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• Ident-paramdeclon-map

Ident-paramdeclon-map-fix

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

Signature

(ident-paramdeclon-map-fix x) → *

Arguments

x — Guard (ident-paramdeclon-mapp x).

Definitions and Theorems

Function: ident-paramdeclon-map-fix

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

Theorem: ident-paramdeclon-mapp-of-ident-paramdeclon-map-fix

(defthm ident-paramdeclon-mapp-of-ident-paramdeclon-map-fix
  (ident-paramdeclon-mapp (ident-paramdeclon-map-fix x)))

Theorem: ident-paramdeclon-map-fix-when-ident-paramdeclon-mapp

(defthm ident-paramdeclon-map-fix-when-ident-paramdeclon-mapp
  (implies (ident-paramdeclon-mapp x)
           (equal (ident-paramdeclon-map-fix x)
                  x)))

Theorem: emptyp-ident-paramdeclon-map-fix

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

Theorem: emptyp-of-ident-paramdeclon-map-fix-to-not-ident-paramdeclon-map-or-emptyp

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

Function: ident-paramdeclon-map-equiv$inline

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

Theorem: ident-paramdeclon-map-equiv-is-an-equivalence

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

Theorem: ident-paramdeclon-map-equiv-implies-equal-ident-paramdeclon-map-fix-1

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

Theorem: ident-paramdeclon-map-fix-under-ident-paramdeclon-map-equiv

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

Theorem: equal-of-ident-paramdeclon-map-fix-1-forward-to-ident-paramdeclon-map-equiv

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

Theorem: equal-of-ident-paramdeclon-map-fix-2-forward-to-ident-paramdeclon-map-equiv

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

Theorem: ident-paramdeclon-map-equiv-of-ident-paramdeclon-map-fix-1-forward

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

Theorem: ident-paramdeclon-map-equiv-of-ident-paramdeclon-map-fix-2-forward

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