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Binop

Binop-fix

Fixing function for binop structures.

Signature
(binop-fix x) → new-x
Arguments: x — Guard (binopp x).
Returns: new-x — Type (binopp new-x).

Definitions and Theorems

Function: binop-fix$inline

(defun binop-fix$inline (x)
  (declare (xargs :guard (binopp x)))
  (let ((__function__ 'binop-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (binop-kind x)
           (:and (cons :and (list)))
           (:or (cons :or (list)))
           (:eq (cons :eq (list)))
           (:ne (cons :ne (list)))
           (:ge (cons :ge (list)))
           (:gt (cons :gt (list)))
           (:le (cons :le (list)))
           (:lt (cons :lt (list)))
           (:bitxor (cons :bitxor (list)))
           (:bitior (cons :bitior (list)))
           (:bitand (cons :bitand (list)))
           (:shl (cons :shl (list)))
           (:shr (cons :shr (list)))
           (:add (cons :add (list)))
           (:sub (cons :sub (list)))
           (:mul (cons :mul (list)))
           (:div (cons :div (list)))
           (:rem (cons :rem (list)))
           (:pow (cons :pow (list)))
           (:nand (cons :nand (list)))
           (:nor (cons :nor (list)))
           (:shl-wrapped (cons :shl-wrapped (list)))
           (:shr-wrapped (cons :shr-wrapped (list)))
           (:add-wrapped (cons :add-wrapped (list)))
           (:sub-wrapped (cons :sub-wrapped (list)))
           (:mul-wrapped (cons :mul-wrapped (list)))
           (:div-wrapped (cons :div-wrapped (list)))
           (:rem-wrapped (cons :rem-wrapped (list)))
           (:pow-wrapped (cons :pow-wrapped (list))))
         :exec x)))

Theorem: binopp-of-binop-fix

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

Theorem: binop-fix-when-binopp

(defthm binop-fix-when-binopp
  (implies (binopp x)
           (equal (binop-fix x) x)))

Function: binop-equiv$inline

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

Theorem: binop-equiv-is-an-equivalence

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

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

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

Theorem: binop-fix-under-binop-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-binop-fix

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