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• Output-item

Output-item-fix

Fixing function for output-item structures.

Signature

(output-item-fix x) → new-x

Arguments

x — Guard (output-itemp x).

Returns

new-x — Type (output-itemp new-x).

Definitions and Theorems

Function: output-item-fix$inline

(defun output-item-fix$inline (x)
 (declare (xargs :guard (output-itemp x)))
 (let ((__function__ 'output-item-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (b* ((get (output-expression-fix (cdr (std::da-nth 0 (cdr x))))))
     (cons :output-item (list (cons 'get get))))
   :exec x)))

Theorem: output-itemp-of-output-item-fix

(defthm output-itemp-of-output-item-fix
  (b* ((new-x (output-item-fix$inline x)))
    (output-itemp new-x))
  :rule-classes :rewrite)

Theorem: output-item-fix-when-output-itemp

(defthm output-item-fix-when-output-itemp
  (implies (output-itemp x)
           (equal (output-item-fix x) x)))

Function: output-item-equiv$inline

(defun output-item-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (output-itemp acl2::x)
                              (output-itemp acl2::y))))
  (equal (output-item-fix acl2::x)
         (output-item-fix acl2::y)))

Theorem: output-item-equiv-is-an-equivalence

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

Theorem: output-item-equiv-implies-equal-output-item-fix-1

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

Theorem: output-item-fix-under-output-item-equiv

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

Theorem: equal-of-output-item-fix-1-forward-to-output-item-equiv

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

Theorem: equal-of-output-item-fix-2-forward-to-output-item-equiv

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

Theorem: output-item-equiv-of-output-item-fix-1-forward

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

Theorem: output-item-equiv-of-output-item-fix-2-forward

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