| import OAI.Combinatorics.ContingencyTables.Rooted.RootedOracleAcceptance |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionBooleanPhase |
|
|
| namespace OAI |
|
|
| /-! # Actual finite-word Metropolis steps using the selected-root oracle -/ |
|
|
| namespace ContingencyTables.RootedBooleanPhase |
|
|
| open CompanionFactorialApplication CompanionOracleDensity CompanionWeightOracle CompanionRationalUpdates |
| open CommonBasesFPRAS FirstSuccess FairBits TwoLiftStates GridBoundary ResidualMixture |
| open CompanionBooleanPhase (State proposal proposal_mass proposalBits proposal_codes stepBits stepWordEquiv) |
| open scoped BigOperators Classical |
|
|
| noncomputable section |
| variable {I J : Type*} [Fintype I] [Fintype J] |
|
|
| / |
| separate words, then uses a third independent word for its rational coin. -/ |
| def step (A : Instance I J) (R : RootedCompletion.Selection A.T) (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 : Pair A → Pair A → ℚ) |
| (S : State A) (bits : Fin (stepBits A C K j k xi) → Bool) : State A := |
| let b := stepWordEquiv A C K j k xi bits |
| if b.1 0 then S else |
| match proposal A S b.2.1 with |
| | none => S |
| | some T => |
| if rationalThresholdAccept k |
| (RootedOracleAcceptance.acceptance A R C K j hK hW xi v w S.val T.val b.2.2.1) b.2.2.2 |
| then T else S |
|
|
| / |
| acceptance experiment. Failed codes and the independent holding bit |
| contribute only to the diagonal. -/ |
| theorem step_eq_iff (A : Instance I J) (R : RootedCompletion.Selection A.T) (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 : Pair A → Pair A → ℚ) |
| (S T : State A) (hST : S ≠ T) (bits : Fin (stepBits A C K j k xi) → Bool) : |
| step A R C K j k hK hW xi v w S bits=T ↔ |
| (stepWordEquiv A C K j k xi bits).1 0=false ∧ |
| proposal A S (stepWordEquiv A C K j k xi bits).2.1=some T ∧ |
| rationalThresholdAccept k |
| (RootedOracleAcceptance.acceptance A R 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 := by |
| dsimp only [step] |
| generalize stepWordEquiv A C K j k xi bits = b |
| cases b.1 0 with |
| | true => simp [hST] |
| | false => |
| cases proposal A S b.2.1 with |
| | none => simp [hST] |
| | some U => |
| by_cases hUT : U=T |
| · subst U |
| simp [hST] |
| · by_cases ha : rationalThresholdAccept k |
| (RootedOracleAcceptance.acceptance A R C K j hK hW xi v w S.val U.val b.2.2.1) b.2.2.2 |
| · simp [hUT,ha] |
| · simp [hST,hUT,ha] |
|
|
| def transitionLaw (A : Instance I J) (R : RootedCompletion.Selection A.T) (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 : Pair A → Pair A → ℚ) (S : State A) : |
| RationalLaw (State A) := |
| mapLaw (uniformLaw (α := Fin (stepBits A C K j k xi) → Bool)) |
| (step A R C K j k hK hW xi v w S) |
|
|
| def walk (A : Instance I J) (R : RootedCompletion.Selection A.T) (C K j k n : ℕ) |
| (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) (xi : ℚ) |
| (v : Grid (I := Fin (dimension A)) K) (w : Pair A → Pair A → ℚ) (S : State A) |
| (bits : Fin (n*stepBits A C K j k xi) → Bool) : State A := |
| wordWalk (step A R C K j k hK hW xi v w) S n |
| (oracleWalkWordEquiv n (stepBits A C K j k xi) bits) |
|
|
| / |
| its law follows from the accepted finite-word walk construction. -/ |
| theorem walk_law (A : Instance I J) (R : RootedCompletion.Selection A.T) (C K j k n : ℕ) |
| (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) (xi : ℚ) |
| (v : Grid (I := Fin (dimension A)) K) (w : Pair A → Pair A → ℚ) (S : State A) : |
| mapLaw (uniformLaw (α := Fin (n*stepBits A C K j k xi) → Bool)) |
| (walk A R C K j k n hK hW xi v w S)= |
| walkLaw (transitionLaw A R C K j k hK hW xi v w) S n := by |
| unfold walk |
| rw [←map_equiv (oracleWalkWordEquiv n (stepBits A C K j k xi)),uniform_equiv] |
| exact wordWalkLaw_eq _ _ _ |
|
|
| end |
| end ContingencyTables.RootedBooleanPhase |
|
|
| end OAI |
|
|
