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Literal

Literal-fix

Fixing function for literal structures.

Signature
(literal-fix x) → new-x
Arguments: x — Guard (literalp x).
Returns: new-x — Type (literalp new-x).

Definitions and Theorems

Function: literal-fix$inline

(defun literal-fix$inline (x)
 (declare (xargs :guard (literalp x)))
 (let ((__function__ 'literal-fix))
  (declare (ignorable __function__))
  (mbe
      :logic
      (case (literal-kind x)
        (:boolean (b* ((value (bool-fix (std::da-nth 0 (cdr x)))))
                    (cons :boolean (list value))))
        (:character (b* ((value (char-fix (std::da-nth 0 (cdr x)))))
                      (cons :character (list value))))
        (:string (b* ((value (str-fix (std::da-nth 0 (cdr x)))))
                   (cons :string (list value))))
        (:integer (b* ((value (nfix (std::da-nth 0 (cdr x)))))
                    (cons :integer (list value)))))
      :exec x)))

Theorem: literalp-of-literal-fix

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

Theorem: literal-fix-when-literalp

(defthm literal-fix-when-literalp
  (implies (literalp x)
           (equal (literal-fix x) x)))

Function: literal-equiv$inline

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

Theorem: literal-equiv-is-an-equivalence

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

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

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

Theorem: literal-fix-under-literal-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-literal-fix

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