Research proof index

OptionT companion to QueryImpl.simulateQ_liftM_eq_of_query: simulating an OracleComp-computation oa lifted into OptionT (OracleComp spec₂') (the shape produced by an OptionT-monadic verifier's let _ ← liftM (queryHelper) binds) agrees, at the run (Option) level, with some-mapping the simulation of oa through a per-query-bridged handler...

Three-layer simulation collapse: if a "reduction" red : QueryImpl spec₁ (StateT σ (OracleComp spec₂)) followed by an inner cached impl : QueryImpl spec₂ (StateT τ n) agrees per-query with a single "combined" handler combined : QueryImpl spec₁ (StateT (σ × τ) n) up to the natural reassociation α × σ × τ ≃ (α × σ) × τ, then the agreement extends to...

Hidden-bit decomposition at the SPMF level: the bias of a coin-flip guessing game equals the distinguishing advantage between the two branches, assuming the coin is fair and both branches have full mass (no failure). This is the SPMF analogue of ProbComp.boolBiasAdvantage_eq_boolDistAdvantage_uniformBool_branch. The ProbComp version holds unconditionall...

withStateOracle commutes with <$>: mapping a function over the surface computation is the same as mapping it over the observed SPMF. This holds because interpret is the bundled monad morphism simulateQ', and the StateT observer fun mx => toSPMF (StateT.run' mx s) preserves <$> even though it is not a full monad morphism: <$> does not thr...

Two conditionally independent executions dominate the square of the corresponding single-execution output probability.

Selector fibers inside an optional output are disjoint, so their finite sum is bounded by the probability of returning any some value.