		Search-engine friendly clone of the ACL2 documentation.

Int+denv

Int+denv-fix

Fixing function for int+denv structures.

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

Definitions and Theorems

Function: int+denv-fix$inline

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

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

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

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

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

Function: int+denv-equiv$inline

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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