YES

(format LCTRS :logic QF_LIA)
(fun sum 1 :sort (Int Int))
(fun test 1 :sort (Int Unit))

(rule (test x_0) (test y_1) :guard (and (> x_0 0) (= x_0 (+ y_1 1))) :vars ((x_0 Int) (y_1 Int)))
(rule (sum x_2) (+ x_2 (sum (- x_2 1))) :guard (< 0 x_2) :vars ((x_2 Int)))
(rule (sum 0) 0)
(rule (sum x_3) 0 :guard (< x_3 0) :vars ((x_3 Int)))

Confluent by Newmans Lemma with proof:
terminating and no critical pairs

Elapsed Time:  15.51 ms