		Search-engine friendly clone of the ACL2 documentation.

Ienv

Ienv-equiv

Basic equivalence relation for ienv structures.

Definitions and Theorems

Function: ienv-equiv$inline

(defun ienv-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (ienvp acl2::x) (ienvp acl2::y))))
  (equal (ienv-fix acl2::x)
         (ienv-fix acl2::y)))

Theorem: ienv-equiv-is-an-equivalence

(defthm ienv-equiv-is-an-equivalence
  (and (booleanp (ienv-equiv x y))
       (ienv-equiv x x)
       (implies (ienv-equiv x y)
                (ienv-equiv y x))
       (implies (and (ienv-equiv x y) (ienv-equiv y z))
                (ienv-equiv x z)))
  :rule-classes (:equivalence))

Theorem: ienv-equiv-implies-equal-ienv-fix-1

(defthm ienv-equiv-implies-equal-ienv-fix-1
  (implies (ienv-equiv acl2::x x-equiv)
           (equal (ienv-fix acl2::x)
                  (ienv-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: ienv-fix-under-ienv-equiv

(defthm ienv-fix-under-ienv-equiv
  (ienv-equiv (ienv-fix acl2::x) acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-ienv-fix-1-forward-to-ienv-equiv

(defthm equal-of-ienv-fix-1-forward-to-ienv-equiv
  (implies (equal (ienv-fix acl2::x) acl2::y)
           (ienv-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-ienv-fix-2-forward-to-ienv-equiv

(defthm equal-of-ienv-fix-2-forward-to-ienv-equiv
  (implies (equal acl2::x (ienv-fix acl2::y))
           (ienv-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: ienv-equiv-of-ienv-fix-1-forward

(defthm ienv-equiv-of-ienv-fix-1-forward
  (implies (ienv-equiv (ienv-fix acl2::x) acl2::y)
           (ienv-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: ienv-equiv-of-ienv-fix-2-forward

(defthm ienv-equiv-of-ienv-fix-2-forward
  (implies (ienv-equiv acl2::x (ienv-fix acl2::y))
           (ienv-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)