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Char/container

Char/container-fix

Fixing function for char/container structures.

Signature
(char/container-fix x) → new-x
Arguments: x — Guard (char/container-p x).
Returns: new-x — Type (char/container-p new-x).

Definitions and Theorems

Function: char/container-fix$inline

(defun char/container-fix$inline (x)
  (declare (xargs :guard (char/container-p x)))
  (let ((__function__ 'char/container-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (char/container-kind x)
           (:char (b* ((get (char-fix (std::da-nth 0 (cdr x)))))
                    (cons :char (list get))))
           (:container (cons :container (list))))
         :exec x)))

Theorem: char/container-p-of-char/container-fix

(defthm char/container-p-of-char/container-fix
  (b* ((new-x (char/container-fix$inline x)))
    (char/container-p new-x))
  :rule-classes :rewrite)

Theorem: char/container-fix-when-char/container-p

(defthm char/container-fix-when-char/container-p
  (implies (char/container-p x)
           (equal (char/container-fix x) x)))

Function: char/container-equiv$inline

(defun char/container-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (char/container-p acl2::x)
                              (char/container-p acl2::y))))
  (equal (char/container-fix acl2::x)
         (char/container-fix acl2::y)))

Theorem: char/container-equiv-is-an-equivalence

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

Theorem: char/container-equiv-implies-equal-char/container-fix-1

(defthm char/container-equiv-implies-equal-char/container-fix-1
  (implies (char/container-equiv acl2::x x-equiv)
           (equal (char/container-fix acl2::x)
                  (char/container-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: char/container-fix-under-char/container-equiv

(defthm char/container-fix-under-char/container-equiv
  (char/container-equiv (char/container-fix acl2::x)
                        acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-char/container-fix-1-forward-to-char/container-equiv

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

Theorem: equal-of-char/container-fix-2-forward-to-char/container-equiv

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

Theorem: char/container-equiv-of-char/container-fix-1-forward

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

Theorem: char/container-equiv-of-char/container-fix-2-forward

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

Theorem: char/container-kind$inline-of-char/container-fix-x

(defthm char/container-kind$inline-of-char/container-fix-x
  (equal (char/container-kind$inline (char/container-fix x))
         (char/container-kind$inline x)))

Theorem: char/container-kind$inline-char/container-equiv-congruence-on-x

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

Theorem: consp-of-char/container-fix

(defthm consp-of-char/container-fix
  (consp (char/container-fix x))
  :rule-classes :type-prescription)