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Variable-context

Variable-context-fix

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

Signature
(variable-context-fix x) → *
Arguments: x — Guard (variable-contextp x).

Definitions and Theorems

Function: variable-context-fix

(defun variable-context-fix (x)
  (declare (xargs :guard (variable-contextp x)))
  (mbe :logic (if (variable-contextp x) x nil)
       :exec x))

Theorem: variable-contextp-of-variable-context-fix

(defthm variable-contextp-of-variable-context-fix
  (variable-contextp (variable-context-fix x)))

Theorem: variable-context-fix-when-variable-contextp

(defthm variable-context-fix-when-variable-contextp
  (implies (variable-contextp x)
           (equal (variable-context-fix x) x)))

Theorem: emptyp-variable-context-fix

(defthm emptyp-variable-context-fix
  (implies (or (omap::emptyp x)
               (not (variable-contextp x)))
           (omap::emptyp (variable-context-fix x))))

Theorem: emptyp-of-variable-context-fix-to-not-variable-context-or-emptyp

(defthm
   emptyp-of-variable-context-fix-to-not-variable-context-or-emptyp
  (equal (omap::emptyp (variable-context-fix x))
         (or (not (variable-contextp x))
             (omap::emptyp x))))

Function: variable-context-equiv$inline

(defun variable-context-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (variable-contextp acl2::x)
                              (variable-contextp acl2::y))))
  (equal (variable-context-fix acl2::x)
         (variable-context-fix acl2::y)))

Theorem: variable-context-equiv-is-an-equivalence

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

Theorem: variable-context-equiv-implies-equal-variable-context-fix-1

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

Theorem: variable-context-fix-under-variable-context-equiv

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

Theorem: equal-of-variable-context-fix-1-forward-to-variable-context-equiv

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

Theorem: equal-of-variable-context-fix-2-forward-to-variable-context-equiv

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

Theorem: variable-context-equiv-of-variable-context-fix-1-forward

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

Theorem: variable-context-equiv-of-variable-context-fix-2-forward

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