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Range

Range-equiv

Basic equivalence relation for range structures.

Definitions and Theorems

Function: range-equiv$inline

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

Theorem: range-equiv-is-an-equivalence

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

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

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

Theorem: range-fix-under-range-equiv

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

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

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

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

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

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

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

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

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