Basic equivalence relation for axi-litlist structures.
Function:
(defun axi-litlist-equiv$inline (x acl2::y) (declare (xargs :guard (and (axi-litlist-p x) (axi-litlist-p acl2::y)))) (equal (axi-litlist-fix x) (axi-litlist-fix acl2::y)))
Theorem:
(defthm axi-litlist-equiv-is-an-equivalence (and (booleanp (axi-litlist-equiv x y)) (axi-litlist-equiv x x) (implies (axi-litlist-equiv x y) (axi-litlist-equiv y x)) (implies (and (axi-litlist-equiv x y) (axi-litlist-equiv y z)) (axi-litlist-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm axi-litlist-equiv-implies-equal-axi-litlist-fix-1 (implies (axi-litlist-equiv x x-equiv) (equal (axi-litlist-fix x) (axi-litlist-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm axi-litlist-fix-under-axi-litlist-equiv (axi-litlist-equiv (axi-litlist-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-axi-litlist-fix-1-forward-to-axi-litlist-equiv (implies (equal (axi-litlist-fix x) acl2::y) (axi-litlist-equiv x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-axi-litlist-fix-2-forward-to-axi-litlist-equiv (implies (equal x (axi-litlist-fix acl2::y)) (axi-litlist-equiv x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm axi-litlist-equiv-of-axi-litlist-fix-1-forward (implies (axi-litlist-equiv (axi-litlist-fix x) acl2::y) (axi-litlist-equiv x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm axi-litlist-equiv-of-axi-litlist-fix-2-forward (implies (axi-litlist-equiv x (axi-litlist-fix acl2::y)) (axi-litlist-equiv x acl2::y)) :rule-classes :forward-chaining)