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• Position

Position-equiv

Basic equivalence relation for position structures.

Definitions and Theorems

Function: position-equiv$inline

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

Theorem: position-equiv-is-an-equivalence

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

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

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

Theorem: position-fix-under-position-equiv

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

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

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

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

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

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

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

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

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