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• Dec-expo-prefix

Dec-expo-prefix-fix

Fixing function for dec-expo-prefix structures.

Signature

(dec-expo-prefix-fix x) → new-x

Arguments

x — Guard (dec-expo-prefixp x).

Returns

new-x — Type (dec-expo-prefixp new-x).

Definitions and Theorems

Function: dec-expo-prefix-fix$inline

(defun dec-expo-prefix-fix$inline (x)
  (declare (xargs :guard (dec-expo-prefixp x)))
  (let ((__function__ 'dec-expo-prefix-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (dec-expo-prefix-kind x)
           (:locase-e (cons :locase-e (list)))
           (:upcase-e (cons :upcase-e (list))))
         :exec x)))

Theorem: dec-expo-prefixp-of-dec-expo-prefix-fix

(defthm dec-expo-prefixp-of-dec-expo-prefix-fix
  (b* ((new-x (dec-expo-prefix-fix$inline x)))
    (dec-expo-prefixp new-x))
  :rule-classes :rewrite)

Theorem: dec-expo-prefix-fix-when-dec-expo-prefixp

(defthm dec-expo-prefix-fix-when-dec-expo-prefixp
  (implies (dec-expo-prefixp x)
           (equal (dec-expo-prefix-fix x) x)))

Function: dec-expo-prefix-equiv$inline

(defun dec-expo-prefix-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (dec-expo-prefixp acl2::x)
                              (dec-expo-prefixp acl2::y))))
  (equal (dec-expo-prefix-fix acl2::x)
         (dec-expo-prefix-fix acl2::y)))

Theorem: dec-expo-prefix-equiv-is-an-equivalence

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

Theorem: dec-expo-prefix-equiv-implies-equal-dec-expo-prefix-fix-1

(defthm dec-expo-prefix-equiv-implies-equal-dec-expo-prefix-fix-1
  (implies (dec-expo-prefix-equiv acl2::x x-equiv)
           (equal (dec-expo-prefix-fix acl2::x)
                  (dec-expo-prefix-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: dec-expo-prefix-fix-under-dec-expo-prefix-equiv

(defthm dec-expo-prefix-fix-under-dec-expo-prefix-equiv
  (dec-expo-prefix-equiv (dec-expo-prefix-fix acl2::x)
                         acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-dec-expo-prefix-fix-1-forward-to-dec-expo-prefix-equiv

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

Theorem: equal-of-dec-expo-prefix-fix-2-forward-to-dec-expo-prefix-equiv

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

Theorem: dec-expo-prefix-equiv-of-dec-expo-prefix-fix-1-forward

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

Theorem: dec-expo-prefix-equiv-of-dec-expo-prefix-fix-2-forward

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

Theorem: dec-expo-prefix-kind$inline-of-dec-expo-prefix-fix-x

(defthm dec-expo-prefix-kind$inline-of-dec-expo-prefix-fix-x
  (equal (dec-expo-prefix-kind$inline (dec-expo-prefix-fix x))
         (dec-expo-prefix-kind$inline x)))

Theorem: dec-expo-prefix-kind$inline-dec-expo-prefix-equiv-congruence-on-x

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

Theorem: consp-of-dec-expo-prefix-fix

(defthm consp-of-dec-expo-prefix-fix
  (consp (dec-expo-prefix-fix x))
  :rule-classes :type-prescription)