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• Partsum-comp

Partsum-comp-equiv

Basic equivalence relation for partsum-comp structures.

Definitions and Theorems

Function: partsum-comp-equiv$inline

(defun partsum-comp-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (partsum-comp-p acl2::x)
                              (partsum-comp-p acl2::y))))
  (equal (partsum-comp-fix acl2::x)
         (partsum-comp-fix acl2::y)))

Theorem: partsum-comp-equiv-is-an-equivalence

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

Theorem: partsum-comp-equiv-implies-equal-partsum-comp-fix-1

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

Theorem: partsum-comp-fix-under-partsum-comp-equiv

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

Theorem: equal-of-partsum-comp-fix-1-forward-to-partsum-comp-equiv

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

Theorem: equal-of-partsum-comp-fix-2-forward-to-partsum-comp-equiv

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

Theorem: partsum-comp-equiv-of-partsum-comp-fix-1-forward

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

Theorem: partsum-comp-equiv-of-partsum-comp-fix-2-forward

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