		Search-engine friendly clone of the ACL2 documentation.

Function-denv

Function-denv-fix

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

Signature
(function-denv-fix x) → *
Arguments: x — Guard (function-denvp x).

Definitions and Theorems

Function: function-denv-fix

(defun function-denv-fix (x)
  (declare (xargs :guard (function-denvp x)))
  (mbe :logic (if (function-denvp x) x nil)
       :exec x))

Theorem: function-denvp-of-function-denv-fix

(defthm function-denvp-of-function-denv-fix
  (function-denvp (function-denv-fix x)))

Theorem: function-denv-fix-when-function-denvp

(defthm function-denv-fix-when-function-denvp
  (implies (function-denvp x)
           (equal (function-denv-fix x) x)))

Theorem: emptyp-function-denv-fix

(defthm emptyp-function-denv-fix
  (implies (or (omap::emptyp x)
               (not (function-denvp x)))
           (omap::emptyp (function-denv-fix x))))

Theorem: emptyp-of-function-denv-fix-to-not-function-denv-or-emptyp

(defthm emptyp-of-function-denv-fix-to-not-function-denv-or-emptyp
  (equal (omap::emptyp (function-denv-fix x))
         (or (not (function-denvp x))
             (omap::emptyp x))))

Function: function-denv-equiv$inline

(defun function-denv-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (function-denvp acl2::x)
                              (function-denvp acl2::y))))
  (equal (function-denv-fix acl2::x)
         (function-denv-fix acl2::y)))

Theorem: function-denv-equiv-is-an-equivalence

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

Theorem: function-denv-equiv-implies-equal-function-denv-fix-1

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

Theorem: function-denv-fix-under-function-denv-equiv

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

Theorem: equal-of-function-denv-fix-1-forward-to-function-denv-equiv

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

Theorem: equal-of-function-denv-fix-2-forward-to-function-denv-equiv

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

Theorem: function-denv-equiv-of-function-denv-fix-1-forward

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

Theorem: function-denv-equiv-of-function-denv-fix-2-forward

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