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Any-nat-map

Any-nat-mapp

Recognizer for any-nat-map.

Signature
(any-nat-mapp x) → *

Definitions and Theorems

Function: any-nat-mapp

(defun any-nat-mapp (x)
  (declare (xargs :guard t))
  (if (atom x)
      (null x)
    (and (consp (car x))
         (any-p (caar x))
         (natp (cdar x))
         (or (null (cdr x))
             (and (consp (cdr x))
                  (consp (cadr x))
                  (fast-<< (caar x) (caadr x))
                  (any-nat-mapp (cdr x)))))))

Theorem: booleanp-of-any-nat-mapp

(defthm booleanp-of-any-nat-mapp
  (booleanp (any-nat-mapp x)))

Theorem: mapp-when-any-nat-mapp

(defthm mapp-when-any-nat-mapp
  (implies (any-nat-mapp x)
           (omap::mapp x))
  :rule-classes (:rewrite :forward-chaining))

Theorem: any-nat-mapp-of-tail

(defthm any-nat-mapp-of-tail
  (implies (any-nat-mapp x)
           (any-nat-mapp (omap::tail x))))

Theorem: any-p-of-head-key-when-any-nat-mapp

(defthm any-p-of-head-key-when-any-nat-mapp
  (implies (and (any-nat-mapp x)
                (not (omap::emptyp x)))
           (any-p (mv-nth 0 (omap::head x)))))

Theorem: natp-of-head-val-when-any-nat-mapp

(defthm natp-of-head-val-when-any-nat-mapp
  (implies (and (any-nat-mapp x)
                (not (omap::emptyp x)))
           (natp (mv-nth 1 (omap::head x)))))

Theorem: any-nat-mapp-of-update

(defthm any-nat-mapp-of-update
  (implies (and (any-nat-mapp x)
                (any-p k)
                (natp v))
           (any-nat-mapp (omap::update k v x))))

Theorem: any-nat-mapp-of-update*

(defthm any-nat-mapp-of-update*
  (implies (and (any-nat-mapp x) (any-nat-mapp y))
           (any-nat-mapp (omap::update* x y))))

Theorem: any-nat-mapp-of-delete

(defthm any-nat-mapp-of-delete
  (implies (any-nat-mapp x)
           (any-nat-mapp (omap::delete k x))))

Theorem: any-nat-mapp-of-delete*

(defthm any-nat-mapp-of-delete*
  (implies (any-nat-mapp x)
           (any-nat-mapp (omap::delete* k x))))

Theorem: any-p-when-assoc-any-nat-mapp-binds-free-x

(defthm any-p-when-assoc-any-nat-mapp-binds-free-x
  (implies (and (omap::assoc k x) (any-nat-mapp x))
           (any-p k)))

Theorem: any-p-of-car-of-assoc-any-nat-mapp

(defthm any-p-of-car-of-assoc-any-nat-mapp
  (implies (and (any-nat-mapp x) (omap::assoc k x))
           (any-p (car (omap::assoc k x)))))

Theorem: natp-of-cdr-of-assoc-any-nat-mapp

(defthm natp-of-cdr-of-assoc-any-nat-mapp
  (implies (and (any-nat-mapp x) (omap::assoc k x))
           (natp (cdr (omap::assoc k x)))))

Theorem: natp-of-lookup-when-any-nat-mapp

(defthm natp-of-lookup-when-any-nat-mapp
  (implies (and (any-nat-mapp x) (omap::assoc k x))
           (natp (omap::lookup k x))))