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• Dec-integer-literal

Dec-integer-literal-equiv

Basic equivalence relation for dec-integer-literal structures.

Definitions and Theorems

Function: dec-integer-literal-equiv$inline

(defun dec-integer-literal-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (dec-integer-literalp acl2::x)
                              (dec-integer-literalp acl2::y))))
  (equal (dec-integer-literal-fix acl2::x)
         (dec-integer-literal-fix acl2::y)))

Theorem: dec-integer-literal-equiv-is-an-equivalence

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

Theorem: dec-integer-literal-equiv-implies-equal-dec-integer-literal-fix-1

(defthm
  dec-integer-literal-equiv-implies-equal-dec-integer-literal-fix-1
  (implies (dec-integer-literal-equiv acl2::x x-equiv)
           (equal (dec-integer-literal-fix acl2::x)
                  (dec-integer-literal-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: dec-integer-literal-fix-under-dec-integer-literal-equiv

(defthm dec-integer-literal-fix-under-dec-integer-literal-equiv
  (dec-integer-literal-equiv (dec-integer-literal-fix acl2::x)
                             acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-dec-integer-literal-fix-1-forward-to-dec-integer-literal-equiv

(defthm
 equal-of-dec-integer-literal-fix-1-forward-to-dec-integer-literal-equiv
 (implies (equal (dec-integer-literal-fix acl2::x)
                 acl2::y)
          (dec-integer-literal-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: equal-of-dec-integer-literal-fix-2-forward-to-dec-integer-literal-equiv

(defthm
 equal-of-dec-integer-literal-fix-2-forward-to-dec-integer-literal-equiv
 (implies (equal acl2::x
                 (dec-integer-literal-fix acl2::y))
          (dec-integer-literal-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: dec-integer-literal-equiv-of-dec-integer-literal-fix-1-forward

(defthm
     dec-integer-literal-equiv-of-dec-integer-literal-fix-1-forward
  (implies
       (dec-integer-literal-equiv (dec-integer-literal-fix acl2::x)
                                  acl2::y)
       (dec-integer-literal-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: dec-integer-literal-equiv-of-dec-integer-literal-fix-2-forward

(defthm
     dec-integer-literal-equiv-of-dec-integer-literal-fix-2-forward
  (implies
       (dec-integer-literal-equiv acl2::x
                                  (dec-integer-literal-fix acl2::y))
       (dec-integer-literal-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)