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• Binding

Binding-fix

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

Signature

(binding-fix x) → *

Arguments

x — Guard (bindingp x).

Definitions and Theorems

Function: binding-fix

(defun binding-fix (x)
  (declare (xargs :guard (bindingp x)))
  (mbe :logic (if (bindingp x) x nil)
       :exec x))

Theorem: bindingp-of-binding-fix

(defthm bindingp-of-binding-fix
  (bindingp (binding-fix x)))

Theorem: binding-fix-when-bindingp

(defthm binding-fix-when-bindingp
  (implies (bindingp x)
           (equal (binding-fix x) x)))

Theorem: emptyp-binding-fix

(defthm emptyp-binding-fix
  (implies (or (omap::emptyp x) (not (bindingp x)))
           (omap::emptyp (binding-fix x))))

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

(defthm emptyp-of-binding-fix-to-not-binding-or-emptyp
  (equal (omap::emptyp (binding-fix x))
         (or (not (bindingp x))
             (omap::emptyp x))))

Function: binding-equiv$inline

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

Theorem: binding-equiv-is-an-equivalence

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

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

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

Theorem: binding-fix-under-binding-equiv

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

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

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

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

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

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

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

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

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