		Search-engine friendly clone of the ACL2 documentation.

Simpadd0-gin

Simpadd0-gin-fix

Fixing function for simpadd0-gin structures.

Signature
(simpadd0-gin-fix x) → new-x
Arguments: x — Guard (simpadd0-ginp x).
Returns: new-x — Type (simpadd0-ginp new-x).

Definitions and Theorems

Function: simpadd0-gin-fix$inline

(defun simpadd0-gin-fix$inline (x)
  (declare (xargs :guard (simpadd0-ginp x)))
  (let ((__function__ 'simpadd0-gin-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((thm-index (acl2::pos-fix (cdr (std::da-nth 0 x))))
              (names-to-avoid
                   (acl2::symbol-list-fix (cdr (std::da-nth 1 x)))))
           (list (cons 'thm-index thm-index)
                 (cons 'names-to-avoid names-to-avoid)))
         :exec x)))

Theorem: simpadd0-ginp-of-simpadd0-gin-fix

(defthm simpadd0-ginp-of-simpadd0-gin-fix
  (b* ((new-x (simpadd0-gin-fix$inline x)))
    (simpadd0-ginp new-x))
  :rule-classes :rewrite)

Theorem: simpadd0-gin-fix-when-simpadd0-ginp

(defthm simpadd0-gin-fix-when-simpadd0-ginp
  (implies (simpadd0-ginp x)
           (equal (simpadd0-gin-fix x) x)))

Function: simpadd0-gin-equiv$inline

(defun simpadd0-gin-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (simpadd0-ginp acl2::x)
                              (simpadd0-ginp acl2::y))))
  (equal (simpadd0-gin-fix acl2::x)
         (simpadd0-gin-fix acl2::y)))

Theorem: simpadd0-gin-equiv-is-an-equivalence

(defthm simpadd0-gin-equiv-is-an-equivalence
  (and (booleanp (simpadd0-gin-equiv x y))
       (simpadd0-gin-equiv x x)
       (implies (simpadd0-gin-equiv x y)
                (simpadd0-gin-equiv y x))
       (implies (and (simpadd0-gin-equiv x y)
                     (simpadd0-gin-equiv y z))
                (simpadd0-gin-equiv x z)))
  :rule-classes (:equivalence))

Theorem: simpadd0-gin-equiv-implies-equal-simpadd0-gin-fix-1

(defthm simpadd0-gin-equiv-implies-equal-simpadd0-gin-fix-1
  (implies (simpadd0-gin-equiv acl2::x x-equiv)
           (equal (simpadd0-gin-fix acl2::x)
                  (simpadd0-gin-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: simpadd0-gin-fix-under-simpadd0-gin-equiv

(defthm simpadd0-gin-fix-under-simpadd0-gin-equiv
  (simpadd0-gin-equiv (simpadd0-gin-fix acl2::x)
                      acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-simpadd0-gin-fix-1-forward-to-simpadd0-gin-equiv

(defthm equal-of-simpadd0-gin-fix-1-forward-to-simpadd0-gin-equiv
  (implies (equal (simpadd0-gin-fix acl2::x)
                  acl2::y)
           (simpadd0-gin-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-simpadd0-gin-fix-2-forward-to-simpadd0-gin-equiv

(defthm equal-of-simpadd0-gin-fix-2-forward-to-simpadd0-gin-equiv
  (implies (equal acl2::x (simpadd0-gin-fix acl2::y))
           (simpadd0-gin-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: simpadd0-gin-equiv-of-simpadd0-gin-fix-1-forward

(defthm simpadd0-gin-equiv-of-simpadd0-gin-fix-1-forward
  (implies (simpadd0-gin-equiv (simpadd0-gin-fix acl2::x)
                               acl2::y)
           (simpadd0-gin-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: simpadd0-gin-equiv-of-simpadd0-gin-fix-2-forward

(defthm simpadd0-gin-equiv-of-simpadd0-gin-fix-2-forward
  (implies (simpadd0-gin-equiv acl2::x (simpadd0-gin-fix acl2::y))
           (simpadd0-gin-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)