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


--------------------------------------------------------------------------------


MAYBE
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Outermost Termination of the given OTRS could not be shown:

(0) OTRS
(1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms]
(2) QTRS
    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
    (4) QDP
    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
    (6) AND
        (7) QDP
            (8) UsableRulesProof [EQUIVALENT, 0 ms]
            (9) QDP
            (10) QReductionProof [EQUIVALENT, 0 ms]
            (11) QDP
            (12) UsableRulesReductionPairsProof [EQUIVALENT, 3 ms]
            (13) QDP
            (14) DependencyGraphProof [EQUIVALENT, 0 ms]
            (15) TRUE
        (16) QDP
            (17) UsableRulesProof [EQUIVALENT, 0 ms]
            (18) QDP
                (19) QReductionProof [EQUIVALENT, 0 ms]
                (20) QDP
                (21) TransformationProof [EQUIVALENT, 0 ms]
                (22) QDP
                (23) TransformationProof [EQUIVALENT, 0 ms]
                (24) QDP
                (25) TransformationProof [EQUIVALENT, 0 ms]
                (26) QDP
                (27) QDPOrderProof [EQUIVALENT, 21 ms]
                (28) QDP
            (29) UsableRulesProof [EQUIVALENT, 0 ms]
            (30) QDP
                (31) QReductionProof [EQUIVALENT, 0 ms]
                (32) QDP
(33) Trivial-Transformation [SOUND, 0 ms]
(34) QTRS
    (35) DependencyPairsProof [EQUIVALENT, 0 ms]
    (36) QDP
    (37) DependencyGraphProof [EQUIVALENT, 0 ms]
    (38) QDP
    (39) UsableRulesProof [EQUIVALENT, 0 ms]
    (40) QDP
    (41) TransformationProof [EQUIVALENT, 0 ms]
    (42) QDP
    (43) TransformationProof [EQUIVALENT, 0 ms]
    (44) QDP
    (45) NonTerminationLoopProof [COMPLETE, 0 ms]
    (46) NO


----------------------------------------

(0)
Obligation:
Term rewrite system R:
The TRS R consists of the following rules:

   cons(x, cons(y, z)) -> big
   inf(x) -> cons(x, inf(s(x)))
   g(x, x) -> b



Outermost Strategy.

----------------------------------------

(1) Thiemann-SpecialC-Transformation (EQUIVALENT)
We applied the Thiemann-SpecialC transformation  to transform the outermost TRS to an innermost TRS.
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   top(go_up(x)) -> top(reduce(x))
   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
   redex_cons(x, cons(y, z)) -> result_cons(big)
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   redex_g(x, x) -> result_g(b)
   check_cons(result_cons(x)) -> go_up(x)
   check_inf(result_inf(x)) -> go_up(x)
   check_g(result_g(x)) -> go_up(x)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
   check_inf(redex_inf(x_1)) -> in_inf_1(reduce(x_1))
   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
   reduce(s(x_1)) -> in_s_1(reduce(x_1))
   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
   in_inf_1(go_up(x_1)) -> go_up(inf(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))

The set Q consists of the following terms:

   top(go_up(x0))
   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   redex_g(x0, x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_inf_1(go_up(x0))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))


----------------------------------------

(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(go_up(x)) -> TOP(reduce(x))
   TOP(go_up(x)) -> REDUCE(x)
   REDUCE(cons(x_1, x_2)) -> CHECK_CONS(redex_cons(x_1, x_2))
   REDUCE(cons(x_1, x_2)) -> REDEX_CONS(x_1, x_2)
   REDUCE(inf(x_1)) -> CHECK_INF(redex_inf(x_1))
   REDUCE(inf(x_1)) -> REDEX_INF(x_1)
   REDUCE(g(x_1, x_2)) -> CHECK_G(redex_g(x_1, x_2))
   REDUCE(g(x_1, x_2)) -> REDEX_G(x_1, x_2)
   CHECK_CONS(redex_cons(x_1, x_2)) -> IN_CONS_1(reduce(x_1), x_2)
   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_1)
   CHECK_CONS(redex_cons(x_1, x_2)) -> IN_CONS_2(x_1, reduce(x_2))
   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_2)
   CHECK_INF(redex_inf(x_1)) -> IN_INF_1(reduce(x_1))
   CHECK_INF(redex_inf(x_1)) -> REDUCE(x_1)
   CHECK_G(redex_g(x_1, x_2)) -> IN_G_1(reduce(x_1), x_2)
   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_1)
   CHECK_G(redex_g(x_1, x_2)) -> IN_G_2(x_1, reduce(x_2))
   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_2)
   REDUCE(s(x_1)) -> IN_S_1(reduce(x_1))
   REDUCE(s(x_1)) -> REDUCE(x_1)

The TRS R consists of the following rules:

   top(go_up(x)) -> top(reduce(x))
   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
   redex_cons(x, cons(y, z)) -> result_cons(big)
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   redex_g(x, x) -> result_g(b)
   check_cons(result_cons(x)) -> go_up(x)
   check_inf(result_inf(x)) -> go_up(x)
   check_g(result_g(x)) -> go_up(x)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
   check_inf(redex_inf(x_1)) -> in_inf_1(reduce(x_1))
   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
   reduce(s(x_1)) -> in_s_1(reduce(x_1))
   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
   in_inf_1(go_up(x_1)) -> go_up(inf(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))

The set Q consists of the following terms:

   top(go_up(x0))
   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   redex_g(x0, x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_inf_1(go_up(x0))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 12 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

----------------------------------------

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_1)
   REDUCE(cons(x_1, x_2)) -> CHECK_CONS(redex_cons(x_1, x_2))
   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_2)
   REDUCE(g(x_1, x_2)) -> CHECK_G(redex_g(x_1, x_2))
   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_1)
   REDUCE(s(x_1)) -> REDUCE(x_1)
   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_2)

The TRS R consists of the following rules:

   top(go_up(x)) -> top(reduce(x))
   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
   redex_cons(x, cons(y, z)) -> result_cons(big)
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   redex_g(x, x) -> result_g(b)
   check_cons(result_cons(x)) -> go_up(x)
   check_inf(result_inf(x)) -> go_up(x)
   check_g(result_g(x)) -> go_up(x)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
   check_inf(redex_inf(x_1)) -> in_inf_1(reduce(x_1))
   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
   reduce(s(x_1)) -> in_s_1(reduce(x_1))
   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
   in_inf_1(go_up(x_1)) -> go_up(inf(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))

The set Q consists of the following terms:

   top(go_up(x0))
   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   redex_g(x0, x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_inf_1(go_up(x0))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(8) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_1)
   REDUCE(cons(x_1, x_2)) -> CHECK_CONS(redex_cons(x_1, x_2))
   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_2)
   REDUCE(g(x_1, x_2)) -> CHECK_G(redex_g(x_1, x_2))
   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_1)
   REDUCE(s(x_1)) -> REDUCE(x_1)
   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_2)

The TRS R consists of the following rules:

   redex_g(x, x) -> result_g(b)
   redex_cons(x, cons(y, z)) -> result_cons(big)

The set Q consists of the following terms:

   top(go_up(x0))
   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   redex_g(x0, x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_inf_1(go_up(x0))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(10) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   top(go_up(x0))
   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_inf_1(go_up(x0))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))


----------------------------------------

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_1)
   REDUCE(cons(x_1, x_2)) -> CHECK_CONS(redex_cons(x_1, x_2))
   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_2)
   REDUCE(g(x_1, x_2)) -> CHECK_G(redex_g(x_1, x_2))
   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_1)
   REDUCE(s(x_1)) -> REDUCE(x_1)
   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_2)

The TRS R consists of the following rules:

   redex_g(x, x) -> result_g(b)
   redex_cons(x, cons(y, z)) -> result_cons(big)

The set Q consists of the following terms:

   redex_cons(x0, cons(x1, x2))
   redex_g(x0, x0)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(12) UsableRulesReductionPairsProof (EQUIVALENT)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

   REDUCE(cons(x_1, x_2)) -> CHECK_CONS(redex_cons(x_1, x_2))
   REDUCE(g(x_1, x_2)) -> CHECK_G(redex_g(x_1, x_2))
   REDUCE(s(x_1)) -> REDUCE(x_1)
The following rules are removed from R:

   redex_cons(x, cons(y, z)) -> result_cons(big)
Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(CHECK_CONS(x_1)) = x_1
   POL(CHECK_G(x_1)) = 2*x_1
   POL(REDUCE(x_1)) = 2*x_1
   POL(b) = 0
   POL(big) = 0
   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(g(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(redex_cons(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(redex_g(x_1, x_2)) = x_1 + x_2
   POL(result_cons(x_1)) = x_1
   POL(result_g(x_1)) = x_1
   POL(s(x_1)) = 2*x_1


----------------------------------------

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_1)
   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_2)
   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_1)
   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_2)

The TRS R consists of the following rules:

   redex_g(x, x) -> result_g(b)

The set Q consists of the following terms:

   redex_cons(x0, cons(x1, x2))
   redex_g(x0, x0)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(14) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
----------------------------------------

(15)
TRUE

----------------------------------------

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(go_up(x)) -> TOP(reduce(x))

The TRS R consists of the following rules:

   top(go_up(x)) -> top(reduce(x))
   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
   redex_cons(x, cons(y, z)) -> result_cons(big)
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   redex_g(x, x) -> result_g(b)
   check_cons(result_cons(x)) -> go_up(x)
   check_inf(result_inf(x)) -> go_up(x)
   check_g(result_g(x)) -> go_up(x)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
   check_inf(redex_inf(x_1)) -> in_inf_1(reduce(x_1))
   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
   reduce(s(x_1)) -> in_s_1(reduce(x_1))
   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
   in_inf_1(go_up(x_1)) -> go_up(inf(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))

The set Q consists of the following terms:

   top(go_up(x0))
   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   redex_g(x0, x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_inf_1(go_up(x0))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(17) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(go_up(x)) -> TOP(reduce(x))

The TRS R consists of the following rules:

   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
   reduce(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   redex_g(x, x) -> result_g(b)
   check_g(result_g(x)) -> go_up(x)
   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   check_inf(result_inf(x)) -> go_up(x)
   redex_cons(x, cons(y, z)) -> result_cons(big)
   check_cons(result_cons(x)) -> go_up(x)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))

The set Q consists of the following terms:

   top(go_up(x0))
   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   redex_g(x0, x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_inf_1(go_up(x0))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(19) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   top(go_up(x0))
   in_inf_1(go_up(x0))


----------------------------------------

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(go_up(x)) -> TOP(reduce(x))

The TRS R consists of the following rules:

   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
   reduce(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   redex_g(x, x) -> result_g(b)
   check_g(result_g(x)) -> go_up(x)
   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   check_inf(result_inf(x)) -> go_up(x)
   redex_cons(x, cons(y, z)) -> result_cons(big)
   check_cons(result_cons(x)) -> go_up(x)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))

The set Q consists of the following terms:

   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   redex_g(x0, x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(21) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]:

   (TOP(go_up(cons(x0, x1))) -> TOP(check_cons(redex_cons(x0, x1))),TOP(go_up(cons(x0, x1))) -> TOP(check_cons(redex_cons(x0, x1))))
   (TOP(go_up(inf(x0))) -> TOP(check_inf(redex_inf(x0))),TOP(go_up(inf(x0))) -> TOP(check_inf(redex_inf(x0))))
   (TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1))),TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1))))
   (TOP(go_up(s(x0))) -> TOP(in_s_1(reduce(x0))),TOP(go_up(s(x0))) -> TOP(in_s_1(reduce(x0))))


----------------------------------------

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(go_up(cons(x0, x1))) -> TOP(check_cons(redex_cons(x0, x1)))
   TOP(go_up(inf(x0))) -> TOP(check_inf(redex_inf(x0)))
   TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1)))
   TOP(go_up(s(x0))) -> TOP(in_s_1(reduce(x0)))

The TRS R consists of the following rules:

   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
   reduce(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   redex_g(x, x) -> result_g(b)
   check_g(result_g(x)) -> go_up(x)
   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   check_inf(result_inf(x)) -> go_up(x)
   redex_cons(x, cons(y, z)) -> result_cons(big)
   check_cons(result_cons(x)) -> go_up(x)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))

The set Q consists of the following terms:

   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   redex_g(x0, x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(23) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule TOP(go_up(inf(x0))) -> TOP(check_inf(redex_inf(x0))) at position [0,0] we obtained the following new rules [LPAR04]:

   (TOP(go_up(inf(x0))) -> TOP(check_inf(result_inf(cons(x0, inf(s(x0)))))),TOP(go_up(inf(x0))) -> TOP(check_inf(result_inf(cons(x0, inf(s(x0)))))))


----------------------------------------

(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(go_up(cons(x0, x1))) -> TOP(check_cons(redex_cons(x0, x1)))
   TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1)))
   TOP(go_up(s(x0))) -> TOP(in_s_1(reduce(x0)))
   TOP(go_up(inf(x0))) -> TOP(check_inf(result_inf(cons(x0, inf(s(x0))))))

The TRS R consists of the following rules:

   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
   reduce(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   redex_g(x, x) -> result_g(b)
   check_g(result_g(x)) -> go_up(x)
   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   check_inf(result_inf(x)) -> go_up(x)
   redex_cons(x, cons(y, z)) -> result_cons(big)
   check_cons(result_cons(x)) -> go_up(x)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))

The set Q consists of the following terms:

   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   redex_g(x0, x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(25) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule TOP(go_up(inf(x0))) -> TOP(check_inf(result_inf(cons(x0, inf(s(x0)))))) at position [0] we obtained the following new rules [LPAR04]:

   (TOP(go_up(inf(x0))) -> TOP(go_up(cons(x0, inf(s(x0))))),TOP(go_up(inf(x0))) -> TOP(go_up(cons(x0, inf(s(x0))))))


----------------------------------------

(26)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(go_up(cons(x0, x1))) -> TOP(check_cons(redex_cons(x0, x1)))
   TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1)))
   TOP(go_up(s(x0))) -> TOP(in_s_1(reduce(x0)))
   TOP(go_up(inf(x0))) -> TOP(go_up(cons(x0, inf(s(x0)))))

The TRS R consists of the following rules:

   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
   reduce(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   redex_g(x, x) -> result_g(b)
   check_g(result_g(x)) -> go_up(x)
   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   check_inf(result_inf(x)) -> go_up(x)
   redex_cons(x, cons(y, z)) -> result_cons(big)
   check_cons(result_cons(x)) -> go_up(x)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))

The set Q consists of the following terms:

   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   redex_g(x0, x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(27) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   TOP(go_up(inf(x0))) -> TOP(go_up(cons(x0, inf(s(x0)))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(TOP(x_1)) = x_1
   POL(b) = 0
   POL(big) = 0
   POL(check_cons(x_1)) = x_1
   POL(check_g(x_1)) = x_1
   POL(check_inf(x_1)) = 1
   POL(cons(x_1, x_2)) = 0
   POL(g(x_1, x_2)) = 0
   POL(go_up(x_1)) = x_1
   POL(in_cons_1(x_1, x_2)) = 0
   POL(in_cons_2(x_1, x_2)) = 0
   POL(in_g_1(x_1, x_2)) = 0
   POL(in_g_2(x_1, x_2)) = 0
   POL(in_s_1(x_1)) = 0
   POL(inf(x_1)) = 1
   POL(redex_cons(x_1, x_2)) = 0
   POL(redex_g(x_1, x_2)) = 0
   POL(redex_inf(x_1)) = x_1
   POL(reduce(x_1)) = 0
   POL(result_cons(x_1)) = x_1
   POL(result_g(x_1)) = x_1
   POL(result_inf(x_1)) = 1
   POL(s(x_1)) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   redex_cons(x, cons(y, z)) -> result_cons(big)
   check_cons(result_cons(x)) -> go_up(x)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
   redex_g(x, x) -> result_g(b)
   check_g(result_g(x)) -> go_up(x)
   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
   reduce(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))


----------------------------------------

(28)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(go_up(cons(x0, x1))) -> TOP(check_cons(redex_cons(x0, x1)))
   TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1)))
   TOP(go_up(s(x0))) -> TOP(in_s_1(reduce(x0)))

The TRS R consists of the following rules:

   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
   reduce(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   redex_g(x, x) -> result_g(b)
   check_g(result_g(x)) -> go_up(x)
   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   check_inf(result_inf(x)) -> go_up(x)
   redex_cons(x, cons(y, z)) -> result_cons(big)
   check_cons(result_cons(x)) -> go_up(x)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))

The set Q consists of the following terms:

   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   redex_g(x0, x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(29) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(30)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(go_up(x)) -> TOP(reduce(x))

The TRS R consists of the following rules:

   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
   reduce(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   redex_g(x, x) -> result_g(b)
   check_g(result_g(x)) -> go_up(x)
   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   check_inf(result_inf(x)) -> go_up(x)
   redex_cons(x, cons(y, z)) -> result_cons(big)
   check_cons(result_cons(x)) -> go_up(x)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))

The set Q consists of the following terms:

   top(go_up(x0))
   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   redex_g(x0, x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_inf_1(go_up(x0))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(31) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   top(go_up(x0))
   in_inf_1(go_up(x0))


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(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(go_up(x)) -> TOP(reduce(x))

The TRS R consists of the following rules:

   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
   reduce(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   redex_g(x, x) -> result_g(b)
   check_g(result_g(x)) -> go_up(x)
   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   check_inf(result_inf(x)) -> go_up(x)
   redex_cons(x, cons(y, z)) -> result_cons(big)
   check_cons(result_cons(x)) -> go_up(x)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))

The set Q consists of the following terms:

   reduce(cons(x0, x1))
   reduce(inf(x0))
   reduce(g(x0, x1))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   redex_g(x0, x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_g(result_g(x0))
   check_cons(redex_cons(x0, x1))
   check_g(redex_g(x0, x1))
   reduce(s(x0))
   in_cons_1(go_up(x0), x1)
   in_cons_2(x0, go_up(x1))
   in_s_1(go_up(x0))
   in_g_1(go_up(x0), x1)
   in_g_2(x0, go_up(x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(33) Trivial-Transformation (SOUND)
We applied the Trivial transformation  to transform the outermost TRS to a standard TRS.
----------------------------------------

(34)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   cons(x, cons(y, z)) -> big
   inf(x) -> cons(x, inf(s(x)))
   g(x, x) -> b

Q is empty.

----------------------------------------

(35) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(36)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   INF(x) -> CONS(x, inf(s(x)))
   INF(x) -> INF(s(x))

The TRS R consists of the following rules:

   cons(x, cons(y, z)) -> big
   inf(x) -> cons(x, inf(s(x)))
   g(x, x) -> b

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(37) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(38)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   INF(x) -> INF(s(x))

The TRS R consists of the following rules:

   cons(x, cons(y, z)) -> big
   inf(x) -> cons(x, inf(s(x)))
   g(x, x) -> b

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(39) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(40)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   INF(x) -> INF(s(x))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(41) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule INF(x) -> INF(s(x)) we obtained the following new rules [LPAR04]:

   (INF(s(z0)) -> INF(s(s(z0))),INF(s(z0)) -> INF(s(s(z0))))


----------------------------------------

(42)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   INF(s(z0)) -> INF(s(s(z0)))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(43) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule INF(s(z0)) -> INF(s(s(z0))) we obtained the following new rules [LPAR04]:

   (INF(s(s(z0))) -> INF(s(s(s(z0)))),INF(s(s(z0))) -> INF(s(s(s(z0)))))


----------------------------------------

(44)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   INF(s(s(z0))) -> INF(s(s(s(z0))))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(45) NonTerminationLoopProof (COMPLETE)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = INF(s(s(z0))) evaluates to  t =INF(s(s(s(z0))))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
* Matcher: [z0 / s(z0)]
* Semiunifier: [ ]

--------------------------------------------------------------------------------
Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from INF(s(s(z0))) to INF(s(s(s(z0)))).




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(46)
NO