		Search-engine friendly clone of the ACL2 documentation.

Fty::basetypes

True-equiv

(true-equiv x y) is a ``degenerate'' equivalence for true-p objects.

Because of the way true-fix works, this is always just true.

Definitions and Theorems

Function: true-equiv$inline

(defun true-equiv$inline (x y)
  (declare (xargs :guard (and (true-p x) (true-p y))))
  (equal (true-fix x) (true-fix y)))

Theorem: true-equiv-is-an-equivalence

(defthm true-equiv-is-an-equivalence
  (and (booleanp (true-equiv x y))
       (true-equiv x x)
       (implies (true-equiv x y)
                (true-equiv y x))
       (implies (and (true-equiv x y) (true-equiv y z))
                (true-equiv x z)))
  :rule-classes (:equivalence))

Theorem: true-equiv-implies-equal-true-fix-1

(defthm true-equiv-implies-equal-true-fix-1
  (implies (true-equiv x x-equiv)
           (equal (true-fix x) (true-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: true-fix-under-true-equiv

(defthm true-fix-under-true-equiv
  (true-equiv (true-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))