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• Scope

Scope-fix

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

Signature

(scope-fix x) → *

Arguments

x — Guard (scopep x).

Definitions and Theorems

Function: scope-fix

(defun scope-fix (x)
  (declare (xargs :guard (scopep x)))
  (mbe :logic (if (scopep x) x nil)
       :exec x))

Theorem: scopep-of-scope-fix

(defthm scopep-of-scope-fix
  (scopep (scope-fix x)))

Theorem: scope-fix-when-scopep

(defthm scope-fix-when-scopep
  (implies (scopep x)
           (equal (scope-fix x) x)))

Theorem: emptyp-scope-fix

(defthm emptyp-scope-fix
  (implies (or (omap::emptyp x) (not (scopep x)))
           (omap::emptyp (scope-fix x))))

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

(defthm emptyp-of-scope-fix-to-not-scope-or-emptyp
  (equal (omap::emptyp (scope-fix x))
         (or (not (scopep x)) (omap::emptyp x))))

Function: scope-equiv$inline

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

Theorem: scope-equiv-is-an-equivalence

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

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

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

Theorem: scope-fix-under-scope-equiv

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

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

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

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

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

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

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

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

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