YES

Prover = TRS(tech=PATTERN_RULES, nb_unfoldings=unlimited, max_nb_unfolded_rules=200)

** BEGIN proof argument **
All the DP problems were proved finite.
As all the involved DP processors are sound,
the TRS under analysis terminates.
** END proof argument **

** BEGIN proof description **
## Searching for a generalized rewrite rule
(a rule whose right-hand side contains a variable
that does not occur in the left-hand side)...
No generalized rewrite rule found!

## Applying the DP framework...
## Round 1:
  ## DP problem: 
  Dependency pairs = [sum1^#(s(_0)) -> sum1^#(_0)]
  TRS = {sum(0) -> 0, sum(s(_0)) -> +(sum(_0),s(_0)), sum1(0) -> 0, sum1(s(_0)) -> s(+(sum1(_0),+(_0,_0)))}
    ## Trying with homeomorphic embeddings... Success!
  This DP problem is finite.
  ## DP problem: 
  Dependency pairs = [sum^#(s(_0)) -> sum^#(_0)]
  TRS = {sum(0) -> 0, sum(s(_0)) -> +(sum(_0),s(_0)), sum1(0) -> 0, sum1(s(_0)) -> s(+(sum1(_0),+(_0,_0)))}
    ## Trying with homeomorphic embeddings... Success!
  This DP problem is finite.

** END proof description **

Proof stopped at iteration 0
Number of unfolded rules generated by this proof = 0
Number of unfolded rules generated by all the parallel proofs = 0