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Term-bvars

Term-bvars-fix

(term-bvars-fix x) is an ACL2::fty alist fixing function that follows the drop-keys strategy.

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

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

Definitions and Theorems

Function: term-bvars-fix$inline

(defun term-bvars-fix$inline (x)
  (declare (xargs :guard (term-bvars-p x)))
  (let ((__function__ 'term-bvars-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (let ((rest (term-bvars-fix (cdr x))))
             (if (and (consp (car x))
                      (fgl-object-p (caar x)))
                 (let ((fty::first-key (caar x))
                       (fty::first-val (nfix (cdar x))))
                   (cons (cons fty::first-key fty::first-val)
                         rest))
               rest)))
         :exec x)))

Theorem: term-bvars-p-of-term-bvars-fix

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

Theorem: term-bvars-fix-when-term-bvars-p

(defthm term-bvars-fix-when-term-bvars-p
  (implies (term-bvars-p x)
           (equal (term-bvars-fix x) x)))

Function: term-bvars-equiv$inline

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

Theorem: term-bvars-equiv-is-an-equivalence

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

Theorem: term-bvars-equiv-implies-equal-term-bvars-fix-1

(defthm term-bvars-equiv-implies-equal-term-bvars-fix-1
  (implies (term-bvars-equiv x x-equiv)
           (equal (term-bvars-fix x)
                  (term-bvars-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: term-bvars-fix-under-term-bvars-equiv

(defthm term-bvars-fix-under-term-bvars-equiv
  (term-bvars-equiv (term-bvars-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-term-bvars-fix-1-forward-to-term-bvars-equiv

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

Theorem: equal-of-term-bvars-fix-2-forward-to-term-bvars-equiv

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

Theorem: term-bvars-equiv-of-term-bvars-fix-1-forward

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

Theorem: term-bvars-equiv-of-term-bvars-fix-2-forward

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

Theorem: cons-of-nfix-v-under-term-bvars-equiv

(defthm cons-of-nfix-v-under-term-bvars-equiv
  (term-bvars-equiv (cons (cons k (nfix v)) x)
                    (cons (cons k v) x)))

Theorem: cons-nat-equiv-congruence-on-v-under-term-bvars-equiv

(defthm cons-nat-equiv-congruence-on-v-under-term-bvars-equiv
  (implies (acl2::nat-equiv v v-equiv)
           (term-bvars-equiv (cons (cons k v) x)
                             (cons (cons k v-equiv) x)))
  :rule-classes :congruence)

Theorem: cons-of-term-bvars-fix-y-under-term-bvars-equiv

(defthm cons-of-term-bvars-fix-y-under-term-bvars-equiv
  (term-bvars-equiv (cons x (term-bvars-fix y))
                    (cons x y)))

Theorem: cons-term-bvars-equiv-congruence-on-y-under-term-bvars-equiv

(defthm cons-term-bvars-equiv-congruence-on-y-under-term-bvars-equiv
  (implies (term-bvars-equiv y y-equiv)
           (term-bvars-equiv (cons x y)
                             (cons x y-equiv)))
  :rule-classes :congruence)

Theorem: term-bvars-fix-of-acons

(defthm term-bvars-fix-of-acons
  (equal (term-bvars-fix (cons (cons a b) x))
         (let ((rest (term-bvars-fix x)))
           (if (and (fgl-object-p a))
               (let ((fty::first-key a)
                     (fty::first-val (nfix b)))
                 (cons (cons fty::first-key fty::first-val)
                       rest))
             rest))))

Theorem: hons-assoc-equal-of-term-bvars-fix

(defthm hons-assoc-equal-of-term-bvars-fix
  (equal (hons-assoc-equal k (term-bvars-fix x))
         (let ((fty::pair (hons-assoc-equal k x)))
           (and (fgl-object-p k)
                fty::pair
                (cons k (nfix (cdr fty::pair)))))))

Theorem: term-bvars-fix-of-append

(defthm term-bvars-fix-of-append
  (equal (term-bvars-fix (append std::a std::b))
         (append (term-bvars-fix std::a)
                 (term-bvars-fix std::b))))

Theorem: consp-car-of-term-bvars-fix

(defthm consp-car-of-term-bvars-fix
  (equal (consp (car (term-bvars-fix x)))
         (consp (term-bvars-fix x))))