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