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