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Pos-set

Pos-sfix

(pos-sfix x) is a usual fty set fixing function.

Signature
(pos-sfix x) → *
Arguments: x — Guard (pos-setp x).

In the logic, we apply pos-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.

Definitions and Theorems

Function: pos-sfix

(defun pos-sfix (x)
  (declare (xargs :guard (pos-setp x)))
  (mbe :logic (if (pos-setp x) x nil)
       :exec x))

Theorem: pos-setp-of-pos-sfix

(defthm pos-setp-of-pos-sfix
  (pos-setp (pos-sfix x)))

Theorem: pos-sfix-when-pos-setp

(defthm pos-sfix-when-pos-setp
  (implies (pos-setp x)
           (equal (pos-sfix x) x)))

Theorem: emptyp-pos-sfix

(defthm emptyp-pos-sfix
  (implies (or (set::emptyp x) (not (pos-setp x)))
           (set::emptyp (pos-sfix x))))

Theorem: emptyp-of-pos-sfix

(defthm emptyp-of-pos-sfix
  (equal (set::emptyp (pos-sfix x))
         (or (not (pos-setp x))
             (set::emptyp x))))

Function: pos-sequiv$inline

(defun pos-sequiv$inline (x y)
  (declare (xargs :guard (and (pos-setp x) (pos-setp y))))
  (equal (pos-sfix x) (pos-sfix y)))

Theorem: pos-sequiv-is-an-equivalence

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

Theorem: pos-sequiv-implies-equal-pos-sfix-1

(defthm pos-sequiv-implies-equal-pos-sfix-1
  (implies (pos-sequiv x x-equiv)
           (equal (pos-sfix x) (pos-sfix x-equiv)))
  :rule-classes (:congruence))

Theorem: pos-sfix-under-pos-sequiv

(defthm pos-sfix-under-pos-sequiv
  (pos-sequiv (pos-sfix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-pos-sfix-1-forward-to-pos-sequiv

(defthm equal-of-pos-sfix-1-forward-to-pos-sequiv
  (implies (equal (pos-sfix x) y)
           (pos-sequiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-pos-sfix-2-forward-to-pos-sequiv

(defthm equal-of-pos-sfix-2-forward-to-pos-sequiv
  (implies (equal x (pos-sfix y))
           (pos-sequiv x y))
  :rule-classes :forward-chaining)

Theorem: pos-sequiv-of-pos-sfix-1-forward

(defthm pos-sequiv-of-pos-sfix-1-forward
  (implies (pos-sequiv (pos-sfix x) y)
           (pos-sequiv x y))
  :rule-classes :forward-chaining)

Theorem: pos-sequiv-of-pos-sfix-2-forward

(defthm pos-sequiv-of-pos-sfix-2-forward
  (implies (pos-sequiv x (pos-sfix y))
           (pos-sequiv x y))
  :rule-classes :forward-chaining)