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Fsm-fix

Fixing function for fsm structures.

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

Definitions and Theorems

Function: fsm-fix$inline

(defun fsm-fix$inline (x)
  (declare (xargs :guard (fsm-p x)))
  (let ((__function__ 'fsm-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((values (svex-alist-fix (cdr (std::da-nth 0 x))))
              (nextstate (svex-alist-fix (cdr (std::da-nth 1 x)))))
           (list (cons 'values values)
                 (cons 'nextstate nextstate)))
         :exec x)))

Theorem: fsm-p-of-fsm-fix

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

Theorem: fsm-fix-when-fsm-p

(defthm fsm-fix-when-fsm-p
  (implies (fsm-p x)
           (equal (fsm-fix x) x)))

Function: fsm-equiv$inline

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

Theorem: fsm-equiv-is-an-equivalence

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

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

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

Theorem: fsm-fix-under-fsm-equiv

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

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

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

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

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

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

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

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

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