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S4vecs

S4vec-fix

Signature
(s4vec-fix x) → new-x
Arguments: x — Guard (s4vec-p x).
Returns: new-x — Type (s4vec-p new-x).

Definitions and Theorems

Function: s4vec-fix

(defun s4vec-fix (x)
  (declare (xargs :guard (s4vec-p x)))
  (let ((__function__ 's4vec-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             (if (integerp x) x '(-1 . 0))
           (b* ((car (sparseint-fix (car x)))
                ((unless (cdr x))
                 (if (integerp car) car (list car)))
                (cdr (sparseint-fix (cdr x)))
                ((when (equal car cdr))
                 (if (integerp car) car (list car))))
             (cons car cdr)))
         :exec x)))

Theorem: s4vec-p-of-s4vec-fix

(defthm s4vec-p-of-s4vec-fix
  (b* ((new-x (s4vec-fix x)))
    (s4vec-p new-x))
  :rule-classes :rewrite)

Theorem: s4vec-fix-when-s4vec-p

(defthm s4vec-fix-when-s4vec-p
  (b* ((?new-x (s4vec-fix x)))
    (implies (s4vec-p x) (equal new-x x))))

Function: s4vec-equiv$inline

(defun s4vec-equiv$inline (x y)
  (declare (xargs :guard (and (s4vec-p x) (s4vec-p y))))
  (equal (s4vec-fix x) (s4vec-fix y)))

Theorem: s4vec-equiv-is-an-equivalence

(defthm s4vec-equiv-is-an-equivalence
  (and (booleanp (s4vec-equiv x y))
       (s4vec-equiv x x)
       (implies (s4vec-equiv x y)
                (s4vec-equiv y x))
       (implies (and (s4vec-equiv x y)
                     (s4vec-equiv y z))
                (s4vec-equiv x z)))
  :rule-classes (:equivalence))

Theorem: s4vec-equiv-implies-equal-s4vec-fix-1

(defthm s4vec-equiv-implies-equal-s4vec-fix-1
  (implies (s4vec-equiv x x-equiv)
           (equal (s4vec-fix x)
                  (s4vec-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: s4vec-fix-under-s4vec-equiv

(defthm s4vec-fix-under-s4vec-equiv
  (s4vec-equiv (s4vec-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-s4vec-fix-1-forward-to-s4vec-equiv

(defthm equal-of-s4vec-fix-1-forward-to-s4vec-equiv
  (implies (equal (s4vec-fix x) y)
           (s4vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-s4vec-fix-2-forward-to-s4vec-equiv

(defthm equal-of-s4vec-fix-2-forward-to-s4vec-equiv
  (implies (equal x (s4vec-fix y))
           (s4vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: s4vec-equiv-of-s4vec-fix-1-forward

(defthm s4vec-equiv-of-s4vec-fix-1-forward
  (implies (s4vec-equiv (s4vec-fix x) y)
           (s4vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: s4vec-equiv-of-s4vec-fix-2-forward

(defthm s4vec-equiv-of-s4vec-fix-2-forward
  (implies (s4vec-equiv x (s4vec-fix y))
           (s4vec-equiv x y))
  :rule-classes :forward-chaining)

Theorem: s4vec-fix-of-s4vec-fix-x

(defthm s4vec-fix-of-s4vec-fix-x
  (equal (s4vec-fix (s4vec-fix x))
         (s4vec-fix x)))

Theorem: s4vec-fix-s4vec-equiv-congruence-on-x

(defthm s4vec-fix-s4vec-equiv-congruence-on-x
  (implies (s4vec-equiv x x-equiv)
           (equal (s4vec-fix x)
                  (s4vec-fix x-equiv)))
  :rule-classes :congruence)