		Search-engine friendly clone of the ACL2 documentation.

Env-block

Env-block-fix

(env-block-fix x) is a usual ACL2::fty omap fixing function.

Signature
(env-block-fix x) → *
Arguments: x — Guard (env-blockp x).

Definitions and Theorems

Function: env-block-fix

(defun env-block-fix (x)
  (declare (xargs :guard (env-blockp x)))
  (mbe :logic (if (env-blockp x) x nil)
       :exec x))

Theorem: env-blockp-of-env-block-fix

(defthm env-blockp-of-env-block-fix
  (env-blockp (env-block-fix x)))

Theorem: env-block-fix-when-env-blockp

(defthm env-block-fix-when-env-blockp
  (implies (env-blockp x)
           (equal (env-block-fix x) x)))

Theorem: emptyp-env-block-fix

(defthm emptyp-env-block-fix
  (implies (or (omap::emptyp x)
               (not (env-blockp x)))
           (omap::emptyp (env-block-fix x))))

Theorem: emptyp-of-env-block-fix-to-not-env-block-or-emptyp

(defthm emptyp-of-env-block-fix-to-not-env-block-or-emptyp
  (equal (omap::emptyp (env-block-fix x))
         (or (not (env-blockp x))
             (omap::emptyp x))))

Function: env-block-equiv$inline

(defun env-block-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (env-blockp acl2::x)
                              (env-blockp acl2::y))))
  (equal (env-block-fix acl2::x)
         (env-block-fix acl2::y)))

Theorem: env-block-equiv-is-an-equivalence

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

Theorem: env-block-equiv-implies-equal-env-block-fix-1

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

Theorem: env-block-fix-under-env-block-equiv

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

Theorem: equal-of-env-block-fix-1-forward-to-env-block-equiv

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

Theorem: equal-of-env-block-fix-2-forward-to-env-block-equiv

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

Theorem: env-block-equiv-of-env-block-fix-1-forward

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

Theorem: env-block-equiv-of-env-block-fix-2-forward

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