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Programdef

Programdef-fix

Fixing function for programdef structures.

Signature
(programdef-fix x) → new-x
Arguments: x — Guard (programdefp x).
Returns: new-x — Type (programdefp new-x).

Definitions and Theorems

Function: programdef-fix$inline

(defun programdef-fix$inline (x)
 (declare (xargs :guard (programdefp x)))
 (let ((__function__ 'programdef-fix))
  (declare (ignorable __function__))
  (mbe
    :logic
    (case (programdef-kind x)
      (:mapping (b* ((get (mapping-fix (std::da-nth 0 (cdr x)))))
                  (cons :mapping (list get))))
      (:interface
           (b* ((get (interface-type-fix (std::da-nth 0 (cdr x)))))
             (cons :interface (list get))))
      (:record (b* ((get (record-type-fix (std::da-nth 0 (cdr x)))))
                 (cons :record (list get))))
      (:closure (b* ((get (closure-fix (std::da-nth 0 (cdr x)))))
                  (cons :closure (list get))))
      (:function (b* ((get (function-fix (std::da-nth 0 (cdr x)))))
                   (cons :function (list get)))))
    :exec x)))

Theorem: programdefp-of-programdef-fix

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

Theorem: programdef-fix-when-programdefp

(defthm programdef-fix-when-programdefp
  (implies (programdefp x)
           (equal (programdef-fix x) x)))

Function: programdef-equiv$inline

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

Theorem: programdef-equiv-is-an-equivalence

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

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

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

Theorem: programdef-fix-under-programdef-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-programdef-fix

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