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Bexpo

Bexpo-fix

Fixing function for bexpo structures.

Signature
(bexpo-fix x) → new-x
Arguments: x — Guard (bexpop x).
Returns: new-x — Type (bexpop new-x).

Definitions and Theorems

Function: bexpo-fix$inline

(defun bexpo-fix$inline (x)
  (declare (xargs :guard (bexpop x)))
  (mbe :logic
       (b* ((prefix (bexprefix-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: bexpop-of-bexpo-fix

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

Theorem: bexpo-fix-when-bexpop

(defthm bexpo-fix-when-bexpop
  (implies (bexpop x)
           (equal (bexpo-fix x) x)))

Function: bexpo-equiv$inline

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

Theorem: bexpo-equiv-is-an-equivalence

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

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

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

Theorem: bexpo-fix-under-bexpo-equiv

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

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

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

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

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

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

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

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

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

Theorem: bexpo-fix$inline-of-bexpo-fix-x

(defthm bexpo-fix$inline-of-bexpo-fix-x
  (equal (bexpo-fix$inline (bexpo-fix x))
         (bexpo-fix$inline x)))

Theorem: bexpo-fix$inline-bexpo-equiv-congruence-on-x

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