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• Bip32-path-set

Bip32-path-sfix

(bip32-path-sfix x) is a usual ACL2::fty set fixing function.

Signature

(bip32-path-sfix x) → *

Arguments

x — Guard (bip32-path-setp x).

In the logic, we apply ubyte32-list-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.

Definitions and Theorems

Function: bip32-path-sfix

(defun bip32-path-sfix (x)
  (declare (xargs :guard (bip32-path-setp x)))
  (mbe :logic (if (bip32-path-setp x) x nil)
       :exec x))

Theorem: bip32-path-setp-of-bip32-path-sfix

(defthm bip32-path-setp-of-bip32-path-sfix
  (bip32-path-setp (bip32-path-sfix x)))

Theorem: bip32-path-sfix-when-bip32-path-setp

(defthm bip32-path-sfix-when-bip32-path-setp
  (implies (bip32-path-setp x)
           (equal (bip32-path-sfix x) x)))

Theorem: emptyp-bip32-path-sfix

(defthm emptyp-bip32-path-sfix
  (implies (or (emptyp x)
               (not (bip32-path-setp x)))
           (emptyp (bip32-path-sfix x))))

Theorem: emptyp-of-bip32-path-sfix

(defthm emptyp-of-bip32-path-sfix
  (equal (emptyp (bip32-path-sfix x))
         (or (not (bip32-path-setp x))
             (emptyp x))))

Function: bip32-path-set-equiv$inline

(defun bip32-path-set-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (bip32-path-setp acl2::x)
                              (bip32-path-setp acl2::y))))
  (equal (bip32-path-sfix acl2::x)
         (bip32-path-sfix acl2::y)))

Theorem: bip32-path-set-equiv-is-an-equivalence

(defthm bip32-path-set-equiv-is-an-equivalence
  (and (booleanp (bip32-path-set-equiv x y))
       (bip32-path-set-equiv x x)
       (implies (bip32-path-set-equiv x y)
                (bip32-path-set-equiv y x))
       (implies (and (bip32-path-set-equiv x y)
                     (bip32-path-set-equiv y z))
                (bip32-path-set-equiv x z)))
  :rule-classes (:equivalence))

Theorem: bip32-path-set-equiv-implies-equal-bip32-path-sfix-1

(defthm bip32-path-set-equiv-implies-equal-bip32-path-sfix-1
  (implies (bip32-path-set-equiv acl2::x x-equiv)
           (equal (bip32-path-sfix acl2::x)
                  (bip32-path-sfix x-equiv)))
  :rule-classes (:congruence))

Theorem: bip32-path-sfix-under-bip32-path-set-equiv

(defthm bip32-path-sfix-under-bip32-path-set-equiv
  (bip32-path-set-equiv (bip32-path-sfix acl2::x)
                        acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-bip32-path-sfix-1-forward-to-bip32-path-set-equiv

(defthm equal-of-bip32-path-sfix-1-forward-to-bip32-path-set-equiv
  (implies (equal (bip32-path-sfix acl2::x)
                  acl2::y)
           (bip32-path-set-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-bip32-path-sfix-2-forward-to-bip32-path-set-equiv

(defthm equal-of-bip32-path-sfix-2-forward-to-bip32-path-set-equiv
  (implies (equal acl2::x (bip32-path-sfix acl2::y))
           (bip32-path-set-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bip32-path-set-equiv-of-bip32-path-sfix-1-forward

(defthm bip32-path-set-equiv-of-bip32-path-sfix-1-forward
  (implies (bip32-path-set-equiv (bip32-path-sfix acl2::x)
                                 acl2::y)
           (bip32-path-set-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bip32-path-set-equiv-of-bip32-path-sfix-2-forward

(defthm bip32-path-set-equiv-of-bip32-path-sfix-2-forward
  (implies (bip32-path-set-equiv acl2::x (bip32-path-sfix acl2::y))
           (bip32-path-set-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)