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• Type-map

Type-map-fix

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

Signature

(type-map-fix x) → *

Arguments

x — Guard (type-mapp x).

Definitions and Theorems

Function: type-map-fix

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

Theorem: type-mapp-of-type-map-fix

(defthm type-mapp-of-type-map-fix
  (type-mapp (type-map-fix x)))

Theorem: type-map-fix-when-type-mapp

(defthm type-map-fix-when-type-mapp
  (implies (type-mapp x)
           (equal (type-map-fix x) x)))

Theorem: emptyp-type-map-fix

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

Theorem: emptyp-of-type-map-fix-to-not-type-map-or-emptyp

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

Function: type-map-equiv$inline

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

Theorem: type-map-equiv-is-an-equivalence

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

Theorem: type-map-equiv-implies-equal-type-map-fix-1

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

Theorem: type-map-fix-under-type-map-equiv

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

Theorem: equal-of-type-map-fix-1-forward-to-type-map-equiv

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

Theorem: equal-of-type-map-fix-2-forward-to-type-map-equiv

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

Theorem: type-map-equiv-of-type-map-fix-1-forward

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

Theorem: type-map-equiv-of-type-map-fix-2-forward

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