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Finalize-input

Finalize-input-fix

Fixing function for finalize-input structures.

Signature
(finalize-input-fix x) → new-x
Arguments: x — Guard (finalize-inputp x).
Returns: new-x — Type (finalize-inputp new-x).

Definitions and Theorems

Function: finalize-input-fix$inline

(defun finalize-input-fix$inline (x)
 (declare (xargs :guard (finalize-inputp x)))
 (let ((__function__ 'finalize-input-fix))
  (declare (ignorable __function__))
  (mbe
      :logic
      (b* ((reg (register-fix (cdr (std::da-nth 0 (cdr x)))))
           (type (finalize-type-fix (cdr (std::da-nth 1 (cdr x))))))
        (cons :finalize-input (list (cons 'reg reg)
                                    (cons 'type type))))
      :exec x)))

Theorem: finalize-inputp-of-finalize-input-fix

(defthm finalize-inputp-of-finalize-input-fix
  (b* ((new-x (finalize-input-fix$inline x)))
    (finalize-inputp new-x))
  :rule-classes :rewrite)

Theorem: finalize-input-fix-when-finalize-inputp

(defthm finalize-input-fix-when-finalize-inputp
  (implies (finalize-inputp x)
           (equal (finalize-input-fix x) x)))

Function: finalize-input-equiv$inline

(defun finalize-input-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (finalize-inputp acl2::x)
                              (finalize-inputp acl2::y))))
  (equal (finalize-input-fix acl2::x)
         (finalize-input-fix acl2::y)))

Theorem: finalize-input-equiv-is-an-equivalence

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

Theorem: finalize-input-equiv-implies-equal-finalize-input-fix-1

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

Theorem: finalize-input-fix-under-finalize-input-equiv

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

Theorem: equal-of-finalize-input-fix-1-forward-to-finalize-input-equiv

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

Theorem: equal-of-finalize-input-fix-2-forward-to-finalize-input-equiv

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

Theorem: finalize-input-equiv-of-finalize-input-fix-1-forward

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

Theorem: finalize-input-equiv-of-finalize-input-fix-2-forward

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