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