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Aexp

Aexp-fix

Fixing function for aexp structures.

Signature
(aexp-fix x) → new-x
Arguments: x — Guard (aexpp x).
Returns: new-x — Type (aexpp new-x).

Definitions and Theorems

Function: aexp-fix$inline

(defun aexp-fix$inline (x)
  (declare (xargs :guard (aexpp x)))
  (let ((__function__ 'aexp-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (aexp-kind x)
           (:const (b* ((value (ifix (std::da-nth 0 (cdr x)))))
                     (cons :const (list value))))
           (:var (b* ((name (str-fix (std::da-nth 0 (cdr x)))))
                   (cons :var (list name))))
           (:add (b* ((left (aexp-fix (std::da-nth 0 (cdr x))))
                      (right (aexp-fix (std::da-nth 1 (cdr x)))))
                   (cons :add (list left right))))
           (:mul (b* ((left (aexp-fix (std::da-nth 0 (cdr x))))
                      (right (aexp-fix (std::da-nth 1 (cdr x)))))
                   (cons :mul (list left right)))))
         :exec x)))

Theorem: aexpp-of-aexp-fix

(defthm aexpp-of-aexp-fix
  (b* ((new-x (aexp-fix$inline x)))
    (aexpp new-x))
  :rule-classes :rewrite)

Theorem: aexp-fix-when-aexpp

(defthm aexp-fix-when-aexpp
  (implies (aexpp x)
           (equal (aexp-fix x) x)))

Function: aexp-equiv$inline

(defun aexp-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (aexpp acl2::x) (aexpp acl2::y))))
  (equal (aexp-fix acl2::x)
         (aexp-fix acl2::y)))

Theorem: aexp-equiv-is-an-equivalence

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

Theorem: aexp-equiv-implies-equal-aexp-fix-1

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

Theorem: aexp-fix-under-aexp-equiv

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

Theorem: equal-of-aexp-fix-1-forward-to-aexp-equiv

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

Theorem: equal-of-aexp-fix-2-forward-to-aexp-equiv

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

Theorem: aexp-equiv-of-aexp-fix-1-forward

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

Theorem: aexp-equiv-of-aexp-fix-2-forward

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

Theorem: aexp-kind$inline-of-aexp-fix-x

(defthm aexp-kind$inline-of-aexp-fix-x
  (equal (aexp-kind$inline (aexp-fix x))
         (aexp-kind$inline x)))

Theorem: aexp-kind$inline-aexp-equiv-congruence-on-x

(defthm aexp-kind$inline-aexp-equiv-congruence-on-x
  (implies (aexp-equiv x x-equiv)
           (equal (aexp-kind$inline x)
                  (aexp-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-aexp-fix

(defthm consp-of-aexp-fix
  (consp (aexp-fix x))
  :rule-classes :type-prescription)