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Usuffix

Usuffix-fix

Fixing function for usuffix structures.

Signature
(usuffix-fix x) → new-x
Arguments: x — Guard (usuffixp x).
Returns: new-x — Type (usuffixp new-x).

Definitions and Theorems

Function: usuffix-fix$inline

(defun usuffix-fix$inline (x)
  (declare (xargs :guard (usuffixp x)))
  (let ((__function__ 'usuffix-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (usuffix-kind x)
           (:locase-u (cons :locase-u (list)))
           (:upcase-u (cons :upcase-u (list))))
         :exec x)))

Theorem: usuffixp-of-usuffix-fix

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

Theorem: usuffix-fix-when-usuffixp

(defthm usuffix-fix-when-usuffixp
  (implies (usuffixp x)
           (equal (usuffix-fix x) x)))

Function: usuffix-equiv$inline

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

Theorem: usuffix-equiv-is-an-equivalence

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

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

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

Theorem: usuffix-fix-under-usuffix-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-usuffix-fix

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