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Lsuffix

Lsuffix-fix

Fixing function for lsuffix structures.

Signature
(lsuffix-fix x) → new-x
Arguments: x — Guard (lsuffixp x).
Returns: new-x — Type (lsuffixp new-x).

Definitions and Theorems

Function: lsuffix-fix$inline

(defun lsuffix-fix$inline (x)
  (declare (xargs :guard (lsuffixp x)))
  (let ((__function__ 'lsuffix-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (lsuffix-kind x)
           (:locase-l (cons :locase-l (list)))
           (:upcase-l (cons :upcase-l (list)))
           (:locase-ll (cons :locase-ll (list)))
           (:upcase-ll (cons :upcase-ll (list))))
         :exec x)))

Theorem: lsuffixp-of-lsuffix-fix

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

Theorem: lsuffix-fix-when-lsuffixp

(defthm lsuffix-fix-when-lsuffixp
  (implies (lsuffixp x)
           (equal (lsuffix-fix x) x)))

Function: lsuffix-equiv$inline

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

Theorem: lsuffix-equiv-is-an-equivalence

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

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

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

Theorem: lsuffix-fix-under-lsuffix-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-lsuffix-fix

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