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Integer-result

Integer-result-equiv

Basic equivalence relation for integer-result structures.

Definitions and Theorems

Function: integer-result-equiv$inline

(defun integer-result-equiv$inline (x y)
  (declare (xargs :guard (and (integer-resultp x)
                              (integer-resultp y))))
  (equal (integer-result-fix x)
         (integer-result-fix y)))

Theorem: integer-result-equiv-is-an-equivalence

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

Theorem: integer-result-equiv-implies-equal-integer-result-fix-1

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

Theorem: integer-result-fix-under-integer-result-equiv

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

Theorem: equal-of-integer-result-fix-1-forward-to-integer-result-equiv

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

Theorem: equal-of-integer-result-fix-2-forward-to-integer-result-equiv

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

Theorem: integer-result-equiv-of-integer-result-fix-1-forward

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

Theorem: integer-result-equiv-of-integer-result-fix-2-forward

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