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