		Search-engine friendly clone of the ACL2 documentation.

Nat-set

Nat-sequiv

Basic equivalence relation for nat-set structures.

Definitions and Theorems

Function: nat-sequiv$inline

(defun nat-sequiv$inline (x y)
  (declare (xargs :guard (and (nat-setp x) (nat-setp y))))
  (equal (nat-sfix x) (nat-sfix y)))

Theorem: nat-sequiv-is-an-equivalence

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

Theorem: nat-sequiv-implies-equal-nat-sfix-1

(defthm nat-sequiv-implies-equal-nat-sfix-1
  (implies (nat-sequiv x x-equiv)
           (equal (nat-sfix x) (nat-sfix x-equiv)))
  :rule-classes (:congruence))

Theorem: nat-sfix-under-nat-sequiv

(defthm nat-sfix-under-nat-sequiv
  (nat-sequiv (nat-sfix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-nat-sfix-1-forward-to-nat-sequiv

(defthm equal-of-nat-sfix-1-forward-to-nat-sequiv
  (implies (equal (nat-sfix x) y)
           (nat-sequiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-nat-sfix-2-forward-to-nat-sequiv

(defthm equal-of-nat-sfix-2-forward-to-nat-sequiv
  (implies (equal x (nat-sfix y))
           (nat-sequiv x y))
  :rule-classes :forward-chaining)

Theorem: nat-sequiv-of-nat-sfix-1-forward

(defthm nat-sequiv-of-nat-sfix-1-forward
  (implies (nat-sequiv (nat-sfix x) y)
           (nat-sequiv x y))
  :rule-classes :forward-chaining)

Theorem: nat-sequiv-of-nat-sfix-2-forward

(defthm nat-sequiv-of-nat-sfix-2-forward
  (implies (nat-sequiv x (nat-sfix y))
           (nat-sequiv x y))
  :rule-classes :forward-chaining)