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Ident-senv

Ident-senv-fix

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

Signature
(ident-senv-fix x) → *
Arguments: x — Guard (ident-senvp x).

Definitions and Theorems

Function: ident-senv-fix

(defun ident-senv-fix (x)
  (declare (xargs :guard (ident-senvp x)))
  (mbe :logic (if (ident-senvp x) x nil)
       :exec x))

Theorem: ident-senvp-of-ident-senv-fix

(defthm ident-senvp-of-ident-senv-fix
  (ident-senvp (ident-senv-fix x)))

Theorem: ident-senv-fix-when-ident-senvp

(defthm ident-senv-fix-when-ident-senvp
  (implies (ident-senvp x)
           (equal (ident-senv-fix x) x)))

Theorem: emptyp-ident-senv-fix

(defthm emptyp-ident-senv-fix
  (implies (or (omap::emptyp x)
               (not (ident-senvp x)))
           (omap::emptyp (ident-senv-fix x))))

Theorem: emptyp-of-ident-senv-fix-to-not-ident-senv-or-emptyp

(defthm emptyp-of-ident-senv-fix-to-not-ident-senv-or-emptyp
  (equal (omap::emptyp (ident-senv-fix x))
         (or (not (ident-senvp x))
             (omap::emptyp x))))

Function: ident-senv-equiv$inline

(defun ident-senv-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (ident-senvp acl2::x)
                              (ident-senvp acl2::y))))
  (equal (ident-senv-fix acl2::x)
         (ident-senv-fix acl2::y)))

Theorem: ident-senv-equiv-is-an-equivalence

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

Theorem: ident-senv-equiv-implies-equal-ident-senv-fix-1

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

Theorem: ident-senv-fix-under-ident-senv-equiv

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

Theorem: equal-of-ident-senv-fix-1-forward-to-ident-senv-equiv

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

Theorem: equal-of-ident-senv-fix-2-forward-to-ident-senv-equiv

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

Theorem: ident-senv-equiv-of-ident-senv-fix-1-forward

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

Theorem: ident-senv-equiv-of-ident-senv-fix-2-forward

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