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Sd-key

Sd-key-fix

Fixing function for sd-key structures.

Signature
(sd-key-fix x) → new-x
Arguments: x — Guard (sd-key-p x).
Returns: new-x — Type (sd-key-p new-x).

Definitions and Theorems

Function: sd-key-fix$inline

(defun sd-key-fix$inline (x)
  (declare (xargs :guard (sd-key-p x)))
  (let ((__function__ 'sd-key-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((pat (str-fix (car (cdr x))))
              (index (maybe-natp-fix (car (cdr (cdr x)))))
              (orig (str-fix (cdr (cdr (cdr x))))))
           (cons :sd-key (cons pat (cons index orig))))
         :exec x)))

Theorem: sd-key-p-of-sd-key-fix

(defthm sd-key-p-of-sd-key-fix
  (b* ((new-x (sd-key-fix$inline x)))
    (sd-key-p new-x))
  :rule-classes :rewrite)

Theorem: sd-key-fix-when-sd-key-p

(defthm sd-key-fix-when-sd-key-p
  (implies (sd-key-p x)
           (equal (sd-key-fix x) x)))

Function: sd-key-equiv$inline

(defun sd-key-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (sd-key-p acl2::x)
                              (sd-key-p acl2::y))))
  (equal (sd-key-fix acl2::x)
         (sd-key-fix acl2::y)))

Theorem: sd-key-equiv-is-an-equivalence

(defthm sd-key-equiv-is-an-equivalence
  (and (booleanp (sd-key-equiv x y))
       (sd-key-equiv x x)
       (implies (sd-key-equiv x y)
                (sd-key-equiv y x))
       (implies (and (sd-key-equiv x y)
                     (sd-key-equiv y z))
                (sd-key-equiv x z)))
  :rule-classes (:equivalence))

Theorem: sd-key-equiv-implies-equal-sd-key-fix-1

(defthm sd-key-equiv-implies-equal-sd-key-fix-1
  (implies (sd-key-equiv acl2::x x-equiv)
           (equal (sd-key-fix acl2::x)
                  (sd-key-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: sd-key-fix-under-sd-key-equiv

(defthm sd-key-fix-under-sd-key-equiv
  (sd-key-equiv (sd-key-fix acl2::x)
                acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sd-key-fix-1-forward-to-sd-key-equiv

(defthm equal-of-sd-key-fix-1-forward-to-sd-key-equiv
  (implies (equal (sd-key-fix acl2::x) acl2::y)
           (sd-key-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-sd-key-fix-2-forward-to-sd-key-equiv

(defthm equal-of-sd-key-fix-2-forward-to-sd-key-equiv
  (implies (equal acl2::x (sd-key-fix acl2::y))
           (sd-key-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: sd-key-equiv-of-sd-key-fix-1-forward

(defthm sd-key-equiv-of-sd-key-fix-1-forward
  (implies (sd-key-equiv (sd-key-fix acl2::x)
                         acl2::y)
           (sd-key-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: sd-key-equiv-of-sd-key-fix-2-forward

(defthm sd-key-equiv-of-sd-key-fix-2-forward
  (implies (sd-key-equiv acl2::x (sd-key-fix acl2::y))
           (sd-key-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)