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Nat-result

Nat-result-fix

Fixing function for nat-result structures.

Signature
(nat-result-fix x) → new-x
Arguments: x — Guard (nat-resultp x).
Returns: new-x — Type (nat-resultp new-x).

Definitions and Theorems

Function: nat-result-fix$inline

(defun nat-result-fix$inline (x)
  (declare (xargs :guard (nat-resultp x)))
  (let ((__function__ 'nat-result-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (nat-result-kind x)
           (:ok (b* ((get (nfix x))) get))
           (:err (b* ((get (fty::reserr-fix x))) get)))
         :exec x)))

Theorem: nat-resultp-of-nat-result-fix

(defthm nat-resultp-of-nat-result-fix
  (b* ((new-x (nat-result-fix$inline x)))
    (nat-resultp new-x))
  :rule-classes :rewrite)

Theorem: nat-result-fix-when-nat-resultp

(defthm nat-result-fix-when-nat-resultp
  (implies (nat-resultp x)
           (equal (nat-result-fix x) x)))

Function: nat-result-equiv$inline

(defun nat-result-equiv$inline (x y)
  (declare (xargs :guard (and (nat-resultp x) (nat-resultp y))))
  (equal (nat-result-fix x)
         (nat-result-fix y)))

Theorem: nat-result-equiv-is-an-equivalence

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

Theorem: nat-result-equiv-implies-equal-nat-result-fix-1

(defthm nat-result-equiv-implies-equal-nat-result-fix-1
  (implies (nat-result-equiv x x-equiv)
           (equal (nat-result-fix x)
                  (nat-result-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: nat-result-fix-under-nat-result-equiv

(defthm nat-result-fix-under-nat-result-equiv
  (nat-result-equiv (nat-result-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-nat-result-fix-1-forward-to-nat-result-equiv

(defthm equal-of-nat-result-fix-1-forward-to-nat-result-equiv
  (implies (equal (nat-result-fix x) y)
           (nat-result-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-nat-result-fix-2-forward-to-nat-result-equiv

(defthm equal-of-nat-result-fix-2-forward-to-nat-result-equiv
  (implies (equal x (nat-result-fix y))
           (nat-result-equiv x y))
  :rule-classes :forward-chaining)

Theorem: nat-result-equiv-of-nat-result-fix-1-forward

(defthm nat-result-equiv-of-nat-result-fix-1-forward
  (implies (nat-result-equiv (nat-result-fix x) y)
           (nat-result-equiv x y))
  :rule-classes :forward-chaining)

Theorem: nat-result-equiv-of-nat-result-fix-2-forward

(defthm nat-result-equiv-of-nat-result-fix-2-forward
  (implies (nat-result-equiv x (nat-result-fix y))
           (nat-result-equiv x y))
  :rule-classes :forward-chaining)

Theorem: nat-result-kind$inline-of-nat-result-fix-x

(defthm nat-result-kind$inline-of-nat-result-fix-x
  (equal (nat-result-kind$inline (nat-result-fix x))
         (nat-result-kind$inline x)))

Theorem: nat-result-kind$inline-nat-result-equiv-congruence-on-x

(defthm nat-result-kind$inline-nat-result-equiv-congruence-on-x
  (implies (nat-result-equiv x x-equiv)
           (equal (nat-result-kind$inline x)
                  (nat-result-kind$inline x-equiv)))
  :rule-classes :congruence)