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Bip32-path-set-closedp-executable-attachment

Bip32-path-set-closedp-exec

Executable version of bip32-path-set-closedp.

Signature
(bip32-path-set-closedp-exec paths) → yes/no
Arguments: paths — Guard (bip32-path-setp paths).
Returns: yes/no — Type (booleanp yes/no).

Roughly, bip32-path-set-closedp has the form

(forall (path prefix)
  (implies (and (in path paths)
                (true-listp prefix)
                (prefixp prefix path))
           (in prefix paths)))

which can be written as

(forall (path)
        (implies (in path paths)
                 (forall (prefix)
                         (implies (and (true-listp prefix)
                                       (prefixp prefix path))
                                  (in prefix paths)))))

We execute this via an outer iteration of path over paths and an inner iteration of prefix over the prefixes of path; these two iterations are realized via recursive functions. Their all-paths argument never changes; it is the initial set of all paths, that the prefixes must be checked to be in. The cur-path argument is the path in the outer iteration, and the cur-paths argument is the subset of paths that remains to process in the outer iteration.

Definitions and Theorems

Function: bip32-path-set-closedp-exec-inner

(defun bip32-path-set-closedp-exec-inner (cur-path all-paths)
  (declare (xargs :guard (and (ubyte32-listp cur-path)
                              (bip32-path-setp all-paths))))
  (and (in (ubyte32-list-fix cur-path)
           (bip32-path-sfix all-paths))
       (or (endp cur-path)
           (bip32-path-set-closedp-exec-inner (butlast cur-path 1)
                                              all-paths))))

Theorem: booleanp-of-bip32-path-set-closedp-exec-inner

(defthm booleanp-of-bip32-path-set-closedp-exec-inner
  (b*
   ((yes/no (bip32-path-set-closedp-exec-inner cur-path all-paths)))
   (booleanp yes/no))
  :rule-classes :rewrite)

Theorem: bip32-path-set-closedp-exec-inner-of-ubyte32-list-fix-cur-path

(defthm
     bip32-path-set-closedp-exec-inner-of-ubyte32-list-fix-cur-path
 (equal
      (bip32-path-set-closedp-exec-inner (ubyte32-list-fix cur-path)
                                         all-paths)
      (bip32-path-set-closedp-exec-inner cur-path all-paths)))

Theorem: bip32-path-set-closedp-exec-inner-ubyte32-list-equiv-congruence-on-cur-path

(defthm
 bip32-path-set-closedp-exec-inner-ubyte32-list-equiv-congruence-on-cur-path
 (implies
  (acl2::ubyte32-list-equiv cur-path cur-path-equiv)
  (equal
      (bip32-path-set-closedp-exec-inner cur-path all-paths)
      (bip32-path-set-closedp-exec-inner cur-path-equiv all-paths)))
 :rule-classes :congruence)

Theorem: bip32-path-set-closedp-exec-inner-of-bip32-path-sfix-all-paths

(defthm
     bip32-path-set-closedp-exec-inner-of-bip32-path-sfix-all-paths
  (equal (bip32-path-set-closedp-exec-inner
              cur-path (bip32-path-sfix all-paths))
         (bip32-path-set-closedp-exec-inner cur-path all-paths)))

Theorem: bip32-path-set-closedp-exec-inner-bip32-path-set-equiv-congruence-on-all-paths

(defthm
 bip32-path-set-closedp-exec-inner-bip32-path-set-equiv-congruence-on-all-paths
 (implies
  (bip32-path-set-equiv all-paths all-paths-equiv)
  (equal
      (bip32-path-set-closedp-exec-inner cur-path all-paths)
      (bip32-path-set-closedp-exec-inner cur-path all-paths-equiv)))
 :rule-classes :congruence)

Function: bip32-path-set-closedp-exec-outer

(defun bip32-path-set-closedp-exec-outer (cur-paths all-paths)
  (declare (xargs :guard (and (bip32-path-setp cur-paths)
                              (bip32-path-setp all-paths))))
  (or (not (mbt (bip32-path-setp cur-paths)))
      (emptyp cur-paths)
      (and (bip32-path-set-closedp-exec-inner (head cur-paths)
                                              all-paths)
           (bip32-path-set-closedp-exec-outer (tail cur-paths)
                                              all-paths))))

Theorem: booleanp-of-bip32-path-set-closedp-exec-outer

(defthm booleanp-of-bip32-path-set-closedp-exec-outer
 (b*
  ((yes/no (bip32-path-set-closedp-exec-outer cur-paths all-paths)))
  (booleanp yes/no))
 :rule-classes :rewrite)

Theorem: bip32-path-set-closedp-exec-outer-of-bip32-path-sfix-cur-paths

(defthm
     bip32-path-set-closedp-exec-outer-of-bip32-path-sfix-cur-paths
 (equal
      (bip32-path-set-closedp-exec-outer (bip32-path-sfix cur-paths)
                                         all-paths)
      (bip32-path-set-closedp-exec-outer cur-paths all-paths)))

Theorem: bip32-path-set-closedp-exec-outer-bip32-path-set-equiv-congruence-on-cur-paths

(defthm
 bip32-path-set-closedp-exec-outer-bip32-path-set-equiv-congruence-on-cur-paths
 (implies
  (bip32-path-set-equiv cur-paths cur-paths-equiv)
  (equal
     (bip32-path-set-closedp-exec-outer cur-paths all-paths)
     (bip32-path-set-closedp-exec-outer cur-paths-equiv all-paths)))
 :rule-classes :congruence)

Theorem: bip32-path-set-closedp-exec-outer-of-bip32-path-sfix-all-paths

(defthm
     bip32-path-set-closedp-exec-outer-of-bip32-path-sfix-all-paths
  (equal (bip32-path-set-closedp-exec-outer
              cur-paths (bip32-path-sfix all-paths))
         (bip32-path-set-closedp-exec-outer cur-paths all-paths)))

Theorem: bip32-path-set-closedp-exec-outer-bip32-path-set-equiv-congruence-on-all-paths

(defthm
 bip32-path-set-closedp-exec-outer-bip32-path-set-equiv-congruence-on-all-paths
 (implies
  (bip32-path-set-equiv all-paths all-paths-equiv)
  (equal
     (bip32-path-set-closedp-exec-outer cur-paths all-paths)
     (bip32-path-set-closedp-exec-outer cur-paths all-paths-equiv)))
 :rule-classes :congruence)

Function: bip32-path-set-closedp-exec

(defun bip32-path-set-closedp-exec (paths)
  (declare (xargs :guard (bip32-path-setp paths)))
  (bip32-path-set-closedp-exec-outer paths paths))

Theorem: booleanp-of-bip32-path-set-closedp-exec

(defthm booleanp-of-bip32-path-set-closedp-exec
  (b* ((yes/no (bip32-path-set-closedp-exec paths)))
    (booleanp yes/no))
  :rule-classes :rewrite)

Theorem: bip32-path-set-closedp-exec-of-bip32-path-sfix-paths

(defthm bip32-path-set-closedp-exec-of-bip32-path-sfix-paths
  (equal (bip32-path-set-closedp-exec (bip32-path-sfix paths))
         (bip32-path-set-closedp-exec paths)))

Theorem: bip32-path-set-closedp-exec-bip32-path-set-equiv-congruence-on-paths

(defthm
 bip32-path-set-closedp-exec-bip32-path-set-equiv-congruence-on-paths
 (implies (bip32-path-set-equiv paths paths-equiv)
          (equal (bip32-path-set-closedp-exec paths)
                 (bip32-path-set-closedp-exec paths-equiv)))
 :rule-classes :congruence)