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String-stringlist-map

String-stringlist-map-fix

(string-stringlist-map-fix x) is a usual fty omap fixing function.

Signature
(string-stringlist-map-fix x) → *
Arguments: x — Guard (string-stringlist-mapp x).

Definitions and Theorems

Function: string-stringlist-map-fix

(defun string-stringlist-map-fix (x)
  (declare (xargs :guard (string-stringlist-mapp x)))
  (mbe :logic (if (string-stringlist-mapp x) x nil)
       :exec x))

Theorem: string-stringlist-mapp-of-string-stringlist-map-fix

(defthm string-stringlist-mapp-of-string-stringlist-map-fix
  (string-stringlist-mapp (string-stringlist-map-fix x)))

Theorem: string-stringlist-map-fix-when-string-stringlist-mapp

(defthm string-stringlist-map-fix-when-string-stringlist-mapp
  (implies (string-stringlist-mapp x)
           (equal (string-stringlist-map-fix x)
                  x)))

Theorem: emptyp-string-stringlist-map-fix

(defthm emptyp-string-stringlist-map-fix
  (implies (or (omap::emptyp x)
               (not (string-stringlist-mapp x)))
           (omap::emptyp (string-stringlist-map-fix x))))

Theorem: emptyp-of-string-stringlist-map-fix-to-not-string-stringlist-map-or-emptyp

(defthm
 emptyp-of-string-stringlist-map-fix-to-not-string-stringlist-map-or-emptyp
 (equal (omap::emptyp (string-stringlist-map-fix x))
        (or (not (string-stringlist-mapp x))
            (omap::emptyp x))))

Function: string-stringlist-map-equiv$inline

(defun string-stringlist-map-equiv$inline (x y)
  (declare (xargs :guard (and (string-stringlist-mapp x)
                              (string-stringlist-mapp y))))
  (equal (string-stringlist-map-fix x)
         (string-stringlist-map-fix y)))

Theorem: string-stringlist-map-equiv-is-an-equivalence

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

Theorem: string-stringlist-map-equiv-implies-equal-string-stringlist-map-fix-1

(defthm
 string-stringlist-map-equiv-implies-equal-string-stringlist-map-fix-1
 (implies (string-stringlist-map-equiv x x-equiv)
          (equal (string-stringlist-map-fix x)
                 (string-stringlist-map-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: string-stringlist-map-fix-under-string-stringlist-map-equiv

(defthm string-stringlist-map-fix-under-string-stringlist-map-equiv
  (string-stringlist-map-equiv (string-stringlist-map-fix x)
                               x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-string-stringlist-map-fix-1-forward-to-string-stringlist-map-equiv

(defthm
 equal-of-string-stringlist-map-fix-1-forward-to-string-stringlist-map-equiv
 (implies (equal (string-stringlist-map-fix x) y)
          (string-stringlist-map-equiv x y))
 :rule-classes :forward-chaining)

Theorem: equal-of-string-stringlist-map-fix-2-forward-to-string-stringlist-map-equiv

(defthm
 equal-of-string-stringlist-map-fix-2-forward-to-string-stringlist-map-equiv
 (implies (equal x (string-stringlist-map-fix y))
          (string-stringlist-map-equiv x y))
 :rule-classes :forward-chaining)

Theorem: string-stringlist-map-equiv-of-string-stringlist-map-fix-1-forward

(defthm
 string-stringlist-map-equiv-of-string-stringlist-map-fix-1-forward
 (implies (string-stringlist-map-equiv (string-stringlist-map-fix x)
                                       y)
          (string-stringlist-map-equiv x y))
 :rule-classes :forward-chaining)

Theorem: string-stringlist-map-equiv-of-string-stringlist-map-fix-2-forward

(defthm
 string-stringlist-map-equiv-of-string-stringlist-map-fix-2-forward
 (implies
      (string-stringlist-map-equiv x (string-stringlist-map-fix y))
      (string-stringlist-map-equiv x y))
 :rule-classes :forward-chaining)