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Exprs-gin

Exprs-gin-equiv

Basic equivalence relation for exprs-gin structures.

Definitions and Theorems

Function: exprs-gin-equiv$inline

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

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

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

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

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

Theorem: exprs-gin-fix-under-exprs-gin-equiv

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

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

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

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

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

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

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

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

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