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Hprefix

Hprefix-fix

Fixing function for hprefix structures.

Signature
(hprefix-fix x) → new-x
Arguments: x — Guard (hprefixp x).
Returns: new-x — Type (hprefixp new-x).

Definitions and Theorems

Function: hprefix-fix$inline

(defun hprefix-fix$inline (x)
  (declare (xargs :guard (hprefixp x)))
  (let ((__function__ 'hprefix-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (hprefix-kind x)
           (:locase-0x (cons :locase-0x (list)))
           (:upcase-0x (cons :upcase-0x (list))))
         :exec x)))

Theorem: hprefixp-of-hprefix-fix

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

Theorem: hprefix-fix-when-hprefixp

(defthm hprefix-fix-when-hprefixp
  (implies (hprefixp x)
           (equal (hprefix-fix x) x)))

Function: hprefix-equiv$inline

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

Theorem: hprefix-equiv-is-an-equivalence

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

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

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

Theorem: hprefix-fix-under-hprefix-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-hprefix-fix

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