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Comm

Comm-equiv

Basic equivalence relation for comm structures.

Definitions and Theorems

Function: comm-equiv$inline

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

Theorem: comm-equiv-is-an-equivalence

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

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

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

Theorem: comm-fix-under-comm-equiv

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

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

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

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

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

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

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

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

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