		Search-engine friendly clone of the ACL2 documentation.

Rulename-set

Rulename-sfix

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

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

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

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

Theorem: rulename-setp-of-rulename-sfix

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

Theorem: rulename-sfix-when-rulename-setp

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

Theorem: emptyp-rulename-sfix

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

Theorem: emptyp-of-rulename-sfix

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

Function: rulename-sequiv$inline

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

Theorem: rulename-sequiv-is-an-equivalence

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

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

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

Theorem: rulename-sfix-under-rulename-sequiv

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

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

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

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

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

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

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

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

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