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Binop

Binop-equiv

Basic equivalence relation for binop structures.

Definitions and Theorems

Function: binop-equiv$inline

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

Theorem: binop-equiv-is-an-equivalence

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

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

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

Theorem: binop-fix-under-binop-equiv

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

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

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

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

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

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

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

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

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