/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

** 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...
## 4 initial DP problems to solve.
## First, we try to decompose these problems into smaller problems.
## Round 1 [4 DP problems]:
  ## DP problem:
  Dependency pairs = [sum^#(cons(_0,cons(_1,_2))) -> sum^#(cons(a(_0,_1,h,h),_2))]
  TRS = {a(h,h,h,_0) -> s(_0), a(_0,_1,s(_2),h) -> a(_0,_1,_2,s(h)), a(_0,_1,s(_2),s(_3)) -> a(_0,_1,_2,a(_0,_1,s(_2),_3)), a(_0,s(_1),h,_2) -> a(_0,_1,_2,_2), a(s(_0),h,h,_1) -> a(_0,_1,h,_1), +(_0,h) -> _0, +(h,_0) -> _0, +(s(_0),s(_1)) -> s(s(+(_0,_1))), +(+(_0,_1),_2) -> +(_0,+(_1,_2)), s(h) -> 1, app(nil,_0) -> _0, app(_0,nil) -> _0, app(cons(_0,_1),_2) -> cons(_0,app(_1,_2)), sum(cons(_0,nil)) -> cons(_0,nil), sum(cons(_0,cons(_1,_2))) -> sum(cons(a(_0,_1,h,h),_2))}
    ## Trying with homeomorphic embeddings... Failed!
    ## Trying with polynomial interpretations... This DP problem is too complex! Aborting!
    ## Trying with lexicographic path orders...
      The constraints are satisfied by the lexicographic path order using
      the argument filtering:
      {app:[0, 1], s:[0], +:[0, 1], cons:[1], a:[0, 1, 2, 3], sum:[0], sum^#:[0]}
      and the  precedence:
      h > [1], cons > [sum^#, sum], app > [cons, sum^#, sum], s > [1, h], + > [1, h, s], a > [1, h, s]
  This DP problem is finite.
  ## DP problem:
  Dependency pairs = [app^#(cons(_0,_1),_2) -> app^#(_1,_2)]
  TRS = {a(h,h,h,_0) -> s(_0), a(_0,_1,s(_2),h) -> a(_0,_1,_2,s(h)), a(_0,_1,s(_2),s(_3)) -> a(_0,_1,_2,a(_0,_1,s(_2),_3)), a(_0,s(_1),h,_2) -> a(_0,_1,_2,_2), a(s(_0),h,h,_1) -> a(_0,_1,h,_1), +(_0,h) -> _0, +(h,_0) -> _0, +(s(_0),s(_1)) -> s(s(+(_0,_1))), +(+(_0,_1),_2) -> +(_0,+(_1,_2)), s(h) -> 1, app(nil,_0) -> _0, app(_0,nil) -> _0, app(cons(_0,_1),_2) -> cons(_0,app(_1,_2)), sum(cons(_0,nil)) -> cons(_0,nil), sum(cons(_0,cons(_1,_2))) -> sum(cons(a(_0,_1,h,h),_2))}
    ## Trying with homeomorphic embeddings... Success!
  This DP problem is finite.
  ## DP problem:
  Dependency pairs = [+^#(s(_0),s(_1)) -> +^#(_0,_1), +^#(+(_0,_1),_2) -> +^#(_0,+(_1,_2)), +^#(+(_0,_1),_2) -> +^#(_1,_2)]
  TRS = {a(h,h,h,_0) -> s(_0), a(_0,_1,s(_2),h) -> a(_0,_1,_2,s(h)), a(_0,_1,s(_2),s(_3)) -> a(_0,_1,_2,a(_0,_1,s(_2),_3)), a(_0,s(_1),h,_2) -> a(_0,_1,_2,_2), a(s(_0),h,h,_1) -> a(_0,_1,h,_1), +(_0,h) -> _0, +(h,_0) -> _0, +(s(_0),s(_1)) -> s(s(+(_0,_1))), +(+(_0,_1),_2) -> +(_0,+(_1,_2)), s(h) -> 1, app(nil,_0) -> _0, app(_0,nil) -> _0, app(cons(_0,_1),_2) -> cons(_0,app(_1,_2)), sum(cons(_0,nil)) -> cons(_0,nil), sum(cons(_0,cons(_1,_2))) -> sum(cons(a(_0,_1,h,h),_2))}
    ## Trying with homeomorphic embeddings... Failed!
    ## Trying with polynomial interpretations... This DP problem is too complex! Aborting!
    ## Trying with lexicographic path orders...
      The constraints are satisfied by the lexicographic path order using
      the argument filtering:
      {app:[0, 1], s:[0], +:[0, 1], cons:[1], a:[0, 1, 2, 3], sum:[0], +^#:[0, 1]}
      and the  precedence:
      h > [1], cons > [sum], app > [cons, sum], s > [1, h], + > [1, h, s], +^# > [1, h, +, s], a > [1, h, s]
  This DP problem is finite.
  ## DP problem:
  Dependency pairs = [a^#(_0,_1,s(_2),h) -> a^#(_0,_1,_2,s(h)), a^#(_0,_1,s(_2),s(_3)) -> a^#(_0,_1,_2,a(_0,_1,s(_2),_3)), a^#(_0,_1,s(_2),s(_3)) -> a^#(_0,_1,s(_2),_3), a^#(_0,s(_1),h,_2) -> a^#(_0,_1,_2,_2), a^#(s(_0),h,h,_1) -> a^#(_0,_1,h,_1)]
  TRS = {a(h,h,h,_0) -> s(_0), a(_0,_1,s(_2),h) -> a(_0,_1,_2,s(h)), a(_0,_1,s(_2),s(_3)) -> a(_0,_1,_2,a(_0,_1,s(_2),_3)), a(_0,s(_1),h,_2) -> a(_0,_1,_2,_2), a(s(_0),h,h,_1) -> a(_0,_1,h,_1), +(_0,h) -> _0, +(h,_0) -> _0, +(s(_0),s(_1)) -> s(s(+(_0,_1))), +(+(_0,_1),_2) -> +(_0,+(_1,_2)), s(h) -> 1, app(nil,_0) -> _0, app(_0,nil) -> _0, app(cons(_0,_1),_2) -> cons(_0,app(_1,_2)), sum(cons(_0,nil)) -> cons(_0,nil), sum(cons(_0,cons(_1,_2))) -> sum(cons(a(_0,_1,h,h),_2))}
    ## Trying with homeomorphic embeddings... Failed!
    ## Trying with polynomial interpretations... This DP problem is too complex! Aborting!
    ## Trying with lexicographic path orders...
      The constraints are satisfied by the lexicographic path order using
      the argument filtering:
      {app:[0, 1], s:[0], +:[0, 1], cons:[1], a:[0, 1, 2, 3], sum:[0], a^#:[0, 1, 2, 3]}
      and the  precedence:
      h > [1], cons > [sum], app > [cons, sum], s > [1, h], + > [1, h, s], a^# > [1, h, s, a], a > [1, h, s]
  This DP problem is finite.
## All the DP problems were proved finite.
As all the involved DP processors are sound,
the TRS under analysis terminates.
Proof run on Linux version 3.10.0-1160.25.1.el7.x86_64 for amd64
using Java version 1.8.0_292
** END proof description **
Total number of generated unfolded rules = 0