Research proof index

Scalar multiplication and mulOperator commute except possibly for c = 0 where the domains of 0 • 𝓜 μ f and 𝓜 μ 0 = 0 may not agree. See mulOperator_smul_eq for equality when c ≠ 0.

The linear equivalence between the Schwartz maps 𝓢(Space d, ℂ) and the Schwartz submodule of SpaceDHilbertSpace d μ. -/ def schwartzEquiv {d : ℕ} (μ : Measure (Space d)) [μ.HasTemperateGrowth] [μ.IsOpenPosMeasure] : 𝓢(Space d, ℂ) ≃ₗ[ℂ] SchwartzSubmodule d μ := LinearEquiv.ofInjective (schwartzIncl μ).toLinearMap (injective_toLp 2 μ) namespace Schwar...

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.