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Simpadd0-gin

Simpadd0-gin-equiv

Basic equivalence relation for simpadd0-gin structures.

Definitions and Theorems

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)