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• Partsum-elt

Partsum-elt-equiv

Basic equivalence relation for partsum-elt structures.

Definitions and Theorems

Function: partsum-elt-equiv$inline

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

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

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

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

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

Theorem: partsum-elt-fix-under-partsum-elt-equiv

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

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

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

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

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

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

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

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

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