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Num-base

Num-base-fix

Fixing function for num-base structures.

Signature
(num-base-fix x) → new-x
Arguments: x — Guard (num-base-p x).
Returns: new-x — Type (num-base-p new-x).

Definitions and Theorems

Function: num-base-fix$inline

(defun num-base-fix$inline (x)
  (declare (xargs :guard (num-base-p x)))
  (let ((__function__ 'num-base-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (num-base-kind x)
           (:dec (cons :dec (list)))
           (:hex (cons :hex (list)))
           (:bin (cons :bin (list))))
         :exec x)))

Theorem: num-base-p-of-num-base-fix

(defthm num-base-p-of-num-base-fix
  (b* ((new-x (num-base-fix$inline x)))
    (num-base-p new-x))
  :rule-classes :rewrite)

Theorem: num-base-fix-when-num-base-p

(defthm num-base-fix-when-num-base-p
  (implies (num-base-p x)
           (equal (num-base-fix x) x)))

Function: num-base-equiv$inline

(defun num-base-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (num-base-p acl2::x)
                              (num-base-p acl2::y))))
  (equal (num-base-fix acl2::x)
         (num-base-fix acl2::y)))

Theorem: num-base-equiv-is-an-equivalence

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

Theorem: num-base-equiv-implies-equal-num-base-fix-1

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

Theorem: num-base-fix-under-num-base-equiv

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

Theorem: equal-of-num-base-fix-1-forward-to-num-base-equiv

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

Theorem: equal-of-num-base-fix-2-forward-to-num-base-equiv

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

Theorem: num-base-equiv-of-num-base-fix-1-forward

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

Theorem: num-base-equiv-of-num-base-fix-2-forward

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

Theorem: num-base-kind$inline-of-num-base-fix-x

(defthm num-base-kind$inline-of-num-base-fix-x
  (equal (num-base-kind$inline (num-base-fix x))
         (num-base-kind$inline x)))

Theorem: num-base-kind$inline-num-base-equiv-congruence-on-x

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

Theorem: consp-of-num-base-fix

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