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System-state

System-state-fix

Fixing function for system-state structures.

Signature
(system-state-fix x) → new-x
Arguments: x — Guard (system-statep x).
Returns: new-x — Type (system-statep new-x).

Definitions and Theorems

Function: system-state-fix$inline

(defun system-state-fix$inline (x)
 (declare (xargs :guard (system-statep x)))
 (let ((__function__ 'system-state-fix))
  (declare (ignorable __function__))
  (mbe
    :logic
    (b* ((validators (validators-state-fix (cdr (std::da-nth 0 x))))
         (network (message-set-fix (cdr (std::da-nth 1 x)))))
      (list (cons 'validators validators)
            (cons 'network network)))
    :exec x)))

Theorem: system-statep-of-system-state-fix

(defthm system-statep-of-system-state-fix
  (b* ((new-x (system-state-fix$inline x)))
    (system-statep new-x))
  :rule-classes :rewrite)

Theorem: system-state-fix-when-system-statep

(defthm system-state-fix-when-system-statep
  (implies (system-statep x)
           (equal (system-state-fix x) x)))

Function: system-state-equiv$inline

(defun system-state-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (system-statep acl2::x)
                              (system-statep acl2::y))))
  (equal (system-state-fix acl2::x)
         (system-state-fix acl2::y)))

Theorem: system-state-equiv-is-an-equivalence

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

Theorem: system-state-equiv-implies-equal-system-state-fix-1

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

Theorem: system-state-fix-under-system-state-equiv

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

Theorem: equal-of-system-state-fix-1-forward-to-system-state-equiv

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

Theorem: equal-of-system-state-fix-2-forward-to-system-state-equiv

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

Theorem: system-state-equiv-of-system-state-fix-1-forward

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

Theorem: system-state-equiv-of-system-state-fix-2-forward

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