/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...
## 5 initial DP problems to solve.
## First, we try to decompose these problems into smaller problems.
## Round 1 [5 DP problems]:
  ## DP problem:
  Dependency pairs = [prod^#(cons(_0,_1)) -> prod^#(_1), prod^#(app(_0,_1)) -> prod^#(_0), prod^#(app(_0,_1)) -> prod^#(_1)]
  TRS = {+(_0,0) -> _0, +(0,_0) -> _0, +(s(_0),s(_1)) -> s(s(+(_0,_1))), +(+(_0,_1),_2) -> +(_0,+(_1,_2)), *(_0,0) -> 0, *(0,_0) -> 0, *(s(_0),s(_1)) -> s(+(*(_0,_1),+(_0,_1))), *(*(_0,_1),_2) -> *(_0,*(_1,_2)), app(nil,_0) -> _0, app(cons(_0,_1),_2) -> cons(_0,app(_1,_2)), sum(nil) -> 0, sum(cons(_0,_1)) -> +(_0,sum(_1)), sum(app(_0,_1)) -> +(sum(_0),sum(_1)), prod(nil) -> s(0), prod(cons(_0,_1)) -> *(_0,prod(_1)), prod(app(_0,_1)) -> *(prod(_0),prod(_1))}
    ## Trying with homeomorphic embeddings... Success!
  This DP problem is finite.
  ## DP problem:
  Dependency pairs = [sum^#(cons(_0,_1)) -> sum^#(_1), sum^#(app(_0,_1)) -> sum^#(_0), sum^#(app(_0,_1)) -> sum^#(_1)]
  TRS = {+(_0,0) -> _0, +(0,_0) -> _0, +(s(_0),s(_1)) -> s(s(+(_0,_1))), +(+(_0,_1),_2) -> +(_0,+(_1,_2)), *(_0,0) -> 0, *(0,_0) -> 0, *(s(_0),s(_1)) -> s(+(*(_0,_1),+(_0,_1))), *(*(_0,_1),_2) -> *(_0,*(_1,_2)), app(nil,_0) -> _0, app(cons(_0,_1),_2) -> cons(_0,app(_1,_2)), sum(nil) -> 0, sum(cons(_0,_1)) -> +(_0,sum(_1)), sum(app(_0,_1)) -> +(sum(_0),sum(_1)), prod(nil) -> s(0), prod(cons(_0,_1)) -> *(_0,prod(_1)), prod(app(_0,_1)) -> *(prod(_0),prod(_1))}
    ## Trying with homeomorphic embeddings... Success!
  This DP problem is finite.
  ## DP problem:
  Dependency pairs = [app^#(cons(_0,_1),_2) -> app^#(_1,_2)]
  TRS = {+(_0,0) -> _0, +(0,_0) -> _0, +(s(_0),s(_1)) -> s(s(+(_0,_1))), +(+(_0,_1),_2) -> +(_0,+(_1,_2)), *(_0,0) -> 0, *(0,_0) -> 0, *(s(_0),s(_1)) -> s(+(*(_0,_1),+(_0,_1))), *(*(_0,_1),_2) -> *(_0,*(_1,_2)), app(nil,_0) -> _0, app(cons(_0,_1),_2) -> cons(_0,app(_1,_2)), sum(nil) -> 0, sum(cons(_0,_1)) -> +(_0,sum(_1)), sum(app(_0,_1)) -> +(sum(_0),sum(_1)), prod(nil) -> s(0), prod(cons(_0,_1)) -> *(_0,prod(_1)), prod(app(_0,_1)) -> *(prod(_0),prod(_1))}
    ## 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 = {+(_0,0) -> _0, +(0,_0) -> _0, +(s(_0),s(_1)) -> s(s(+(_0,_1))), +(+(_0,_1),_2) -> +(_0,+(_1,_2)), *(_0,0) -> 0, *(0,_0) -> 0, *(s(_0),s(_1)) -> s(+(*(_0,_1),+(_0,_1))), *(*(_0,_1),_2) -> *(_0,*(_1,_2)), app(nil,_0) -> _0, app(cons(_0,_1),_2) -> cons(_0,app(_1,_2)), sum(nil) -> 0, sum(cons(_0,_1)) -> +(_0,sum(_1)), sum(app(_0,_1)) -> +(sum(_0),sum(_1)), prod(nil) -> s(0), prod(cons(_0,_1)) -> *(_0,prod(_1)), prod(app(_0,_1)) -> *(prod(_0),prod(_1))}
    ## 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:
      {s:[0], sum:[0], *:[0, 1], app:[0, 1], cons:[0, 1], +:[0, 1], prod:[0], *^#:[0, 1]}
      and the  precedence:
      * > [s, +], app > [s, sum, *, cons, +, prod], cons > [s, sum, *, +, prod], *^# > [s, *, +], + > [s], nil > [0, s]
  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 = {+(_0,0) -> _0, +(0,_0) -> _0, +(s(_0),s(_1)) -> s(s(+(_0,_1))), +(+(_0,_1),_2) -> +(_0,+(_1,_2)), *(_0,0) -> 0, *(0,_0) -> 0, *(s(_0),s(_1)) -> s(+(*(_0,_1),+(_0,_1))), *(*(_0,_1),_2) -> *(_0,*(_1,_2)), app(nil,_0) -> _0, app(cons(_0,_1),_2) -> cons(_0,app(_1,_2)), sum(nil) -> 0, sum(cons(_0,_1)) -> +(_0,sum(_1)), sum(app(_0,_1)) -> +(sum(_0),sum(_1)), prod(nil) -> s(0), prod(cons(_0,_1)) -> *(_0,prod(_1)), prod(app(_0,_1)) -> *(prod(_0),prod(_1))}
    ## 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:
      {s:[0], sum:[0], *:[0, 1], app:[0, 1], cons:[0, 1], +:[0, 1], prod:[0], +^#:[0, 1]}
      and the  precedence:
      +^# > [s, +], * > [s, +], app > [s, sum, *, cons, +, prod], cons > [s, sum, *, +, prod], + > [s], nil > [0, 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