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Eval-state

Eval-state-fix

Fixing function for eval-state structures.

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

Definitions and Theorems

Function: eval-state-fix$inline

(defun eval-state-fix$inline (x)
 (declare (xargs :guard (eval-state-p x)))
 (let ((__function__ 'eval-state-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (case (eval-state-kind x)
     (:init
          (b* ((function (symbol-value-fix (std::da-nth 0 (cdr x))))
               (arguments (value-list-fix (std::da-nth 1 (cdr x)))))
            (cons :init (list function arguments))))
     (:trans (b* ((stack (stack-fix (std::da-nth 0 (cdr x)))))
               (cons :trans (list stack))))
     (:final (b* ((result (value-fix (std::da-nth 0 (cdr x)))))
               (cons :final (list result))))
     (:error (cons :error (list))))
   :exec x)))

Theorem: eval-state-p-of-eval-state-fix

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

Theorem: eval-state-fix-when-eval-state-p

(defthm eval-state-fix-when-eval-state-p
  (implies (eval-state-p x)
           (equal (eval-state-fix x) x)))

Function: eval-state-equiv$inline

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

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

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

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

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

Theorem: eval-state-fix-under-eval-state-equiv

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

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

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

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

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

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

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

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

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

Theorem: eval-state-kind$inline-of-eval-state-fix-x

(defthm eval-state-kind$inline-of-eval-state-fix-x
  (equal (eval-state-kind$inline (eval-state-fix x))
         (eval-state-kind$inline x)))

Theorem: eval-state-kind$inline-eval-state-equiv-congruence-on-x

(defthm eval-state-kind$inline-eval-state-equiv-congruence-on-x
  (implies (eval-state-equiv x x-equiv)
           (equal (eval-state-kind$inline x)
                  (eval-state-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-eval-state-fix

(defthm consp-of-eval-state-fix
  (consp (eval-state-fix x))
  :rule-classes :type-prescription)