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Constrel-set

Constrel-sfix

(constrel-sfix x) is a usual ACL2::fty set fixing function.

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

In the logic, we apply constrel-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: constrel-sfix

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

Theorem: constrel-setp-of-constrel-sfix

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

Theorem: constrel-sfix-when-constrel-setp

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

Theorem: emptyp-constrel-sfix

(defthm emptyp-constrel-sfix
  (implies (or (emptyp x) (not (constrel-setp x)))
           (emptyp (constrel-sfix x))))

Theorem: emptyp-of-constrel-sfix

(defthm emptyp-of-constrel-sfix
  (equal (emptyp (constrel-sfix x))
         (or (not (constrel-setp x))
             (emptyp x))))

Function: constrel-sequiv$inline

(defun constrel-sequiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (constrel-setp acl2::x)
                              (constrel-setp acl2::y))))
  (equal (constrel-sfix acl2::x)
         (constrel-sfix acl2::y)))

Theorem: constrel-sequiv-is-an-equivalence

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

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

(defthm constrel-sequiv-implies-equal-constrel-sfix-1
  (implies (constrel-sequiv acl2::x x-equiv)
           (equal (constrel-sfix acl2::x)
                  (constrel-sfix x-equiv)))
  :rule-classes (:congruence))

Theorem: constrel-sfix-under-constrel-sequiv

(defthm constrel-sfix-under-constrel-sequiv
  (constrel-sequiv (constrel-sfix acl2::x)
                   acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

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

(defthm equal-of-constrel-sfix-1-forward-to-constrel-sequiv
  (implies (equal (constrel-sfix acl2::x) acl2::y)
           (constrel-sequiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

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

(defthm equal-of-constrel-sfix-2-forward-to-constrel-sequiv
  (implies (equal acl2::x (constrel-sfix acl2::y))
           (constrel-sequiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

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

(defthm constrel-sequiv-of-constrel-sfix-1-forward
  (implies (constrel-sequiv (constrel-sfix acl2::x)
                            acl2::y)
           (constrel-sequiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

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

(defthm constrel-sequiv-of-constrel-sfix-2-forward
  (implies (constrel-sequiv acl2::x (constrel-sfix acl2::y))
           (constrel-sequiv acl2::x acl2::y))
  :rule-classes :forward-chaining)