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• Vars+modes

Vars+modes-fix

Fixing function for vars+modes structures.

Signature

(vars+modes-fix x) → new-x

Arguments

x — Guard (vars+modes-p x).

Returns

new-x — Type (vars+modes-p new-x).

Definitions and Theorems

Function: vars+modes-fix$inline

(defun vars+modes-fix$inline (x)
 (declare (xargs :guard (vars+modes-p x)))
 (let ((__function__ 'vars+modes-fix))
  (declare (ignorable __function__))
  (mbe
      :logic
      (b* ((vars (identifier-set-fix (cdr (std::da-nth 0 (cdr x)))))
           (modes (mode-set-fix (cdr (std::da-nth 1 (cdr x))))))
        (cons :vars+modes (list (cons 'vars vars)
                                (cons 'modes modes))))
      :exec x)))

Theorem: vars+modes-p-of-vars+modes-fix

(defthm vars+modes-p-of-vars+modes-fix
  (b* ((new-x (vars+modes-fix$inline x)))
    (vars+modes-p new-x))
  :rule-classes :rewrite)

Theorem: vars+modes-fix-when-vars+modes-p

(defthm vars+modes-fix-when-vars+modes-p
  (implies (vars+modes-p x)
           (equal (vars+modes-fix x) x)))

Function: vars+modes-equiv$inline

(defun vars+modes-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (vars+modes-p acl2::x)
                              (vars+modes-p acl2::y))))
  (equal (vars+modes-fix acl2::x)
         (vars+modes-fix acl2::y)))

Theorem: vars+modes-equiv-is-an-equivalence

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

Theorem: vars+modes-equiv-implies-equal-vars+modes-fix-1

(defthm vars+modes-equiv-implies-equal-vars+modes-fix-1
  (implies (vars+modes-equiv acl2::x x-equiv)
           (equal (vars+modes-fix acl2::x)
                  (vars+modes-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: vars+modes-fix-under-vars+modes-equiv

(defthm vars+modes-fix-under-vars+modes-equiv
  (vars+modes-equiv (vars+modes-fix acl2::x)
                    acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-vars+modes-fix-1-forward-to-vars+modes-equiv

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

Theorem: equal-of-vars+modes-fix-2-forward-to-vars+modes-equiv

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

Theorem: vars+modes-equiv-of-vars+modes-fix-1-forward

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

Theorem: vars+modes-equiv-of-vars+modes-fix-2-forward

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