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