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• System

System-fix

Fixing function for system structures.

Signature

(system-fix x) → new-x

Arguments

x — Guard (systemp x).

Returns

new-x — Type (systemp new-x).

Definitions and Theorems

Function: system-fix$inline

(defun system-fix$inline (x)
 (declare (xargs :guard (systemp x)))
 (let ((__function__ 'system-fix))
  (declare (ignorable __function__))
  (mbe
     :logic
     (b* ((definitions
               (definition-list-fix (cdr (std::da-nth 0 (cdr x)))))
          (constraints
               (constraint-list-fix (cdr (std::da-nth 1 (cdr x))))))
       (cons :system (list (cons 'definitions definitions)
                           (cons 'constraints constraints))))
     :exec x)))

Theorem: systemp-of-system-fix

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

Theorem: system-fix-when-systemp

(defthm system-fix-when-systemp
  (implies (systemp x)
           (equal (system-fix x) x)))

Function: system-equiv$inline

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

Theorem: system-equiv-is-an-equivalence

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

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

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

Theorem: system-fix-under-system-equiv

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

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

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

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

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

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

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

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

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