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Bexp

Bexp-fix

Fixing function for bexp structures.

Signature
(bexp-fix x) → new-x
Arguments: x — Guard (bexpp x).
Returns: new-x — Type (bexpp new-x).

Definitions and Theorems

Function: bexp-fix$inline

(defun bexp-fix$inline (x)
 (declare (xargs :guard (bexpp x)))
 (let ((__function__ 'bexp-fix))
  (declare (ignorable __function__))
  (mbe
    :logic
    (case (bexp-kind x)
      (:const (b* ((value (acl2::bool-fix (std::da-nth 0 (cdr x)))))
                (cons :const (list value))))
      (:equal (b* ((left (aexp-fix (std::da-nth 0 (cdr x))))
                   (right (aexp-fix (std::da-nth 1 (cdr x)))))
                (cons :equal (list left right))))
      (:less (b* ((left (aexp-fix (std::da-nth 0 (cdr x))))
                  (right (aexp-fix (std::da-nth 1 (cdr x)))))
               (cons :less (list left right))))
      (:not (b* ((arg (bexp-fix (std::da-nth 0 (cdr x)))))
              (cons :not (list arg))))
      (:and (b* ((left (bexp-fix (std::da-nth 0 (cdr x))))
                 (right (bexp-fix (std::da-nth 1 (cdr x)))))
              (cons :and (list left right)))))
    :exec x)))

Theorem: bexpp-of-bexp-fix

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

Theorem: bexp-fix-when-bexpp

(defthm bexp-fix-when-bexpp
  (implies (bexpp x)
           (equal (bexp-fix x) x)))

Function: bexp-equiv$inline

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

Theorem: bexp-equiv-is-an-equivalence

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

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

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

Theorem: bexp-fix-under-bexp-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-bexp-fix

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