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• Int

Int-equiv

Basic equivalence relation for int structures.

Definitions and Theorems

Function: int-equiv$inline

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

Theorem: int-equiv-is-an-equivalence

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

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

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

Theorem: int-fix-under-int-equiv

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

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

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

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

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

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

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

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

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