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Eprefix

Eprefix-fix

Fixing function for eprefix structures.

Signature
(eprefix-fix x) → new-x
Arguments: x — Guard (eprefixp x).
Returns: new-x — Type (eprefixp new-x).

Definitions and Theorems

Function: eprefix-fix$inline

(defun eprefix-fix$inline (x)
  (declare (xargs :guard (eprefixp x)))
  (let ((__function__ 'eprefix-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (eprefix-kind x)
           (:locase-u8 (cons :locase-u8 (list)))
           (:locase-u (cons :locase-u (list)))
           (:upcase-u (cons :upcase-u (list)))
           (:upcase-l (cons :upcase-l (list))))
         :exec x)))

Theorem: eprefixp-of-eprefix-fix

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

Theorem: eprefix-fix-when-eprefixp

(defthm eprefix-fix-when-eprefixp
  (implies (eprefixp x)
           (equal (eprefix-fix x) x)))

Function: eprefix-equiv$inline

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

Theorem: eprefix-equiv-is-an-equivalence

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

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

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

Theorem: eprefix-fix-under-eprefix-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-eprefix-fix

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