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• Funtable

Funtable-fix

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

Signature

(funtable-fix x) → *

Arguments

x — Guard (funtablep x).

Definitions and Theorems

Function: funtable-fix

(defun funtable-fix (x)
  (declare (xargs :guard (funtablep x)))
  (mbe :logic (if (funtablep x) x nil)
       :exec x))

Theorem: funtablep-of-funtable-fix

(defthm funtablep-of-funtable-fix
  (funtablep (funtable-fix x)))

Theorem: funtable-fix-when-funtablep

(defthm funtable-fix-when-funtablep
  (implies (funtablep x)
           (equal (funtable-fix x) x)))

Theorem: emptyp-funtable-fix

(defthm emptyp-funtable-fix
  (implies (or (omap::emptyp x)
               (not (funtablep x)))
           (omap::emptyp (funtable-fix x))))

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

(defthm emptyp-of-funtable-fix-to-not-funtable-or-emptyp
  (equal (omap::emptyp (funtable-fix x))
         (or (not (funtablep x))
             (omap::emptyp x))))

Function: funtable-equiv$inline

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

Theorem: funtable-equiv-is-an-equivalence

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

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

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

Theorem: funtable-fix-under-funtable-equiv

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

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

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

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

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

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

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

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

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