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Expression

Expression-fix

Fixing function for expression structures.

Signature
(expression-fix x) → new-x
Arguments: x — Guard (expressionp x).
Returns: new-x — Type (expressionp new-x).

Definitions and Theorems

Function: expression-fix$inline

(defun expression-fix$inline (x)
 (declare (xargs :guard (expressionp x)))
 (let ((__function__ 'expression-fix))
  (declare (ignorable __function__))
  (mbe :logic
       (case (expression-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* ((arg1 (expression-fix (std::da-nth 0 (cdr x))))
                    (arg2 (expression-fix (std::da-nth 1 (cdr x)))))
                 (cons :add (list arg1 arg2))))
         (:mul (b* ((arg1 (expression-fix (std::da-nth 0 (cdr x))))
                    (arg2 (expression-fix (std::da-nth 1 (cdr x)))))
                 (cons :mul (list arg1 arg2)))))
       :exec x)))

Theorem: expressionp-of-expression-fix

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

Theorem: expression-fix-when-expressionp

(defthm expression-fix-when-expressionp
  (implies (expressionp x)
           (equal (expression-fix x) x)))

Function: expression-equiv$inline

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

Theorem: expression-equiv-is-an-equivalence

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

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

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

Theorem: expression-fix-under-expression-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-expression-fix

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