(tag-env-fix x) is a usual ACL2::fty omap fixing function.
(tag-env-fix x) → *
Function:
(defun tag-env-fix (x) (declare (xargs :guard (tag-envp x))) (mbe :logic (if (tag-envp x) x nil) :exec x))
Theorem:
(defthm tag-envp-of-tag-env-fix (tag-envp (tag-env-fix x)))
Theorem:
(defthm tag-env-fix-when-tag-envp (implies (tag-envp x) (equal (tag-env-fix x) x)))
Theorem:
(defthm emptyp-tag-env-fix (implies (or (omap::emptyp x) (not (tag-envp x))) (omap::emptyp (tag-env-fix x))))
Theorem:
(defthm emptyp-of-tag-env-fix-to-not-tag-env-or-emptyp (equal (omap::emptyp (tag-env-fix x)) (or (not (tag-envp x)) (omap::emptyp x))))
Function:
(defun tag-env-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (tag-envp acl2::x) (tag-envp acl2::y)))) (equal (tag-env-fix acl2::x) (tag-env-fix acl2::y)))
Theorem:
(defthm tag-env-equiv-is-an-equivalence (and (booleanp (tag-env-equiv x y)) (tag-env-equiv x x) (implies (tag-env-equiv x y) (tag-env-equiv y x)) (implies (and (tag-env-equiv x y) (tag-env-equiv y z)) (tag-env-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm tag-env-equiv-implies-equal-tag-env-fix-1 (implies (tag-env-equiv acl2::x x-equiv) (equal (tag-env-fix acl2::x) (tag-env-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm tag-env-fix-under-tag-env-equiv (tag-env-equiv (tag-env-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-tag-env-fix-1-forward-to-tag-env-equiv (implies (equal (tag-env-fix acl2::x) acl2::y) (tag-env-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-tag-env-fix-2-forward-to-tag-env-equiv (implies (equal acl2::x (tag-env-fix acl2::y)) (tag-env-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm tag-env-equiv-of-tag-env-fix-1-forward (implies (tag-env-equiv (tag-env-fix acl2::x) acl2::y) (tag-env-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm tag-env-equiv-of-tag-env-fix-2-forward (implies (tag-env-equiv acl2::x (tag-env-fix acl2::y)) (tag-env-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)