi) : State A :=
if rest.1 0 then S else
match U with
| none => S
| some T =>
if rationalThresholdAccept k
(CompanionOracleAcceptance.acceptance A C K j hK hW xi v w S.val T.val rest.2.1)
rest.2.2 then T else S
def step (A : Instance I J) (C K j k : ℕ)
(hK : 1≤K) (hW : 100*A.d*K≤A.W) (xi : ℚ)
(v : Grid (I := Fin (dimension A)) K) (w : CompanionRationalUpdates.Pair A → CompanionRationalUpdates.Pair A → ℚ)
(S : State A) (bits : Fin (stepBits A C K j k xi) → Bool) : State A :=
let b := wordSplit A C K j k xi bits
finish A C K j k hK hW xi v w S (CompanionArithmeticProposal.proposal A S b.1) b.2
private lemma source_step_eq_finish (A : Instance I J) (C K j k : ℕ)
(hK : 1≤K) (hW : 100*A.d*K≤A.W) (xi : ℚ)
(v : Grid (I := Fin (dimension A)) K) (w : CompanionRationalUpdates.Pair A → CompanionRationalUpdates.Pair A → ℚ)
(S : State A) :
(fun bits => finish A C K j k hK hW xi v w S
(CompanionBooleanPhase.proposal A S (wordSplit A C K j k xi bits).1)
(wordSplit A C K j k xi bits).2)=
CompanionBooleanPhase.step A C K j k hK hW xi v w S := by
funext bits
simp only [finish,wordSplit,regroup,CompanionBooleanPhase.step,Equiv.trans_apply,Equiv.coe_fn_mk]
by_cases hlazy : (stepWordEquiv A C K j k xi bits).1 0 = true
· simp only [hlazy,ite_true]
· simp only [hlazy]
cases CompanionBooleanPhase.proposal A S
(stepWordEquiv A C K j k xi bits).2.1 with
| none => simp only
| some T =>
simp only
by_cases haccept : rationalThresholdAccept k
(CompanionOracleAcceptance.acceptance A C K j hK hW xi v w S.val T.val
(stepWordEquiv A C K j k xi bits).2.2.1)
(stepWordEquiv A C K j k xi bits).2.2.2
· simp only [haccept,ite_true]
· simp only [haccept,ite_false]
/-- Exact equality of complete one-step laws, including all holding
probability. Weight calls and the acceptance coin have not been reused. -/
theorem step_law (A : Instance I J) (C K j k : ℕ)
(hK : 1≤K) (hW : 100*A.d*K≤A.W) (xi : ℚ)
(v : Grid (I := Fin (dimension A)) K) (w : CompanionRationalUpdates.Pair A → CompanionRationalUpdates.Pair A → ℚ)
(S : State A) :
mapLaw (uniformLaw (α := Fin (stepBits A C K j k xi) → Bool))
(step A C K j k hK hW xi v w S)=
transitionLaw A C K j k hK hW xi v w S := by
have hnew := word_finish_law (wordSplit A C K j k xi)
(CompanionArithmeticProposal.proposal A S) (finish A C K j k hK hW xi v w S)
have hold := word_finish_law (wordSplit A C K j k xi)
(CompanionBooleanPhase.proposal A S) (finish A C K j k hK hW xi v w S)
change mapLaw _ (fun bits => finish A C K j k hK hW xi v w S
(CompanionArithmeticProposal.proposal A S (wordSplit A C K j k xi bits).1)
(wordSplit A C K j k xi bits).2)=_
rw [hnew,CompanionArithmeticProposal.proposal_law]
rw [source_step_eq_finish] at hold
exact hold.symm
/-- The full recorded finite-word path therefore retains the proved
joint-history distribution, rather than merely its terminal marginal. -/
def recordedWalk (A : Instance I J) (C K j k N n : ℕ)
(hK : 1≤K) (hW : 100*A.d*K≤A.W) (xi : ℚ)
(v : Grid (I := Fin (dimension A)) K) (w : CompanionRationalUpdates.Pair A → CompanionRationalUpdates.Pair A → ℚ)
(S : State A) (bits : Fin (n*stepBits A C K j k xi) → Bool) : PathState (State A) N :=
wordWalk (fun a b => pathPut a (step A C K j k hK hW xi v w (pathRead a) b))
(pathInitial N S) n (oracleWalkWordEquiv n (stepBits A C K j k xi) bits)
theorem recordedWalk_law (A : Instance I J) (C K j k N n : ℕ)
(hK : 1≤K) (hW : 100*A.d*K≤A.W) (xi : ℚ)
(v : Grid (I := Fin (dimension A)) K) (w : CompanionRationalUpdates.Pair A → CompanionRationalUpdates.Pair A → ℚ)
(S : State A) :
mapLaw (uniformLaw (α := Fin (n*stepBits A C K j k xi) → Bool))
(recordedWalk A C K j k N n hK hW xi v w S)=
pathLaw A C K j k N n hK hW xi v w S := by
unfold recordedWalk pathLaw
rw [←map_equiv (oracleWalkWordEquiv n (stepBits A C K j k xi)),uniform_equiv]
rw [show mapLaw (uniformLaw (α := Fin n → Fin (stepBits A C K j k xi) → Bool))
(wordWalk (fun a b => pathPut a (step A C K j k hK hW xi v w (pathRead a) b))
(pathInitial N S) n)=
wordWalkLaw (fun a b => pathPut a (step A C K j k hK hW xi v w (pathRead a) b))
(pathInitial N S) n from rfl,wordWalkLaw_eq]
congr 1
funext a
rw [←mapLaw_comp,step_law]
rfl
end
end ContingencyTables.CompanionArithmeticStep
end OAI
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