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• Api+senv

Api+senv-fix

Fixing function for api+senv structures.

Signature

(api+senv-fix x) → new-x

Arguments

x — Guard (api+senv-p x).

Returns

new-x — Type (api+senv-p new-x).

Definitions and Theorems

Function: api+senv-fix$inline

(defun api+senv-fix$inline (x)
  (declare (xargs :guard (api+senv-p x)))
  (let ((__function__ 'api+senv-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((api (api-fix (cdr (std::da-nth 0 (cdr x)))))
              (senv (senv-fix (cdr (std::da-nth 1 (cdr x))))))
           (cons :api+senv (list (cons 'api api)
                                 (cons 'senv senv))))
         :exec x)))

Theorem: api+senv-p-of-api+senv-fix

(defthm api+senv-p-of-api+senv-fix
  (b* ((new-x (api+senv-fix$inline x)))
    (api+senv-p new-x))
  :rule-classes :rewrite)

Theorem: api+senv-fix-when-api+senv-p

(defthm api+senv-fix-when-api+senv-p
  (implies (api+senv-p x)
           (equal (api+senv-fix x) x)))

Function: api+senv-equiv$inline

(defun api+senv-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (api+senv-p acl2::x)
                              (api+senv-p acl2::y))))
  (equal (api+senv-fix acl2::x)
         (api+senv-fix acl2::y)))

Theorem: api+senv-equiv-is-an-equivalence

(defthm api+senv-equiv-is-an-equivalence
  (and (booleanp (api+senv-equiv x y))
       (api+senv-equiv x x)
       (implies (api+senv-equiv x y)
                (api+senv-equiv y x))
       (implies (and (api+senv-equiv x y)
                     (api+senv-equiv y z))
                (api+senv-equiv x z)))
  :rule-classes (:equivalence))

Theorem: api+senv-equiv-implies-equal-api+senv-fix-1

(defthm api+senv-equiv-implies-equal-api+senv-fix-1
  (implies (api+senv-equiv acl2::x x-equiv)
           (equal (api+senv-fix acl2::x)
                  (api+senv-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: api+senv-fix-under-api+senv-equiv

(defthm api+senv-fix-under-api+senv-equiv
  (api+senv-equiv (api+senv-fix acl2::x)
                  acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-api+senv-fix-1-forward-to-api+senv-equiv

(defthm equal-of-api+senv-fix-1-forward-to-api+senv-equiv
  (implies (equal (api+senv-fix acl2::x) acl2::y)
           (api+senv-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-api+senv-fix-2-forward-to-api+senv-equiv

(defthm equal-of-api+senv-fix-2-forward-to-api+senv-equiv
  (implies (equal acl2::x (api+senv-fix acl2::y))
           (api+senv-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: api+senv-equiv-of-api+senv-fix-1-forward

(defthm api+senv-equiv-of-api+senv-fix-1-forward
  (implies (api+senv-equiv (api+senv-fix acl2::x)
                           acl2::y)
           (api+senv-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: api+senv-equiv-of-api+senv-fix-2-forward

(defthm api+senv-equiv-of-api+senv-fix-2-forward
  (implies (api+senv-equiv acl2::x (api+senv-fix acl2::y))
           (api+senv-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)