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• Dec/oct/hex-const

Dec/oct/hex-const-fix

Fixing function for dec/oct/hex-const structures.

Signature

(dec/oct/hex-const-fix x) → new-x

Arguments

x — Guard (dec/oct/hex-constp x).

Returns

new-x — Type (dec/oct/hex-constp new-x).

Definitions and Theorems

Function: dec/oct/hex-const-fix$inline

(defun dec/oct/hex-const-fix$inline (x)
 (declare (xargs :guard (dec/oct/hex-constp x)))
 (mbe
  :logic
  (case (dec/oct/hex-const-kind x)
   (:dec (b* ((value (pos-fix (cdr x))))
           (cons :dec value)))
   (:oct (b* ((leading-zeros (pos-fix (car (cdr x))))
              (value (nfix (cdr (cdr x)))))
           (cons :oct (cons leading-zeros value))))
   (:hex (b* ((prefix (hprefix-fix (car (cdr x))))
              (digits (str::hex-digit-char-list-fix (cdr (cdr x)))))
           (cons :hex (cons prefix digits)))))
  :exec x))

Theorem: dec/oct/hex-constp-of-dec/oct/hex-const-fix

(defthm dec/oct/hex-constp-of-dec/oct/hex-const-fix
  (b* ((new-x (dec/oct/hex-const-fix$inline x)))
    (dec/oct/hex-constp new-x))
  :rule-classes :rewrite)

Theorem: dec/oct/hex-const-fix-when-dec/oct/hex-constp

(defthm dec/oct/hex-const-fix-when-dec/oct/hex-constp
  (implies (dec/oct/hex-constp x)
           (equal (dec/oct/hex-const-fix x) x)))

Function: dec/oct/hex-const-equiv$inline

(defun dec/oct/hex-const-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (dec/oct/hex-constp acl2::x)
                              (dec/oct/hex-constp acl2::y))))
  (equal (dec/oct/hex-const-fix acl2::x)
         (dec/oct/hex-const-fix acl2::y)))

Theorem: dec/oct/hex-const-equiv-is-an-equivalence

(defthm dec/oct/hex-const-equiv-is-an-equivalence
  (and (booleanp (dec/oct/hex-const-equiv x y))
       (dec/oct/hex-const-equiv x x)
       (implies (dec/oct/hex-const-equiv x y)
                (dec/oct/hex-const-equiv y x))
       (implies (and (dec/oct/hex-const-equiv x y)
                     (dec/oct/hex-const-equiv y z))
                (dec/oct/hex-const-equiv x z)))
  :rule-classes (:equivalence))

Theorem: dec/oct/hex-const-equiv-implies-equal-dec/oct/hex-const-fix-1

(defthm
      dec/oct/hex-const-equiv-implies-equal-dec/oct/hex-const-fix-1
  (implies (dec/oct/hex-const-equiv acl2::x x-equiv)
           (equal (dec/oct/hex-const-fix acl2::x)
                  (dec/oct/hex-const-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: dec/oct/hex-const-fix-under-dec/oct/hex-const-equiv

(defthm dec/oct/hex-const-fix-under-dec/oct/hex-const-equiv
  (dec/oct/hex-const-equiv (dec/oct/hex-const-fix acl2::x)
                           acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-dec/oct/hex-const-fix-1-forward-to-dec/oct/hex-const-equiv

(defthm
 equal-of-dec/oct/hex-const-fix-1-forward-to-dec/oct/hex-const-equiv
 (implies (equal (dec/oct/hex-const-fix acl2::x)
                 acl2::y)
          (dec/oct/hex-const-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: equal-of-dec/oct/hex-const-fix-2-forward-to-dec/oct/hex-const-equiv

(defthm
 equal-of-dec/oct/hex-const-fix-2-forward-to-dec/oct/hex-const-equiv
 (implies (equal acl2::x (dec/oct/hex-const-fix acl2::y))
          (dec/oct/hex-const-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: dec/oct/hex-const-equiv-of-dec/oct/hex-const-fix-1-forward

(defthm dec/oct/hex-const-equiv-of-dec/oct/hex-const-fix-1-forward
  (implies (dec/oct/hex-const-equiv (dec/oct/hex-const-fix acl2::x)
                                    acl2::y)
           (dec/oct/hex-const-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: dec/oct/hex-const-equiv-of-dec/oct/hex-const-fix-2-forward

(defthm dec/oct/hex-const-equiv-of-dec/oct/hex-const-fix-2-forward
 (implies
   (dec/oct/hex-const-equiv acl2::x (dec/oct/hex-const-fix acl2::y))
   (dec/oct/hex-const-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: dec/oct/hex-const-kind$inline-of-dec/oct/hex-const-fix-x

(defthm dec/oct/hex-const-kind$inline-of-dec/oct/hex-const-fix-x
  (equal (dec/oct/hex-const-kind$inline (dec/oct/hex-const-fix x))
         (dec/oct/hex-const-kind$inline x)))

Theorem: dec/oct/hex-const-kind$inline-dec/oct/hex-const-equiv-congruence-on-x

(defthm
 dec/oct/hex-const-kind$inline-dec/oct/hex-const-equiv-congruence-on-x
 (implies (dec/oct/hex-const-equiv x x-equiv)
          (equal (dec/oct/hex-const-kind$inline x)
                 (dec/oct/hex-const-kind$inline x-equiv)))
 :rule-classes :congruence)

Theorem: consp-of-dec/oct/hex-const-fix

(defthm consp-of-dec/oct/hex-const-fix
  (consp (dec/oct/hex-const-fix x))
  :rule-classes :type-prescription)

Theorem: dec/oct/hex-const-fix$inline-of-dec/oct/hex-const-fix-x

(defthm dec/oct/hex-const-fix$inline-of-dec/oct/hex-const-fix-x
  (equal (dec/oct/hex-const-fix$inline (dec/oct/hex-const-fix x))
         (dec/oct/hex-const-fix$inline x)))

Theorem: dec/oct/hex-const-fix$inline-dec/oct/hex-const-equiv-congruence-on-x

(defthm
 dec/oct/hex-const-fix$inline-dec/oct/hex-const-equiv-congruence-on-x
 (implies (dec/oct/hex-const-equiv x x-equiv)
          (equal (dec/oct/hex-const-fix$inline x)
                 (dec/oct/hex-const-fix$inline x-equiv)))
 :rule-classes :congruence)