def internal_all_eq_not_any_not

bend-mathlib@0.8.1.0 · list.bend:2030

import Base

2030def internal_all_eq_not_any_not(~a: Quant, ~A: Kind(a), ~f: A -> Bool, xs: List<a, A>) -> {List.all(~a, ~A, ~f, xs) == Bool.not(List.any(~a, ~A, ~(x => Bool.not(f(x))), xs)) : Bool}:
2031  match xs:
2032    case Nil{}:
2033      {==}
2034    case h <> t:
2035      +v = f(h)
2036      %internal_all_cons_test(~a, ~A, ~f, h, t, v, {==}) : {_ == Bool.not(List.any(~a, ~A, ~(x => Bool.not(f(x))), h <> t)) : Bool}
2037      %internal_any_cons_test(~a, ~A, ~(x => Bool.not(f(x))), h, t, Bool.not(v), {==}) : {Bool.and(v, List.all(~a, ~A, ~f, t)) == Bool.not(_) : Bool}
2038      %Equal.sym(Bool, List.all(~a, ~A, ~f, t), Bool.not(List.any(~a, ~A, ~(x => Bool.not(f(x))), t)), internal_all_eq_not_any_not(~a, ~A, ~f, t)) : {Bool.and(v, _) == Bool.not(Bool.or(Bool.not(v), List.any(~a, ~A, ~(x => Bool.not(f(x))), t))) : Bool}
2039      internal_and_not_eq_not_or_not(v, List.any(~a, ~A, ~(x => Bool.not(f(x))), t))