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