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Gin-equiv

Basic equivalence relation for gin structures.

Definitions and Theorems

Function: gin-equiv$inline

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

Theorem: gin-equiv-is-an-equivalence

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

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

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

Theorem: gin-fix-under-gin-equiv

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

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

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

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

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

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

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

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

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