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Ext-declon

Ext-declon-fix

Fixing function for ext-declon structures.

Signature
(ext-declon-fix x) → new-x
Arguments: x — Guard (ext-declonp x).
Returns: new-x — Type (ext-declonp new-x).

Definitions and Theorems

Function: ext-declon-fix$inline

(defun ext-declon-fix$inline (x)
  (declare (xargs :guard (ext-declonp x)))
  (let ((__function__ 'ext-declon-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (ext-declon-kind x)
           (:fundef (b* ((get (fundef-fix (std::da-nth 0 (cdr x)))))
                      (cons :fundef (list get))))
           (:fun-declon
                (b* ((get (fun-declon-fix (std::da-nth 0 (cdr x)))))
                  (cons :fun-declon (list get))))
           (:obj-declon
                (b* ((get (obj-declon-fix (std::da-nth 0 (cdr x)))))
                  (cons :obj-declon (list get))))
           (:tag-declon
                (b* ((get (tag-declon-fix (std::da-nth 0 (cdr x)))))
                  (cons :tag-declon (list get)))))
         :exec x)))

Theorem: ext-declonp-of-ext-declon-fix

(defthm ext-declonp-of-ext-declon-fix
  (b* ((new-x (ext-declon-fix$inline x)))
    (ext-declonp new-x))
  :rule-classes :rewrite)

Theorem: ext-declon-fix-when-ext-declonp

(defthm ext-declon-fix-when-ext-declonp
  (implies (ext-declonp x)
           (equal (ext-declon-fix x) x)))

Function: ext-declon-equiv$inline

(defun ext-declon-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (ext-declonp acl2::x)
                              (ext-declonp acl2::y))))
  (equal (ext-declon-fix acl2::x)
         (ext-declon-fix acl2::y)))

Theorem: ext-declon-equiv-is-an-equivalence

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

Theorem: ext-declon-equiv-implies-equal-ext-declon-fix-1

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

Theorem: ext-declon-fix-under-ext-declon-equiv

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

Theorem: equal-of-ext-declon-fix-1-forward-to-ext-declon-equiv

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

Theorem: equal-of-ext-declon-fix-2-forward-to-ext-declon-equiv

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

Theorem: ext-declon-equiv-of-ext-declon-fix-1-forward

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

Theorem: ext-declon-equiv-of-ext-declon-fix-2-forward

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

Theorem: ext-declon-kind$inline-of-ext-declon-fix-x

(defthm ext-declon-kind$inline-of-ext-declon-fix-x
  (equal (ext-declon-kind$inline (ext-declon-fix x))
         (ext-declon-kind$inline x)))

Theorem: ext-declon-kind$inline-ext-declon-equiv-congruence-on-x

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

Theorem: consp-of-ext-declon-fix

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