Research proof index

First-moment / Markov bound (support-restricted cost). Variant of probEvent_le_tsum_probOutput_mul_cost whose c ≥ 1 hypothesis need only hold on the support of mx.

The stateful-responder probability bridge: running an adversary against the responder built from a StateT σ ProbComp handler (ofStateQueryImpl impl) is exactly the evaluation distribution of running it against impl itself , the responder's SPMF program-run is 𝒟 of the ProbComp program-run, jointly in the returned value and the final sta...

A WriterT query log can be replayed as a StateT input log. For computations over a sum spec + loggedSpec, this theorem compares two implementations: * left queries in spec are forwarded unchanged; * right queries in loggedSpec are either handled with withLogging, producing a QueryLog loggedSpec, or with appendInputLog, appending just the q...

Induction principle for proj (simulateQ (so.postInsert nx) oa) parametric in a motive OracleComp spec β → m β → Prop. The recursion structure of proj_simulateQ_postInsert is exposed as two cases mirroring OracleComp.inductionOn: in pure x the projected term reduces to pure x, and in query t >>= k it reduces to so t >>= k' for some continua...

State-dependent ε-perturbed identical-until-bad with output bad flag.

parSum congruence on both sides from per-factor handler equivalences with explicit initial states.