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Svar-widths

Svar-widths-fix

(svar-widths-fix x) is an fty alist fixing function that follows the fix-keys strategy.

Signature
(svar-widths-fix x) → fty::newx
Arguments: x — Guard (svar-widths-p x).
Returns: fty::newx — Type (svar-widths-p fty::newx).

Note that in the execution this is just an inline identity function.

Definitions and Theorems

Function: svar-widths-fix$inline

(defun svar-widths-fix$inline (x)
  (declare (xargs :guard (svar-widths-p x)))
  (let ((__function__ 'svar-widths-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (if (consp (car x))
               (cons (cons (svar-fix (caar x))
                           (ifix (cdar x)))
                     (svar-widths-fix (cdr x)))
             (svar-widths-fix (cdr x))))
         :exec x)))

Theorem: svar-widths-p-of-svar-widths-fix

(defthm svar-widths-p-of-svar-widths-fix
  (b* ((fty::newx (svar-widths-fix$inline x)))
    (svar-widths-p fty::newx))
  :rule-classes :rewrite)

Theorem: svar-widths-fix-when-svar-widths-p

(defthm svar-widths-fix-when-svar-widths-p
  (implies (svar-widths-p x)
           (equal (svar-widths-fix x) x)))

Function: svar-widths-equiv$inline

(defun svar-widths-equiv$inline (x y)
  (declare (xargs :guard (and (svar-widths-p x)
                              (svar-widths-p y))))
  (equal (svar-widths-fix x)
         (svar-widths-fix y)))

Theorem: svar-widths-equiv-is-an-equivalence

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

Theorem: svar-widths-equiv-implies-equal-svar-widths-fix-1

(defthm svar-widths-equiv-implies-equal-svar-widths-fix-1
  (implies (svar-widths-equiv x x-equiv)
           (equal (svar-widths-fix x)
                  (svar-widths-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: svar-widths-fix-under-svar-widths-equiv

(defthm svar-widths-fix-under-svar-widths-equiv
  (svar-widths-equiv (svar-widths-fix x)
                     x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-svar-widths-fix-1-forward-to-svar-widths-equiv

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

Theorem: equal-of-svar-widths-fix-2-forward-to-svar-widths-equiv

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

Theorem: svar-widths-equiv-of-svar-widths-fix-1-forward

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

Theorem: svar-widths-equiv-of-svar-widths-fix-2-forward

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

Theorem: cons-of-svar-fix-k-under-svar-widths-equiv

(defthm cons-of-svar-fix-k-under-svar-widths-equiv
  (svar-widths-equiv (cons (cons (svar-fix acl2::k) acl2::v)
                           x)
                     (cons (cons acl2::k acl2::v) x)))

Theorem: cons-svar-equiv-congruence-on-k-under-svar-widths-equiv

(defthm cons-svar-equiv-congruence-on-k-under-svar-widths-equiv
  (implies (svar-equiv acl2::k k-equiv)
           (svar-widths-equiv (cons (cons acl2::k acl2::v) x)
                              (cons (cons k-equiv acl2::v) x)))
  :rule-classes :congruence)

Theorem: cons-of-ifix-v-under-svar-widths-equiv

(defthm cons-of-ifix-v-under-svar-widths-equiv
  (svar-widths-equiv (cons (cons acl2::k (ifix acl2::v)) x)
                     (cons (cons acl2::k acl2::v) x)))

Theorem: cons-int-equiv-congruence-on-v-under-svar-widths-equiv

(defthm cons-int-equiv-congruence-on-v-under-svar-widths-equiv
  (implies (int-equiv acl2::v v-equiv)
           (svar-widths-equiv (cons (cons acl2::k acl2::v) x)
                              (cons (cons acl2::k v-equiv) x)))
  :rule-classes :congruence)

Theorem: cons-of-svar-widths-fix-y-under-svar-widths-equiv

(defthm cons-of-svar-widths-fix-y-under-svar-widths-equiv
  (svar-widths-equiv (cons x (svar-widths-fix y))
                     (cons x y)))

Theorem: cons-svar-widths-equiv-congruence-on-y-under-svar-widths-equiv

(defthm
     cons-svar-widths-equiv-congruence-on-y-under-svar-widths-equiv
  (implies (svar-widths-equiv y y-equiv)
           (svar-widths-equiv (cons x y)
                              (cons x y-equiv)))
  :rule-classes :congruence)

Theorem: svar-widths-fix-of-acons

(defthm svar-widths-fix-of-acons
  (equal (svar-widths-fix (cons (cons acl2::a acl2::b) x))
         (cons (cons (svar-fix acl2::a) (ifix acl2::b))
               (svar-widths-fix x))))

Theorem: svar-widths-fix-of-append

(defthm svar-widths-fix-of-append
  (equal (svar-widths-fix (append std::a std::b))
         (append (svar-widths-fix std::a)
                 (svar-widths-fix std::b))))

Theorem: consp-car-of-svar-widths-fix

(defthm consp-car-of-svar-widths-fix
  (equal (consp (car (svar-widths-fix x)))
         (consp (svar-widths-fix x))))