Basic equivalence relation for calist structures.
Function:
(defun calist-equiv$inline (x y) (declare (xargs :guard (and (calistp x) (calistp y)))) (equal (calist-fix x) (calist-fix y)))
Theorem:
(defthm calist-equiv-is-an-equivalence (and (booleanp (calist-equiv x y)) (calist-equiv x x) (implies (calist-equiv x y) (calist-equiv y x)) (implies (and (calist-equiv x y) (calist-equiv y z)) (calist-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm calist-equiv-implies-equal-calist-fix-1 (implies (calist-equiv x x-equiv) (equal (calist-fix x) (calist-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm calist-fix-under-calist-equiv (calist-equiv (calist-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-calist-fix-1-forward-to-calist-equiv (implies (equal (calist-fix x) y) (calist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-calist-fix-2-forward-to-calist-equiv (implies (equal x (calist-fix y)) (calist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm calist-equiv-of-calist-fix-1-forward (implies (calist-equiv (calist-fix x) y) (calist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm calist-equiv-of-calist-fix-2-forward (implies (calist-equiv x (calist-fix y)) (calist-equiv x y)) :rule-classes :forward-chaining)