Reduced ADF

The reduced ADF of an ADF $D = (A, L, C)$ under a two-valued interpretation $v$ is $D_{\downarrow v} = (A', L \cap (A' \times A'), C')$ with $A' = v^{-1}(\mathsf{t})$, where each $C'_a$ is $C_a$ with every $b$ such that $v(b) = \mathsf{f}$ substituted by $\bot$.