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Binary-op

Binary-op-equiv

Basic equivalence relation for binary-op structures.

Definitions and Theorems

Function: binary-op-equiv$inline

(defun binary-op-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (binary-opp acl2::x)
                              (binary-opp acl2::y))))
  (equal (binary-op-fix acl2::x)
         (binary-op-fix acl2::y)))

Theorem: binary-op-equiv-is-an-equivalence

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

Theorem: binary-op-equiv-implies-equal-binary-op-fix-1

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

Theorem: binary-op-fix-under-binary-op-equiv

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

Theorem: equal-of-binary-op-fix-1-forward-to-binary-op-equiv

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

Theorem: equal-of-binary-op-fix-2-forward-to-binary-op-equiv

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

Theorem: binary-op-equiv-of-binary-op-fix-1-forward

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

Theorem: binary-op-equiv-of-binary-op-fix-2-forward

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