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Initer

Initer-fix

Fixing function for initer structures.

Signature
(initer-fix x) → new-x
Arguments: x — Guard (initerp x).
Returns: new-x — Type (initerp new-x).

Definitions and Theorems

Function: initer-fix$inline

(defun initer-fix$inline (x)
 (declare (xargs :guard (initerp x)))
 (let ((__function__ 'initer-fix))
   (declare (ignorable __function__))
   (mbe :logic
        (case (initer-kind x)
          (:single (b* ((get (expr-fix (std::da-nth 0 (cdr x)))))
                     (cons :single (list get))))
          (:list (b* ((get (expr-list-fix (std::da-nth 0 (cdr x)))))
                   (cons :list (list get)))))
        :exec x)))

Theorem: initerp-of-initer-fix

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

Theorem: initer-fix-when-initerp

(defthm initer-fix-when-initerp
  (implies (initerp x)
           (equal (initer-fix x) x)))

Function: initer-equiv$inline

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

Theorem: initer-equiv-is-an-equivalence

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

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

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

Theorem: initer-fix-under-initer-equiv

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

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

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

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

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

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

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

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

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

Theorem: initer-kind$inline-of-initer-fix-x

(defthm initer-kind$inline-of-initer-fix-x
  (equal (initer-kind$inline (initer-fix x))
         (initer-kind$inline x)))

Theorem: initer-kind$inline-initer-equiv-congruence-on-x

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

Theorem: consp-of-initer-fix

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