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Ex-outs

Ex-outs-fix

Fixing function for ex-outs structures.

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

Definitions and Theorems

Function: ex-outs-fix$inline

(defun ex-outs-fix$inline (x)
  (declare (xargs :guard (ex-outs-p x)))
  (let ((acl2::__function__ 'ex-outs-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (b* ((expanded-term-lst
                   (pseudo-term-list-fix (cdr (std::da-nth 0 x))))
              (expanded-fn-lst
                   (pseudo-term-alist-fix (cdr (std::da-nth 1 x)))))
           (list (cons 'expanded-term-lst
                       expanded-term-lst)
                 (cons 'expanded-fn-lst
                       expanded-fn-lst)))
         :exec x)))

Theorem: ex-outs-p-of-ex-outs-fix

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

Theorem: ex-outs-fix-when-ex-outs-p

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

Function: ex-outs-equiv$inline

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

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

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

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

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

Theorem: ex-outs-fix-under-ex-outs-equiv

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

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

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

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

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

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

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

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

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