Research proof index

Extensionality lemma for accCube.

[⊗_{i=0}^{ϑ-1}(1-r_i, r_i)] · [ - u₀ -; ...; - u_{2^ϑ-1} - ] - [⊗_{i=0}^{ϑ-2}(1-r_i, r_i)] · ([(1-r_{ϑ-1}) · U₀] + [r_{ϑ-1} · U₁])

Taking the first component of the WriterT output recovers the original computation, when the query implementation preserves the underlying oracle behavior (hso).

Loop-invariant rule for bounded left folds over related input lists.

Simulating an OracleComp through a stateful implementation in monad m can only shrink the support: any output reachable after simulation was already reachable in the original computation (where oracle queries may return any value). This is the support-level analogue of evalDist_simulateQ_run'_eq_evalDist.

The two-term Wirtinger chain rule for dWirtingerAntiDir, the anti-holomorphic dual of dWirtingerDir_comp: ∂̄_v(g∘f) = (∂g/∂f)·∂̄_v f + (∂g/∂f̄)·∂̄_v f̄. Same outer ∂g/∂f, ∂g/∂f̄ coefficients, now each multiplying its anti-holomorphic inner derivative ∂̄_v f, ∂̄_v f̄; same proof as dWirtingerDir_comp.