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Dec-digit-char-list

Dec-digit-char-list-equiv

Basic equivalence relation for dec-digit-char-list structures.

Definitions and Theorems

Function: dec-digit-char-list-equiv$inline

(defun dec-digit-char-list-equiv$inline (x y)
  (declare (xargs :guard (and (dec-digit-char-listp x)
                              (dec-digit-char-listp y))))
  (equal (dec-digit-char-list-fix x)
         (dec-digit-char-list-fix y)))

Theorem: dec-digit-char-list-equiv-is-an-equivalence

(defthm dec-digit-char-list-equiv-is-an-equivalence
  (and (booleanp (dec-digit-char-list-equiv x y))
       (dec-digit-char-list-equiv x x)
       (implies (dec-digit-char-list-equiv x y)
                (dec-digit-char-list-equiv y x))
       (implies (and (dec-digit-char-list-equiv x y)
                     (dec-digit-char-list-equiv y z))
                (dec-digit-char-list-equiv x z)))
  :rule-classes (:equivalence))

Theorem: dec-digit-char-list-equiv-implies-equal-dec-digit-char-list-fix-1

(defthm
  dec-digit-char-list-equiv-implies-equal-dec-digit-char-list-fix-1
  (implies (dec-digit-char-list-equiv x x-equiv)
           (equal (dec-digit-char-list-fix x)
                  (dec-digit-char-list-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: dec-digit-char-list-fix-under-dec-digit-char-list-equiv

(defthm dec-digit-char-list-fix-under-dec-digit-char-list-equiv
  (dec-digit-char-list-equiv (dec-digit-char-list-fix x)
                             x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-dec-digit-char-list-fix-1-forward-to-dec-digit-char-list-equiv

(defthm
 equal-of-dec-digit-char-list-fix-1-forward-to-dec-digit-char-list-equiv
 (implies (equal (dec-digit-char-list-fix x) y)
          (dec-digit-char-list-equiv x y))
 :rule-classes :forward-chaining)

Theorem: equal-of-dec-digit-char-list-fix-2-forward-to-dec-digit-char-list-equiv

(defthm
 equal-of-dec-digit-char-list-fix-2-forward-to-dec-digit-char-list-equiv
 (implies (equal x (dec-digit-char-list-fix y))
          (dec-digit-char-list-equiv x y))
 :rule-classes :forward-chaining)

Theorem: dec-digit-char-list-equiv-of-dec-digit-char-list-fix-1-forward

(defthm
     dec-digit-char-list-equiv-of-dec-digit-char-list-fix-1-forward
  (implies (dec-digit-char-list-equiv (dec-digit-char-list-fix x)
                                      y)
           (dec-digit-char-list-equiv x y))
  :rule-classes :forward-chaining)

Theorem: dec-digit-char-list-equiv-of-dec-digit-char-list-fix-2-forward

(defthm
     dec-digit-char-list-equiv-of-dec-digit-char-list-fix-2-forward
  (implies (dec-digit-char-list-equiv x (dec-digit-char-list-fix y))
           (dec-digit-char-list-equiv x y))
  :rule-classes :forward-chaining)