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Lstate

Lstate-fix

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

Signature
(lstate-fix x) → *
Arguments: x — Guard (lstatep x).

Definitions and Theorems

Function: lstate-fix

(defun lstate-fix (x)
  (declare (xargs :guard (lstatep x)))
  (mbe :logic (if (lstatep x) x nil)
       :exec x))

Theorem: lstatep-of-lstate-fix

(defthm lstatep-of-lstate-fix
  (lstatep (lstate-fix x)))

Theorem: lstate-fix-when-lstatep

(defthm lstate-fix-when-lstatep
  (implies (lstatep x)
           (equal (lstate-fix x) x)))

Theorem: emptyp-lstate-fix

(defthm emptyp-lstate-fix
  (implies (or (omap::emptyp x) (not (lstatep x)))
           (omap::emptyp (lstate-fix x))))

Theorem: emptyp-of-lstate-fix-to-not-lstate-or-emptyp

(defthm emptyp-of-lstate-fix-to-not-lstate-or-emptyp
  (equal (omap::emptyp (lstate-fix x))
         (or (not (lstatep x))
             (omap::emptyp x))))

Function: lstate-equiv$inline

(defun lstate-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (lstatep acl2::x)
                              (lstatep acl2::y))))
  (equal (lstate-fix acl2::x)
         (lstate-fix acl2::y)))

Theorem: lstate-equiv-is-an-equivalence

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

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

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

Theorem: lstate-fix-under-lstate-equiv

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

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

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

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

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

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

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

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

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