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Simpadd0-gout

Simpadd0-gout-fix

Fixing function for simpadd0-gout structures.

Signature
(simpadd0-gout-fix x) → new-x
Arguments: x — Guard (simpadd0-goutp x).
Returns: new-x — Type (simpadd0-goutp new-x).

Definitions and Theorems

Function: simpadd0-gout-fix$inline

(defun simpadd0-gout-fix$inline (x)
 (declare (xargs :guard (simpadd0-goutp x)))
 (let ((__function__ 'simpadd0-gout-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (b*
    ((events
         (acl2::pseudo-event-form-list-fix (cdr (std::da-nth 0 x))))
     (thm-name (acl2::symbol-fix (cdr (std::da-nth 1 x))))
     (thm-index (pos-fix (cdr (std::da-nth 2 x))))
     (names-to-avoid
          (acl2::symbol-list-fix (cdr (std::da-nth 3 x))))
     (vartys (ident-type-map-fix (cdr (std::da-nth 4 x))))
     (diffp (acl2::bool-fix (cdr (std::da-nth 5 x)))))
    (list (cons 'events events)
          (cons 'thm-name thm-name)
          (cons 'thm-index thm-index)
          (cons 'names-to-avoid names-to-avoid)
          (cons 'vartys vartys)
          (cons 'diffp diffp)))
   :exec x)))

Theorem: simpadd0-goutp-of-simpadd0-gout-fix

(defthm simpadd0-goutp-of-simpadd0-gout-fix
  (b* ((new-x (simpadd0-gout-fix$inline x)))
    (simpadd0-goutp new-x))
  :rule-classes :rewrite)

Theorem: simpadd0-gout-fix-when-simpadd0-goutp

(defthm simpadd0-gout-fix-when-simpadd0-goutp
  (implies (simpadd0-goutp x)
           (equal (simpadd0-gout-fix x) x)))

Function: simpadd0-gout-equiv$inline

(defun simpadd0-gout-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (simpadd0-goutp acl2::x)
                              (simpadd0-goutp acl2::y))))
  (equal (simpadd0-gout-fix acl2::x)
         (simpadd0-gout-fix acl2::y)))

Theorem: simpadd0-gout-equiv-is-an-equivalence

(defthm simpadd0-gout-equiv-is-an-equivalence
  (and (booleanp (simpadd0-gout-equiv x y))
       (simpadd0-gout-equiv x x)
       (implies (simpadd0-gout-equiv x y)
                (simpadd0-gout-equiv y x))
       (implies (and (simpadd0-gout-equiv x y)
                     (simpadd0-gout-equiv y z))
                (simpadd0-gout-equiv x z)))
  :rule-classes (:equivalence))

Theorem: simpadd0-gout-equiv-implies-equal-simpadd0-gout-fix-1

(defthm simpadd0-gout-equiv-implies-equal-simpadd0-gout-fix-1
  (implies (simpadd0-gout-equiv acl2::x x-equiv)
           (equal (simpadd0-gout-fix acl2::x)
                  (simpadd0-gout-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: simpadd0-gout-fix-under-simpadd0-gout-equiv

(defthm simpadd0-gout-fix-under-simpadd0-gout-equiv
  (simpadd0-gout-equiv (simpadd0-gout-fix acl2::x)
                       acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-simpadd0-gout-fix-1-forward-to-simpadd0-gout-equiv

(defthm equal-of-simpadd0-gout-fix-1-forward-to-simpadd0-gout-equiv
  (implies (equal (simpadd0-gout-fix acl2::x)
                  acl2::y)
           (simpadd0-gout-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-simpadd0-gout-fix-2-forward-to-simpadd0-gout-equiv

(defthm equal-of-simpadd0-gout-fix-2-forward-to-simpadd0-gout-equiv
  (implies (equal acl2::x (simpadd0-gout-fix acl2::y))
           (simpadd0-gout-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: simpadd0-gout-equiv-of-simpadd0-gout-fix-1-forward

(defthm simpadd0-gout-equiv-of-simpadd0-gout-fix-1-forward
  (implies (simpadd0-gout-equiv (simpadd0-gout-fix acl2::x)
                                acl2::y)
           (simpadd0-gout-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: simpadd0-gout-equiv-of-simpadd0-gout-fix-2-forward

(defthm simpadd0-gout-equiv-of-simpadd0-gout-fix-2-forward
  (implies (simpadd0-gout-equiv acl2::x (simpadd0-gout-fix acl2::y))
           (simpadd0-gout-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)