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Linkage

Linkage-fix

Fixing function for linkage structures.

Signature
(linkage-fix x) → new-x
Arguments: x — Guard (linkagep x).
Returns: new-x — Type (linkagep new-x).

Definitions and Theorems

Function: linkage-fix$inline

(defun linkage-fix$inline (x)
  (declare (xargs :guard (linkagep x)))
  (let ((__function__ 'linkage-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (linkage-kind x)
           (:external (cons :external (list)))
           (:internal (cons :internal (list)))
           (:none (cons :none (list))))
         :exec x)))

Theorem: linkagep-of-linkage-fix

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

Theorem: linkage-fix-when-linkagep

(defthm linkage-fix-when-linkagep
  (implies (linkagep x)
           (equal (linkage-fix x) x)))

Function: linkage-equiv$inline

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

Theorem: linkage-equiv-is-an-equivalence

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

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

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

Theorem: linkage-fix-under-linkage-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-linkage-fix

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