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Uint

Uint-equiv

Basic equivalence relation for uint structures.

Definitions and Theorems

Function: uint-equiv$inline

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

Theorem: uint-equiv-is-an-equivalence

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

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

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

Theorem: uint-fix-under-uint-equiv

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

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

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

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

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

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

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

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

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