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• Block-item

Block-item-equiv

Basic equivalence relation for block-item structures.

Definitions and Theorems

Function: block-item-equiv$inline

(defun block-item-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (block-itemp acl2::x)
                              (block-itemp acl2::y))))
  (equal (block-item-fix acl2::x)
         (block-item-fix acl2::y)))

Theorem: block-item-equiv-is-an-equivalence

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

Theorem: block-item-equiv-implies-equal-block-item-fix-1

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

Theorem: block-item-fix-under-block-item-equiv

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

Theorem: equal-of-block-item-fix-1-forward-to-block-item-equiv

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

Theorem: equal-of-block-item-fix-2-forward-to-block-item-equiv

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

Theorem: block-item-equiv-of-block-item-fix-1-forward

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

Theorem: block-item-equiv-of-block-item-fix-2-forward

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