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Rangelist

Rangelist-equiv

Basic equivalence relation for rangelist structures.

Definitions and Theorems

Function: rangelist-equiv$inline

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

Theorem: rangelist-equiv-is-an-equivalence

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

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

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

Theorem: rangelist-fix-under-rangelist-equiv

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

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

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

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

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

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

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

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

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