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