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 = [qsort^#(.(_0,_1)) -> qsort^#(lowers(_0,_1)), qsort^#(.(_0,_1)) -> qsort^#(greaters(_0,_1))]
  TRS = {qsort(nil) -> nil, qsort(.(_0,_1)) -> ++(qsort(lowers(_0,_1)),.(_0,qsort(greaters(_0,_1)))), lowers(_0,nil) -> nil, lowers(_0,.(_1,_2)) -> if(<=(_1,_0),.(_1,lowers(_0,_2)),lowers(_0,_2)), greaters(_0,nil) -> nil, greaters(_0,.(_1,_2)) -> if(<=(_1,_0),greaters(_0,_2),.(_1,greaters(_0,_2)))}
    ## Trying with homeomorphic embeddings... Failed!
    ## Trying with polynomial interpretations... Too many coefficients (32)! Aborting!
    ## Trying with lexicographic path orders...
      The constraints are satisfied by the lexicographic path order using the precedence:
      qsort > [++], lowers > [nil, if, <=], greaters > [nil, if, <=], . > [qsort, ++, lowers, greaters, nil, if, <=]
      and the argument filtering:
      {qsort:[0], ++:[0, 1], if:[0], .:[0], <=:[1], greaters:[0], lowers:[0], qsort^#:[0]}
  This DP problem is finite.
  ## DP problem: 
  Dependency pairs = [greaters^#(_0,.(_1,_2)) -> greaters^#(_0,_2), greaters^#(_0,.(_1,_2)) -> greaters^#(_0,_2)]
  TRS = {qsort(nil) -> nil, qsort(.(_0,_1)) -> ++(qsort(lowers(_0,_1)),.(_0,qsort(greaters(_0,_1)))), lowers(_0,nil) -> nil, lowers(_0,.(_1,_2)) -> if(<=(_1,_0),.(_1,lowers(_0,_2)),lowers(_0,_2)), greaters(_0,nil) -> nil, greaters(_0,.(_1,_2)) -> if(<=(_1,_0),greaters(_0,_2),.(_1,greaters(_0,_2)))}
    ## Trying with homeomorphic embeddings... Success!
  This DP problem is finite.
  ## DP problem: 
  Dependency pairs = [lowers^#(_0,.(_1,_2)) -> lowers^#(_0,_2), lowers^#(_0,.(_1,_2)) -> lowers^#(_0,_2)]
  TRS = {qsort(nil) -> nil, qsort(.(_0,_1)) -> ++(qsort(lowers(_0,_1)),.(_0,qsort(greaters(_0,_1)))), lowers(_0,nil) -> nil, lowers(_0,.(_1,_2)) -> if(<=(_1,_0),.(_1,lowers(_0,_2)),lowers(_0,_2)), greaters(_0,nil) -> nil, greaters(_0,.(_1,_2)) -> if(<=(_1,_0),greaters(_0,_2),.(_1,greaters(_0,_2)))}
    ## 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 = 2