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