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Ex-args

Ex-args-fix

Fixing function for ex-args structures.

Signature
(ex-args-fix x) → new-x
Arguments: x — Guard (ex-args-p x).
Returns: new-x — Type (ex-args-p new-x).

Definitions and Theorems

Function: ex-args-fix$inline

(defun ex-args-fix$inline (x)
 (declare (xargs :guard (ex-args-p x)))
 (let ((acl2::__function__ 'ex-args-fix))
  (declare (ignorable acl2::__function__))
  (mbe
    :logic
    (b*
      ((term-lst (pseudo-term-list-fix (cdr (std::da-nth 0 x))))
       (fn-lst (func-alist-fix (cdr (std::da-nth 1 x))))
       (fn-lvls (sym-nat-alist-fix (cdr (std::da-nth 2 x))))
       (wrld-fn-len (nfix (cdr (std::da-nth 3 x))))
       (expand-lst (pseudo-term-alist-fix (cdr (std::da-nth 4 x)))))
      (list (cons 'term-lst term-lst)
            (cons 'fn-lst fn-lst)
            (cons 'fn-lvls fn-lvls)
            (cons 'wrld-fn-len wrld-fn-len)
            (cons 'expand-lst expand-lst)))
    :exec x)))

Theorem: ex-args-p-of-ex-args-fix

(defthm ex-args-p-of-ex-args-fix
  (b* ((new-x (ex-args-fix$inline x)))
    (ex-args-p new-x))
  :rule-classes :rewrite)

Theorem: ex-args-fix-when-ex-args-p

(defthm ex-args-fix-when-ex-args-p
  (implies (ex-args-p x)
           (equal (ex-args-fix x) x)))

Function: ex-args-equiv$inline

(defun ex-args-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (ex-args-p acl2::x)
                              (ex-args-p acl2::y))))
  (equal (ex-args-fix acl2::x)
         (ex-args-fix acl2::y)))

Theorem: ex-args-equiv-is-an-equivalence

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

Theorem: ex-args-equiv-implies-equal-ex-args-fix-1

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

Theorem: ex-args-fix-under-ex-args-equiv

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

Theorem: equal-of-ex-args-fix-1-forward-to-ex-args-equiv

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

Theorem: equal-of-ex-args-fix-2-forward-to-ex-args-equiv

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

Theorem: ex-args-equiv-of-ex-args-fix-1-forward

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

Theorem: ex-args-equiv-of-ex-args-fix-2-forward

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