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Nati

Nati-equiv

Basic equivalence relation for nati structures.

Definitions and Theorems

Function: nati-equiv$inline

(defun nati-equiv$inline (x y)
  (declare (xargs :guard (and (natip x) (natip y))))
  (equal (nati-fix x) (nati-fix y)))

Theorem: nati-equiv-is-an-equivalence

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

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

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

Theorem: nati-fix-under-nati-equiv

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

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

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

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

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

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

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

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

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