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Dec-expo

Dec-expo-fix

Fixing function for dec-expo structures.

Signature
(dec-expo-fix x) → new-x
Arguments: x — Guard (dec-expop x).
Returns: new-x — Type (dec-expop new-x).

Definitions and Theorems

Function: dec-expo-fix$inline

(defun dec-expo-fix$inline (x)
  (declare (xargs :guard (dec-expop x)))
  (mbe :logic
       (b* ((prefix (dec-expo-prefix-fix (car x)))
            (sign? (sign-option-fix (car (cdr x))))
            (digits (str::dec-digit-char-list-fix (cdr (cdr x)))))
         (cons prefix (cons sign? digits)))
       :exec x))

Theorem: dec-expop-of-dec-expo-fix

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

Theorem: dec-expo-fix-when-dec-expop

(defthm dec-expo-fix-when-dec-expop
  (implies (dec-expop x)
           (equal (dec-expo-fix x) x)))

Function: dec-expo-equiv$inline

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: dec-expo-fix$inline-dec-expo-equiv-congruence-on-x

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