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Pexprs-gout

Pexprs-gout-fix

Fixing function for pexprs-gout structures.

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

Definitions and Theorems

Function: pexprs-gout-fix$inline

(defun pexprs-gout-fix$inline (x)
 (declare (xargs :guard (pexprs-goutp x)))
 (let ((__function__ 'pexprs-gout-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (b*
    ((exprs (expr-list-fix (cdr (std::da-nth 0 x))))
     (types (type-list-fix (cdr (std::da-nth 1 x))))
     (terms (pseudo-term-list-fix (cdr (std::da-nth 2 x))))
     (events
         (acl2::pseudo-event-form-list-fix (cdr (std::da-nth 3 x))))
     (thm-names (symbol-list-fix (cdr (std::da-nth 4 x))))
     (thm-index (acl2::pos-fix (cdr (std::da-nth 5 x))))
     (names-to-avoid (symbol-list-fix (cdr (std::da-nth 6 x)))))
    (list (cons 'exprs exprs)
          (cons 'types types)
          (cons 'terms terms)
          (cons 'events events)
          (cons 'thm-names thm-names)
          (cons 'thm-index thm-index)
          (cons 'names-to-avoid names-to-avoid)))
   :exec x)))

Theorem: pexprs-goutp-of-pexprs-gout-fix

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

Theorem: pexprs-gout-fix-when-pexprs-goutp

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

Function: pexprs-gout-equiv$inline

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

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

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

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

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

Theorem: pexprs-gout-fix-under-pexprs-gout-equiv

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

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

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

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

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

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

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

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

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