law all_filter_sym

bend-mathlib@0.8.1.0 · list.bend:2766

import Base

2766# All over a filter checks q only where p holds: all q (filter p xs) = all (not p or q) xs, reversed to rewrite toward the simple side.
2767law all_filter_sym:
2768  for ~A: Data
2769  for ~p: A -> Bool
2770  for ~q: A -> Bool
2771  for xs: List<&2, A>
2772  {List.all(~&2, ~A, ~(+x => Bool.or(Bool.not(p(x)), q(x))), xs) == List.all(~&2, ~A, ~q, List.filter(~A, ~p, xs)) : Bool}