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• Bip32-path-set-closedp-executable-attachment

Bip32-path-set-closedp-exec-correctness

Correctness of bip32-path-set-closedp-exec.

We prove that bip32-path-set-closedp-exec is equivalent to bip32-path-set-closedp.

We first prove that each one implies the other, and then the equality follows from the fact that these functions are boolean-valued.

To prove that bip32-path-set-closedp-exec implies bip32-path-set-closedp, first recall the nested quantification of the latter illustrated in the documentation of bip32-path-set-closedp-exec: we prove that bip32-path-set-closedp-exec-inner implies the inner matrix, and that bip32-path-set-closedp-exec-outer implies the outer matrix. From these two lemmas, the desired theorem follows.

To prove that bip32-path-set-closedp implies bip32-path-set-closedp-exec, we first prove that it implies bip32-path-set-closedp-exec-inner and bip32-path-set-closedp-exec-outer.

Definitions and Theorems

Theorem: bip32-path-set-closedp-exec-inner-implies-inner-matrix

(defthm bip32-path-set-closedp-exec-inner-implies-inner-matrix
 (implies
      (and (bip32-path-set-closedp-exec-inner cur-path all-paths)
           (ubyte32-listp cur-path)
           (true-listp prefix)
           (prefixp prefix cur-path))
      (in prefix all-paths))
 :rule-classes
 ((:rewrite
    :corollary
    (implies
         (and (prefixp prefix cur-path)
              (bip32-path-set-closedp-exec-inner cur-path all-paths)
              (ubyte32-listp cur-path)
              (true-listp prefix))
         (in prefix all-paths)))))

Theorem: bip32-path-set-closedp-exec-outer-implies-outer-matrix

(defthm bip32-path-set-closedp-exec-outer-implies-outer-matrix
  (implies
       (and (bip32-path-set-closedp-exec-outer cur-paths all-paths)
            (in cur-path (bip32-path-sfix cur-paths)))
       (bip32-path-set-closedp-exec-inner cur-path all-paths)))

Theorem: bip32-path-set-closedp-when-bip32-path-set-closedp-exec

(defthm bip32-path-set-closedp-when-bip32-path-set-closedp-exec
  (implies (bip32-path-set-closedp-exec paths)
           (bip32-path-set-closedp paths)))

Theorem: bip32-path-set-closedp-exec-inner-when-bip32-path-set-closedp

(defthm
      bip32-path-set-closedp-exec-inner-when-bip32-path-set-closedp
  (implies (and (bip32-path-set-closedp all-paths)
                (bip32-path-setp all-paths)
                (ubyte32-listp cur-path)
                (in cur-path all-paths))
           (bip32-path-set-closedp-exec-inner cur-path all-paths)))

Theorem: bip32-path-set-closedp-exec-outer-when-bip32-path-set-closedp

(defthm
      bip32-path-set-closedp-exec-outer-when-bip32-path-set-closedp
  (implies (and (bip32-path-set-closedp all-paths)
                (bip32-path-setp all-paths)
                (subset cur-paths all-paths))
           (bip32-path-set-closedp-exec-outer cur-paths all-paths)))

Theorem: bip32-path-set-closedp-exec-when-bip32-path-set-closedp

(defthm bip32-path-set-closedp-exec-when-bip32-path-set-closedp
  (implies (bip32-path-set-closedp paths)
           (bip32-path-set-closedp-exec paths)))

Theorem: bip32-path-set-closedp-exec-is-bip32-path-set-closedp

(defthm bip32-path-set-closedp-exec-is-bip32-path-set-closedp
  (equal (bip32-path-set-closedp-exec paths)
         (bip32-path-set-closedp paths)))