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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)))
  (mbe :logic
       (case (binop-kind x)
         (:mul (cons :mul nil))
         (:div (cons :div nil))
         (:rem (cons :rem nil))
         (:add (cons :add nil))
         (:sub (cons :sub nil))
         (:shl (cons :shl nil))
         (:shr (cons :shr nil))
         (:lt (cons :lt nil))
         (:gt (cons :gt nil))
         (:le (cons :le nil))
         (:ge (cons :ge nil))
         (:eq (cons :eq nil))
         (:ne (cons :ne nil))
         (:bitand (cons :bitand nil))
         (:bitxor (cons :bitxor nil))
         (:bitior (cons :bitior nil))
         (:logand (cons :logand nil))
         (:logor (cons :logor nil))
         (:asg (cons :asg nil))
         (:asg-mul (cons :asg-mul nil))
         (:asg-div (cons :asg-div nil))
         (:asg-rem (cons :asg-rem nil))
         (:asg-add (cons :asg-add nil))
         (:asg-sub (cons :asg-sub nil))
         (:asg-shl (cons :asg-shl nil))
         (:asg-shr (cons :asg-shr nil))
         (:asg-and (cons :asg-and nil))
         (:asg-xor (cons :asg-xor nil))
         (:asg-ior (cons :asg-ior nil)))
       :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)

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

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

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

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