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