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