Research proof index

Instantiating symbolic capabilities is associative. Substituting impl₁ into a profile c, and then substituting impl₂ into the resulting profile, is the same as substituting the composite implementation fun k ↦ (impl₁ k).instantiate impl₂ into c directly.

runObs on a lifted base-spec computation: the trace is 1 (monoid identity).

Truncating the seed at oracle i₀ to only the first k entries does not change the distribution when averaging over seeds from generateSeed.

If every value x in the support of gen satisfies Pr[= a | f x] ≥ 1 - δ, then the overall probability satisfies Pr[= a | gen >>= f] ≥ 1 - δ. This reduces a "for all keys" completeness statement to per-key bounds.

Phase B (freshness-drop) bound. The CMA advantage is bounded above by the success probability of the same experiment with the freshness check dropped. Both unforgeableExp and unforgeableExpNoFresh factor as runtime.evalDist (joint >>= ...) sharing the same prefix joint. The hypothesis h_pull packages the runtime-specific factoring step that...

Failing companion to simulateQ_optionT_forIn_yield_pure_some: if each loop body, under simulateQ, resolves to pure (some (ForInStep.yield init)) when its per-element condition cond a holds and to pure none otherwise, and some element of the list fails its condition, then the whole OptionT-monadic forIn resolves to pure none (the failure...