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Namevalue+denv

Namevalue+denv-fix

Fixing function for namevalue+denv structures.

Signature
(namevalue+denv-fix x) → new-x
Arguments: x — Guard (namevalue+denv-p x).
Returns: new-x — Type (namevalue+denv-p new-x).

Definitions and Theorems

Function: namevalue+denv-fix$inline

(defun namevalue+denv-fix$inline (x)
 (declare (xargs :guard (namevalue+denv-p x)))
 (let ((__function__ 'namevalue+denv-fix))
  (declare (ignorable __function__))
  (mbe
     :logic
     (b* ((namevalue (name+value-fix (cdr (std::da-nth 0 (cdr x)))))
          (denv (denv-fix (cdr (std::da-nth 1 (cdr x))))))
       (cons :namevalue+denv (list (cons 'namevalue namevalue)
                                   (cons 'denv denv))))
     :exec x)))

Theorem: namevalue+denv-p-of-namevalue+denv-fix

(defthm namevalue+denv-p-of-namevalue+denv-fix
  (b* ((new-x (namevalue+denv-fix$inline x)))
    (namevalue+denv-p new-x))
  :rule-classes :rewrite)

Theorem: namevalue+denv-fix-when-namevalue+denv-p

(defthm namevalue+denv-fix-when-namevalue+denv-p
  (implies (namevalue+denv-p x)
           (equal (namevalue+denv-fix x) x)))

Function: namevalue+denv-equiv$inline

(defun namevalue+denv-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (namevalue+denv-p acl2::x)
                              (namevalue+denv-p acl2::y))))
  (equal (namevalue+denv-fix acl2::x)
         (namevalue+denv-fix acl2::y)))

Theorem: namevalue+denv-equiv-is-an-equivalence

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

Theorem: namevalue+denv-equiv-implies-equal-namevalue+denv-fix-1

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

Theorem: namevalue+denv-fix-under-namevalue+denv-equiv

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

Theorem: equal-of-namevalue+denv-fix-1-forward-to-namevalue+denv-equiv

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

Theorem: equal-of-namevalue+denv-fix-2-forward-to-namevalue+denv-equiv

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

Theorem: namevalue+denv-equiv-of-namevalue+denv-fix-1-forward

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

Theorem: namevalue+denv-equiv-of-namevalue+denv-fix-2-forward

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