		Search-engine friendly clone of the ACL2 documentation.

Jbinop

Jbinop-fix

Fixing function for jbinop structures.

Signature
(jbinop-fix x) → new-x
Arguments: x — Guard (jbinopp x).
Returns: new-x — Type (jbinopp new-x).

Definitions and Theorems

Function: jbinop-fix$inline

(defun jbinop-fix$inline (x)
  (declare (xargs :guard (jbinopp x)))
  (let ((__function__ 'jbinop-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (jbinop-kind x)
           (:mul (cons :mul (list)))
           (:div (cons :div (list)))
           (:rem (cons :rem (list)))
           (:add (cons :add (list)))
           (:sub (cons :sub (list)))
           (:shl (cons :shl (list)))
           (:sshr (cons :sshr (list)))
           (:ushr (cons :ushr (list)))
           (:lt (cons :lt (list)))
           (:gt (cons :gt (list)))
           (:le (cons :le (list)))
           (:ge (cons :ge (list)))
           (:eq (cons :eq (list)))
           (:ne (cons :ne (list)))
           (:and (cons :and (list)))
           (:xor (cons :xor (list)))
           (:ior (cons :ior (list)))
           (:condand (cons :condand (list)))
           (:condor (cons :condor (list)))
           (:asg (cons :asg (list)))
           (:asg-mul (cons :asg-mul (list)))
           (:asg-div (cons :asg-div (list)))
           (:asg-rem (cons :asg-rem (list)))
           (:asg-add (cons :asg-add (list)))
           (:asg-sub (cons :asg-sub (list)))
           (:asg-shl (cons :asg-shl (list)))
           (:asg-sshr (cons :asg-sshr (list)))
           (:asg-ushr (cons :asg-ushr (list)))
           (:asg-and (cons :asg-and (list)))
           (:asg-xor (cons :asg-xor (list)))
           (:asg-ior (cons :asg-ior (list))))
         :exec x)))

Theorem: jbinopp-of-jbinop-fix

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

Theorem: jbinop-fix-when-jbinopp

(defthm jbinop-fix-when-jbinopp
  (implies (jbinopp x)
           (equal (jbinop-fix x) x)))

Function: jbinop-equiv$inline

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

Theorem: jbinop-equiv-is-an-equivalence

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

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

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

Theorem: jbinop-fix-under-jbinop-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-jbinop-fix

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