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Filepath-filedata-map

Filepath-filedata-map-fix

(filepath-filedata-map-fix x) is a usual ACL2::fty omap fixing function.

Signature
(filepath-filedata-map-fix x) → *
Arguments: x — Guard (filepath-filedata-mapp x).

Definitions and Theorems

Function: filepath-filedata-map-fix

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

Theorem: filepath-filedata-mapp-of-filepath-filedata-map-fix

(defthm filepath-filedata-mapp-of-filepath-filedata-map-fix
  (filepath-filedata-mapp (filepath-filedata-map-fix x)))

Theorem: filepath-filedata-map-fix-when-filepath-filedata-mapp

(defthm filepath-filedata-map-fix-when-filepath-filedata-mapp
  (implies (filepath-filedata-mapp x)
           (equal (filepath-filedata-map-fix x)
                  x)))

Theorem: emptyp-filepath-filedata-map-fix

(defthm emptyp-filepath-filedata-map-fix
  (implies (or (omap::emptyp x)
               (not (filepath-filedata-mapp x)))
           (omap::emptyp (filepath-filedata-map-fix x))))

Theorem: emptyp-of-filepath-filedata-map-fix-to-not-filepath-filedata-map-or-emptyp

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

Function: filepath-filedata-map-equiv$inline

(defun filepath-filedata-map-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (filepath-filedata-mapp acl2::x)
                              (filepath-filedata-mapp acl2::y))))
  (equal (filepath-filedata-map-fix acl2::x)
         (filepath-filedata-map-fix acl2::y)))

Theorem: filepath-filedata-map-equiv-is-an-equivalence

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

Theorem: filepath-filedata-map-equiv-implies-equal-filepath-filedata-map-fix-1

(defthm
 filepath-filedata-map-equiv-implies-equal-filepath-filedata-map-fix-1
 (implies (filepath-filedata-map-equiv acl2::x x-equiv)
          (equal (filepath-filedata-map-fix acl2::x)
                 (filepath-filedata-map-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: filepath-filedata-map-fix-under-filepath-filedata-map-equiv

(defthm filepath-filedata-map-fix-under-filepath-filedata-map-equiv
  (filepath-filedata-map-equiv (filepath-filedata-map-fix acl2::x)
                               acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-filepath-filedata-map-fix-1-forward-to-filepath-filedata-map-equiv

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

Theorem: equal-of-filepath-filedata-map-fix-2-forward-to-filepath-filedata-map-equiv

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

Theorem: filepath-filedata-map-equiv-of-filepath-filedata-map-fix-1-forward

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

Theorem: filepath-filedata-map-equiv-of-filepath-filedata-map-fix-2-forward

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