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Binary-op

Binary-op-fix

Fixing function for binary-op structures.

Signature
(binary-op-fix x) → new-x
Arguments: x — Guard (binary-opp x).
Returns: new-x — Type (binary-opp new-x).

Definitions and Theorems

Function: binary-op-fix$inline

(defun binary-op-fix$inline (x)
  (declare (xargs :guard (binary-opp x)))
  (let ((__function__ 'binary-op-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (binary-op-kind x)
           (:eq (cons :eq (list)))
           (:ne (cons :ne (list)))
           (:gt (cons :gt (list)))
           (:ge (cons :ge (list)))
           (:lt (cons :lt (list)))
           (:le (cons :le (list)))
           (:and (cons :and (list)))
           (:or (cons :or (list)))
           (:implies (cons :implies (list)))
           (:implied (cons :implied (list)))
           (:iff (cons :iff (list)))
           (:add (cons :add (list)))
           (:sub (cons :sub (list)))
           (:mul (cons :mul (list)))
           (:div (cons :div (list)))
           (:rem (cons :rem (list))))
         :exec x)))

Theorem: binary-opp-of-binary-op-fix

(defthm binary-opp-of-binary-op-fix
  (b* ((new-x (binary-op-fix$inline x)))
    (binary-opp new-x))
  :rule-classes :rewrite)

Theorem: binary-op-fix-when-binary-opp

(defthm binary-op-fix-when-binary-opp
  (implies (binary-opp x)
           (equal (binary-op-fix x) x)))

Function: binary-op-equiv$inline

(defun binary-op-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (binary-opp acl2::x)
                              (binary-opp acl2::y))))
  (equal (binary-op-fix acl2::x)
         (binary-op-fix acl2::y)))

Theorem: binary-op-equiv-is-an-equivalence

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

Theorem: binary-op-equiv-implies-equal-binary-op-fix-1

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

Theorem: binary-op-fix-under-binary-op-equiv

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

Theorem: equal-of-binary-op-fix-1-forward-to-binary-op-equiv

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

Theorem: equal-of-binary-op-fix-2-forward-to-binary-op-equiv

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

Theorem: binary-op-equiv-of-binary-op-fix-1-forward

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

Theorem: binary-op-equiv-of-binary-op-fix-2-forward

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

Theorem: binary-op-kind$inline-of-binary-op-fix-x

(defthm binary-op-kind$inline-of-binary-op-fix-x
  (equal (binary-op-kind$inline (binary-op-fix x))
         (binary-op-kind$inline x)))

Theorem: binary-op-kind$inline-binary-op-equiv-congruence-on-x

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

Theorem: consp-of-binary-op-fix

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