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Toplevel

Toplevel-fix

Fixing function for toplevel structures.

Signature
(toplevel-fix x) → new-x
Arguments: x — Guard (toplevelp x).
Returns: new-x — Type (toplevelp new-x).

Definitions and Theorems

Function: toplevel-fix$inline

(defun toplevel-fix$inline (x)
 (declare (xargs :guard (toplevelp x)))
 (let ((__function__ 'toplevel-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (case (toplevel-kind x)
    (:type (b* ((get (type-definition-fix (std::da-nth 0 (cdr x)))))
             (cons :type (list get))))
    (:types (b* ((get (type-recursion-fix (std::da-nth 0 (cdr x)))))
              (cons :types (list get))))
    (:function
       (b* ((get (function-definition-fix (std::da-nth 0 (cdr x)))))
         (cons :function (list get))))
    (:functions
        (b* ((get (function-recursion-fix (std::da-nth 0 (cdr x)))))
          (cons :functions (list get))))
    (:specification
      (b*
        ((get (function-specification-fix (std::da-nth 0 (cdr x)))))
        (cons :specification (list get))))
    (:theorem (b* ((get (theorem-fix (std::da-nth 0 (cdr x)))))
                (cons :theorem (list get))))
    (:transform (b* ((get (transform-fix (std::da-nth 0 (cdr x)))))
                  (cons :transform (list get)))))
   :exec x)))

Theorem: toplevelp-of-toplevel-fix

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

Theorem: toplevel-fix-when-toplevelp

(defthm toplevel-fix-when-toplevelp
  (implies (toplevelp x)
           (equal (toplevel-fix x) x)))

Function: toplevel-equiv$inline

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

Theorem: toplevel-equiv-is-an-equivalence

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

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

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

Theorem: toplevel-fix-under-toplevel-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-toplevel-fix

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