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• Function-set

Function-set-fix

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

Signature

(function-set-fix x) → *

Arguments

x — Guard (function-setp x).

In the logic, we apply function-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: function-set-fix

(defun function-set-fix (x)
  (declare (xargs :guard (function-setp x)))
  (mbe :logic (if (function-setp x) x nil)
       :exec x))

Theorem: function-setp-of-function-set-fix

(defthm function-setp-of-function-set-fix
  (function-setp (function-set-fix x)))

Theorem: function-set-fix-when-function-setp

(defthm function-set-fix-when-function-setp
  (implies (function-setp x)
           (equal (function-set-fix x) x)))

Theorem: emptyp-function-set-fix

(defthm emptyp-function-set-fix
  (implies (or (emptyp x) (not (function-setp x)))
           (emptyp (function-set-fix x))))

Theorem: emptyp-of-function-set-fix

(defthm emptyp-of-function-set-fix
  (equal (emptyp (function-set-fix x))
         (or (not (function-setp x))
             (emptyp x))))

Function: function-set-equiv$inline

(defun function-set-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (function-setp acl2::x)
                              (function-setp acl2::y))))
  (equal (function-set-fix acl2::x)
         (function-set-fix acl2::y)))

Theorem: function-set-equiv-is-an-equivalence

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

Theorem: function-set-equiv-implies-equal-function-set-fix-1

(defthm function-set-equiv-implies-equal-function-set-fix-1
  (implies (function-set-equiv acl2::x x-equiv)
           (equal (function-set-fix acl2::x)
                  (function-set-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: function-set-fix-under-function-set-equiv

(defthm function-set-fix-under-function-set-equiv
  (function-set-equiv (function-set-fix acl2::x)
                      acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-function-set-fix-1-forward-to-function-set-equiv

(defthm equal-of-function-set-fix-1-forward-to-function-set-equiv
  (implies (equal (function-set-fix acl2::x)
                  acl2::y)
           (function-set-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-function-set-fix-2-forward-to-function-set-equiv

(defthm equal-of-function-set-fix-2-forward-to-function-set-equiv
  (implies (equal acl2::x (function-set-fix acl2::y))
           (function-set-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: function-set-equiv-of-function-set-fix-1-forward

(defthm function-set-equiv-of-function-set-fix-1-forward
  (implies (function-set-equiv (function-set-fix acl2::x)
                               acl2::y)
           (function-set-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: function-set-equiv-of-function-set-fix-2-forward

(defthm function-set-equiv-of-function-set-fix-2-forward
  (implies (function-set-equiv acl2::x (function-set-fix acl2::y))
           (function-set-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)