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Digits-any-base

Dab-base

Fixtype for dab-basep.

Definitions and Theorems

Function: dab-base-equiv$inline

(defun dab-base-equiv$inline (x y)
  (declare (xargs :guard (and (dab-basep x) (dab-basep y))))
  (equal (dab-base-fix x)
         (dab-base-fix y)))

Theorem: dab-base-equiv-is-an-equivalence

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

Theorem: dab-base-equiv-implies-equal-dab-base-fix-1

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

Theorem: dab-base-fix-under-dab-base-equiv

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

Theorem: equal-of-dab-base-fix-1-forward-to-dab-base-equiv

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

Theorem: equal-of-dab-base-fix-2-forward-to-dab-base-equiv

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

Theorem: dab-base-equiv-of-dab-base-fix-1-forward

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

Theorem: dab-base-equiv-of-dab-base-fix-2-forward

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