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Svex-alist-compose

Signature

(svex-alist-compose x a) → xx

Arguments

x — Guard (svex-alist-p x).

a — Guard (svex-alist-p a).

Returns

xx — Type (svex-alist-p xx).

Definitions and Theorems

Function: svex-alist-compose

(defun svex-alist-compose (x a)
  (declare (xargs :guard (and (svex-alist-p x)
                              (svex-alist-p a))))
  (let ((__function__ 'svex-alist-compose))
    (declare (ignorable __function__))
    (if (atom x)
        nil
      (mbe :logic
           (if (mbt (and (consp (car x)) (svar-p (caar x))))
               (svex-acons (caar x)
                           (svex-compose (cdar x) a)
                           (svex-alist-compose (cdr x) a))
             (svex-alist-compose (cdr x) a))
           :exec (acl2::with-local-nrev
                      (svex-alist-compose-nrev x a acl2::nrev))))))

Theorem: svex-alist-p-of-svex-alist-compose

(defthm svex-alist-p-of-svex-alist-compose
  (b* ((xx (svex-alist-compose x a)))
    (svex-alist-p xx))
  :rule-classes :rewrite)

Theorem: svex-alist-compose-of-svex-alist-fix-x

(defthm svex-alist-compose-of-svex-alist-fix-x
  (equal (svex-alist-compose (svex-alist-fix x)
                             a)
         (svex-alist-compose x a)))

Theorem: svex-alist-compose-svex-alist-equiv-congruence-on-x

(defthm svex-alist-compose-svex-alist-equiv-congruence-on-x
  (implies (svex-alist-equiv x x-equiv)
           (equal (svex-alist-compose x a)
                  (svex-alist-compose x-equiv a)))
  :rule-classes :congruence)

Theorem: svex-alist-compose-of-svex-alist-fix-a

(defthm svex-alist-compose-of-svex-alist-fix-a
  (equal (svex-alist-compose x (svex-alist-fix a))
         (svex-alist-compose x a)))

Theorem: svex-alist-compose-svex-alist-equiv-congruence-on-a

(defthm svex-alist-compose-svex-alist-equiv-congruence-on-a
  (implies (svex-alist-equiv a a-equiv)
           (equal (svex-alist-compose x a)
                  (svex-alist-compose x a-equiv)))
  :rule-classes :congruence)

Theorem: svex-alist-eval-of-svex-compose

(defthm svex-alist-eval-of-svex-compose
  (equal (svex-alist-eval (svex-alist-compose x subst)
                          env)
         (svex-alist-eval x
                          (append (svex-alist-eval subst env)
                                  env))))

Theorem: vars-of-svex-alist-compose

(defthm vars-of-svex-alist-compose
  (implies
       (and (not (member v (svex-alist-vars x)))
            (not (member v (svex-alist-vars a))))
       (not (member v
                    (svex-alist-vars (svex-alist-compose x a))))))

Theorem: svex-lookup-of-svex-alist-compose

(defthm svex-lookup-of-svex-alist-compose
  (equal (svex-lookup v (svex-alist-compose x a))
         (let ((x-look (svex-lookup v x)))
           (and x-look (svex-compose x-look a)))))