/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) Trivial-Transformation [SOUND, 0 ms]
(2) QTRS
    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
    (4) QDP
    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
    (6) QDP
    (7) UsableRulesProof [EQUIVALENT, 0 ms]
    (8) QDP
    (9) TransformationProof [EQUIVALENT, 0 ms]
    (10) QDP
    (11) TransformationProof [EQUIVALENT, 0 ms]
    (12) QDP
    (13) NonTerminationLoopProof [COMPLETE, 2 ms]
    (14) NO
(15) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms]
(16) QTRS
    (17) DependencyPairsProof [EQUIVALENT, 0 ms]
    (18) QDP
    (19) DependencyGraphProof [EQUIVALENT, 0 ms]
    (20) AND
        (21) QDP
            (22) UsableRulesProof [EQUIVALENT, 0 ms]
            (23) QDP
            (24) QReductionProof [EQUIVALENT, 0 ms]
            (25) QDP
            (26) UsableRulesReductionPairsProof [EQUIVALENT, 7 ms]
            (27) QDP
            (28) DependencyGraphProof [EQUIVALENT, 0 ms]
            (29) TRUE
        (30) QDP
            (31) UsableRulesProof [EQUIVALENT, 0 ms]
            (32) QDP
                (33) QReductionProof [EQUIVALENT, 0 ms]
                (34) QDP
                (35) TransformationProof [EQUIVALENT, 0 ms]
                (36) QDP
                (37) TransformationProof [EQUIVALENT, 0 ms]
                (38) QDP
                (39) TransformationProof [EQUIVALENT, 0 ms]
                (40) QDP
                (41) QDPOrderProof [EQUIVALENT, 14 ms]
                (42) QDP
            (43) UsableRulesProof [EQUIVALENT, 0 ms]
            (44) QDP
                (45) QReductionProof [EQUIVALENT, 0 ms]
                (46) QDP
(47) Raffelsieper-Zantema-Transformation [SOUND, 0 ms]
(48) QTRS
    (49) DependencyPairsProof [EQUIVALENT, 0 ms]
    (50) QDP
    (51) DependencyGraphProof [EQUIVALENT, 0 ms]
    (52) AND
        (53) QDP
            (54) UsableRulesProof [EQUIVALENT, 0 ms]
            (55) QDP
            (56) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (57) YES
        (58) QDP
            (59) TransformationProof [EQUIVALENT, 0 ms]
            (60) QDP
            (61) DependencyGraphProof [EQUIVALENT, 0 ms]
            (62) QDP
            (63) QDPOrderProof [EQUIVALENT, 12 ms]
            (64) QDP
            (65) QDPOrderProof [EQUIVALENT, 218 ms]
            (66) QDP


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

(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)))



Outermost Strategy.

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

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

(2)
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)))

Q is empty.

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

(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:

   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)))

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

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

(6)
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)))

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

(7) 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.
----------------------------------------

(8)
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.
----------------------------------------

(9) 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))))


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

(10)
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.
----------------------------------------

(11) 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)))))


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

(12)
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.
----------------------------------------

(13) 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)))).




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

(14)
NO

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

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

(16)
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))
   redex_cons(x, cons(y, z)) -> result_cons(big)
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   check_cons(result_cons(x)) -> go_up(x)
   check_inf(result_inf(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))
   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))

The set Q consists of the following terms:

   top(go_up(x0))
   reduce(cons(x0, x1))
   reduce(inf(x0))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))


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

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

(18)
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)
   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)
   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))
   redex_cons(x, cons(y, z)) -> result_cons(big)
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   check_cons(result_cons(x)) -> go_up(x)
   check_inf(result_inf(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))
   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))

The set Q consists of the following terms:

   top(go_up(x0))
   reduce(cons(x0, x1))
   reduce(inf(x0))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))

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

(19) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 9 less nodes.
----------------------------------------

(20)
Complex Obligation (AND)

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

(21)
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(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))
   redex_cons(x, cons(y, z)) -> result_cons(big)
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   check_cons(result_cons(x)) -> go_up(x)
   check_inf(result_inf(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))
   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))

The set Q consists of the following terms:

   top(go_up(x0))
   reduce(cons(x0, x1))
   reduce(inf(x0))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))

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

(22) 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.
----------------------------------------

(23)
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(s(x_1)) -> REDUCE(x_1)

The TRS R consists of the following rules:

   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))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))

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

(24) 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))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))


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

(25)
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(s(x_1)) -> REDUCE(x_1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

   redex_cons(x0, cons(x1, x2))

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

(26) 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(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)) = 2*x_1
   POL(REDUCE(x_1)) = 2*x_1
   POL(big) = 0
   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(redex_cons(x_1, x_2)) = 2*x_1 + x_2
   POL(result_cons(x_1)) = x_1
   POL(s(x_1)) = 2*x_1


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

(27)
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)

R is empty.
The set Q consists of the following terms:

   redex_cons(x0, cons(x1, x2))

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

(28) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
----------------------------------------

(29)
TRUE

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

(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:

   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))
   redex_cons(x, cons(y, z)) -> result_cons(big)
   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
   check_cons(result_cons(x)) -> go_up(x)
   check_inf(result_inf(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))
   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))

The set Q consists of the following terms:

   top(go_up(x0))
   reduce(cons(x0, x1))
   reduce(inf(x0))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))

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

(31) 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.
----------------------------------------

(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(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   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))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))

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

(33) 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))


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

(34)
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(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   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))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))

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

(35) 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(s(x0))) -> TOP(in_s_1(reduce(x0))),TOP(go_up(s(x0))) -> TOP(in_s_1(reduce(x0))))


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

(36)
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(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(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   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))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))

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

(37) 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)))))))


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

(38)
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(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(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   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))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))

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

(39) 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))))))


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

(40)
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(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(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   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))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))

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

(41) 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(big) = 0
   POL(check_cons(x_1)) = x_1
   POL(check_inf(x_1)) = 1
   POL(cons(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_s_1(x_1)) = 0
   POL(inf(x_1)) = 1
   POL(redex_cons(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_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))
   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(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))


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

(42)
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(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(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   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))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))

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

(43) 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.
----------------------------------------

(44)
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(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   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))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))

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

(45) 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))


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

(46)
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(s(x_1)) -> in_s_1(reduce(x_1))
   in_s_1(go_up(x_1)) -> go_up(s(x_1))
   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))
   redex_cons(x0, cons(x1, x2))
   redex_inf(x0)
   check_cons(result_cons(x0))
   check_inf(result_inf(x0))
   check_cons(redex_cons(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))

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

(47) Raffelsieper-Zantema-Transformation (SOUND)
We applied the Raffelsieper-Zantema transformation  to transform the outermost TRS to a standard TRS.
----------------------------------------

(48)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   down(cons(x, cons(y, z))) -> up(big)
   down(inf(x)) -> up(cons(x, inf(s(x))))
   top(up(x)) -> top(down(x))
   down(s(y3)) -> s_flat(down(y3))
   down(cons(y32, big)) -> cons_flat(down(y32), block(big))
   down(cons(y32, big)) -> cons_flat(block(y32), down(big))
   down(cons(y37, inf(y38))) -> cons_flat(down(y37), block(inf(y38)))
   down(cons(y37, inf(y38))) -> cons_flat(block(y37), down(inf(y38)))
   down(cons(y48, s(y49))) -> cons_flat(down(y48), block(s(y49)))
   down(cons(y48, s(y49))) -> cons_flat(block(y48), down(s(y49)))
   down(cons(y59, fresh_constant)) -> cons_flat(down(y59), block(fresh_constant))
   down(cons(y59, fresh_constant)) -> cons_flat(block(y59), down(fresh_constant))
   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
   inf_flat(up(x_1)) -> up(inf(x_1))
   s_flat(up(x_1)) -> up(s(x_1))

Q is empty.

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

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

(50)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(up(x)) -> TOP(down(x))
   TOP(up(x)) -> DOWN(x)
   DOWN(s(y3)) -> S_FLAT(down(y3))
   DOWN(s(y3)) -> DOWN(y3)
   DOWN(cons(y32, big)) -> CONS_FLAT(down(y32), block(big))
   DOWN(cons(y32, big)) -> DOWN(y32)
   DOWN(cons(y32, big)) -> CONS_FLAT(block(y32), down(big))
   DOWN(cons(y32, big)) -> DOWN(big)
   DOWN(cons(y37, inf(y38))) -> CONS_FLAT(down(y37), block(inf(y38)))
   DOWN(cons(y37, inf(y38))) -> DOWN(y37)
   DOWN(cons(y37, inf(y38))) -> CONS_FLAT(block(y37), down(inf(y38)))
   DOWN(cons(y37, inf(y38))) -> DOWN(inf(y38))
   DOWN(cons(y48, s(y49))) -> CONS_FLAT(down(y48), block(s(y49)))
   DOWN(cons(y48, s(y49))) -> DOWN(y48)
   DOWN(cons(y48, s(y49))) -> CONS_FLAT(block(y48), down(s(y49)))
   DOWN(cons(y48, s(y49))) -> DOWN(s(y49))
   DOWN(cons(y59, fresh_constant)) -> CONS_FLAT(down(y59), block(fresh_constant))
   DOWN(cons(y59, fresh_constant)) -> DOWN(y59)
   DOWN(cons(y59, fresh_constant)) -> CONS_FLAT(block(y59), down(fresh_constant))
   DOWN(cons(y59, fresh_constant)) -> DOWN(fresh_constant)

The TRS R consists of the following rules:

   down(cons(x, cons(y, z))) -> up(big)
   down(inf(x)) -> up(cons(x, inf(s(x))))
   top(up(x)) -> top(down(x))
   down(s(y3)) -> s_flat(down(y3))
   down(cons(y32, big)) -> cons_flat(down(y32), block(big))
   down(cons(y32, big)) -> cons_flat(block(y32), down(big))
   down(cons(y37, inf(y38))) -> cons_flat(down(y37), block(inf(y38)))
   down(cons(y37, inf(y38))) -> cons_flat(block(y37), down(inf(y38)))
   down(cons(y48, s(y49))) -> cons_flat(down(y48), block(s(y49)))
   down(cons(y48, s(y49))) -> cons_flat(block(y48), down(s(y49)))
   down(cons(y59, fresh_constant)) -> cons_flat(down(y59), block(fresh_constant))
   down(cons(y59, fresh_constant)) -> cons_flat(block(y59), down(fresh_constant))
   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
   inf_flat(up(x_1)) -> up(inf(x_1))
   s_flat(up(x_1)) -> up(s(x_1))

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

(51) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 13 less nodes.
----------------------------------------

(52)
Complex Obligation (AND)

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

(53)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   DOWN(cons(y32, big)) -> DOWN(y32)
   DOWN(s(y3)) -> DOWN(y3)
   DOWN(cons(y37, inf(y38))) -> DOWN(y37)
   DOWN(cons(y48, s(y49))) -> DOWN(y48)
   DOWN(cons(y48, s(y49))) -> DOWN(s(y49))
   DOWN(cons(y59, fresh_constant)) -> DOWN(y59)

The TRS R consists of the following rules:

   down(cons(x, cons(y, z))) -> up(big)
   down(inf(x)) -> up(cons(x, inf(s(x))))
   top(up(x)) -> top(down(x))
   down(s(y3)) -> s_flat(down(y3))
   down(cons(y32, big)) -> cons_flat(down(y32), block(big))
   down(cons(y32, big)) -> cons_flat(block(y32), down(big))
   down(cons(y37, inf(y38))) -> cons_flat(down(y37), block(inf(y38)))
   down(cons(y37, inf(y38))) -> cons_flat(block(y37), down(inf(y38)))
   down(cons(y48, s(y49))) -> cons_flat(down(y48), block(s(y49)))
   down(cons(y48, s(y49))) -> cons_flat(block(y48), down(s(y49)))
   down(cons(y59, fresh_constant)) -> cons_flat(down(y59), block(fresh_constant))
   down(cons(y59, fresh_constant)) -> cons_flat(block(y59), down(fresh_constant))
   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
   inf_flat(up(x_1)) -> up(inf(x_1))
   s_flat(up(x_1)) -> up(s(x_1))

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

(54) 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.
----------------------------------------

(55)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   DOWN(cons(y32, big)) -> DOWN(y32)
   DOWN(s(y3)) -> DOWN(y3)
   DOWN(cons(y37, inf(y38))) -> DOWN(y37)
   DOWN(cons(y48, s(y49))) -> DOWN(y48)
   DOWN(cons(y48, s(y49))) -> DOWN(s(y49))
   DOWN(cons(y59, fresh_constant)) -> DOWN(y59)

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

(56) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*DOWN(cons(y48, s(y49))) -> DOWN(s(y49))
The graph contains the following edges 1 > 1


*DOWN(s(y3)) -> DOWN(y3)
The graph contains the following edges 1 > 1


*DOWN(cons(y32, big)) -> DOWN(y32)
The graph contains the following edges 1 > 1


*DOWN(cons(y37, inf(y38))) -> DOWN(y37)
The graph contains the following edges 1 > 1


*DOWN(cons(y48, s(y49))) -> DOWN(y48)
The graph contains the following edges 1 > 1


*DOWN(cons(y59, fresh_constant)) -> DOWN(y59)
The graph contains the following edges 1 > 1


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

(57)
YES

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

(58)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(up(x)) -> TOP(down(x))

The TRS R consists of the following rules:

   down(cons(x, cons(y, z))) -> up(big)
   down(inf(x)) -> up(cons(x, inf(s(x))))
   top(up(x)) -> top(down(x))
   down(s(y3)) -> s_flat(down(y3))
   down(cons(y32, big)) -> cons_flat(down(y32), block(big))
   down(cons(y32, big)) -> cons_flat(block(y32), down(big))
   down(cons(y37, inf(y38))) -> cons_flat(down(y37), block(inf(y38)))
   down(cons(y37, inf(y38))) -> cons_flat(block(y37), down(inf(y38)))
   down(cons(y48, s(y49))) -> cons_flat(down(y48), block(s(y49)))
   down(cons(y48, s(y49))) -> cons_flat(block(y48), down(s(y49)))
   down(cons(y59, fresh_constant)) -> cons_flat(down(y59), block(fresh_constant))
   down(cons(y59, fresh_constant)) -> cons_flat(block(y59), down(fresh_constant))
   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
   inf_flat(up(x_1)) -> up(inf(x_1))
   s_flat(up(x_1)) -> up(s(x_1))

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

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

   (TOP(up(cons(x0, cons(x1, x2)))) -> TOP(up(big)),TOP(up(cons(x0, cons(x1, x2)))) -> TOP(up(big)))
   (TOP(up(inf(x0))) -> TOP(up(cons(x0, inf(s(x0))))),TOP(up(inf(x0))) -> TOP(up(cons(x0, inf(s(x0))))))
   (TOP(up(s(x0))) -> TOP(s_flat(down(x0))),TOP(up(s(x0))) -> TOP(s_flat(down(x0))))
   (TOP(up(cons(x0, big))) -> TOP(cons_flat(down(x0), block(big))),TOP(up(cons(x0, big))) -> TOP(cons_flat(down(x0), block(big))))
   (TOP(up(cons(x0, big))) -> TOP(cons_flat(block(x0), down(big))),TOP(up(cons(x0, big))) -> TOP(cons_flat(block(x0), down(big))))
   (TOP(up(cons(x0, inf(x1)))) -> TOP(cons_flat(down(x0), block(inf(x1)))),TOP(up(cons(x0, inf(x1)))) -> TOP(cons_flat(down(x0), block(inf(x1)))))
   (TOP(up(cons(x0, inf(x1)))) -> TOP(cons_flat(block(x0), down(inf(x1)))),TOP(up(cons(x0, inf(x1)))) -> TOP(cons_flat(block(x0), down(inf(x1)))))
   (TOP(up(cons(x0, s(x1)))) -> TOP(cons_flat(down(x0), block(s(x1)))),TOP(up(cons(x0, s(x1)))) -> TOP(cons_flat(down(x0), block(s(x1)))))
   (TOP(up(cons(x0, s(x1)))) -> TOP(cons_flat(block(x0), down(s(x1)))),TOP(up(cons(x0, s(x1)))) -> TOP(cons_flat(block(x0), down(s(x1)))))
   (TOP(up(cons(x0, fresh_constant))) -> TOP(cons_flat(down(x0), block(fresh_constant))),TOP(up(cons(x0, fresh_constant))) -> TOP(cons_flat(down(x0), block(fresh_constant))))
   (TOP(up(cons(x0, fresh_constant))) -> TOP(cons_flat(block(x0), down(fresh_constant))),TOP(up(cons(x0, fresh_constant))) -> TOP(cons_flat(block(x0), down(fresh_constant))))


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

(60)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(up(cons(x0, cons(x1, x2)))) -> TOP(up(big))
   TOP(up(inf(x0))) -> TOP(up(cons(x0, inf(s(x0)))))
   TOP(up(s(x0))) -> TOP(s_flat(down(x0)))
   TOP(up(cons(x0, big))) -> TOP(cons_flat(down(x0), block(big)))
   TOP(up(cons(x0, big))) -> TOP(cons_flat(block(x0), down(big)))
   TOP(up(cons(x0, inf(x1)))) -> TOP(cons_flat(down(x0), block(inf(x1))))
   TOP(up(cons(x0, inf(x1)))) -> TOP(cons_flat(block(x0), down(inf(x1))))
   TOP(up(cons(x0, s(x1)))) -> TOP(cons_flat(down(x0), block(s(x1))))
   TOP(up(cons(x0, s(x1)))) -> TOP(cons_flat(block(x0), down(s(x1))))
   TOP(up(cons(x0, fresh_constant))) -> TOP(cons_flat(down(x0), block(fresh_constant)))
   TOP(up(cons(x0, fresh_constant))) -> TOP(cons_flat(block(x0), down(fresh_constant)))

The TRS R consists of the following rules:

   down(cons(x, cons(y, z))) -> up(big)
   down(inf(x)) -> up(cons(x, inf(s(x))))
   top(up(x)) -> top(down(x))
   down(s(y3)) -> s_flat(down(y3))
   down(cons(y32, big)) -> cons_flat(down(y32), block(big))
   down(cons(y32, big)) -> cons_flat(block(y32), down(big))
   down(cons(y37, inf(y38))) -> cons_flat(down(y37), block(inf(y38)))
   down(cons(y37, inf(y38))) -> cons_flat(block(y37), down(inf(y38)))
   down(cons(y48, s(y49))) -> cons_flat(down(y48), block(s(y49)))
   down(cons(y48, s(y49))) -> cons_flat(block(y48), down(s(y49)))
   down(cons(y59, fresh_constant)) -> cons_flat(down(y59), block(fresh_constant))
   down(cons(y59, fresh_constant)) -> cons_flat(block(y59), down(fresh_constant))
   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
   inf_flat(up(x_1)) -> up(inf(x_1))
   s_flat(up(x_1)) -> up(s(x_1))

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

(61) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
----------------------------------------

(62)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(up(inf(x0))) -> TOP(up(cons(x0, inf(s(x0)))))
   TOP(up(cons(x0, inf(x1)))) -> TOP(cons_flat(down(x0), block(inf(x1))))
   TOP(up(s(x0))) -> TOP(s_flat(down(x0)))
   TOP(up(cons(x0, big))) -> TOP(cons_flat(down(x0), block(big)))
   TOP(up(cons(x0, inf(x1)))) -> TOP(cons_flat(block(x0), down(inf(x1))))
   TOP(up(cons(x0, s(x1)))) -> TOP(cons_flat(down(x0), block(s(x1))))
   TOP(up(cons(x0, s(x1)))) -> TOP(cons_flat(block(x0), down(s(x1))))
   TOP(up(cons(x0, fresh_constant))) -> TOP(cons_flat(down(x0), block(fresh_constant)))

The TRS R consists of the following rules:

   down(cons(x, cons(y, z))) -> up(big)
   down(inf(x)) -> up(cons(x, inf(s(x))))
   top(up(x)) -> top(down(x))
   down(s(y3)) -> s_flat(down(y3))
   down(cons(y32, big)) -> cons_flat(down(y32), block(big))
   down(cons(y32, big)) -> cons_flat(block(y32), down(big))
   down(cons(y37, inf(y38))) -> cons_flat(down(y37), block(inf(y38)))
   down(cons(y37, inf(y38))) -> cons_flat(block(y37), down(inf(y38)))
   down(cons(y48, s(y49))) -> cons_flat(down(y48), block(s(y49)))
   down(cons(y48, s(y49))) -> cons_flat(block(y48), down(s(y49)))
   down(cons(y59, fresh_constant)) -> cons_flat(down(y59), block(fresh_constant))
   down(cons(y59, fresh_constant)) -> cons_flat(block(y59), down(fresh_constant))
   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
   inf_flat(up(x_1)) -> up(inf(x_1))
   s_flat(up(x_1)) -> up(s(x_1))

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

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


The following pairs can be oriented strictly and are deleted.

   TOP(up(inf(x0))) -> TOP(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(big) = 0
   POL(block(x_1)) = 0
   POL(cons(x_1, x_2)) = 0
   POL(cons_flat(x_1, x_2)) = 0
   POL(down(x_1)) = 0
   POL(fresh_constant) = 0
   POL(inf(x_1)) = 1
   POL(s(x_1)) = 0
   POL(s_flat(x_1)) = 0
   POL(up(x_1)) = x_1

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

   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
   s_flat(up(x_1)) -> up(s(x_1))
   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))


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

(64)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(up(cons(x0, inf(x1)))) -> TOP(cons_flat(down(x0), block(inf(x1))))
   TOP(up(s(x0))) -> TOP(s_flat(down(x0)))
   TOP(up(cons(x0, big))) -> TOP(cons_flat(down(x0), block(big)))
   TOP(up(cons(x0, inf(x1)))) -> TOP(cons_flat(block(x0), down(inf(x1))))
   TOP(up(cons(x0, s(x1)))) -> TOP(cons_flat(down(x0), block(s(x1))))
   TOP(up(cons(x0, s(x1)))) -> TOP(cons_flat(block(x0), down(s(x1))))
   TOP(up(cons(x0, fresh_constant))) -> TOP(cons_flat(down(x0), block(fresh_constant)))

The TRS R consists of the following rules:

   down(cons(x, cons(y, z))) -> up(big)
   down(inf(x)) -> up(cons(x, inf(s(x))))
   top(up(x)) -> top(down(x))
   down(s(y3)) -> s_flat(down(y3))
   down(cons(y32, big)) -> cons_flat(down(y32), block(big))
   down(cons(y32, big)) -> cons_flat(block(y32), down(big))
   down(cons(y37, inf(y38))) -> cons_flat(down(y37), block(inf(y38)))
   down(cons(y37, inf(y38))) -> cons_flat(block(y37), down(inf(y38)))
   down(cons(y48, s(y49))) -> cons_flat(down(y48), block(s(y49)))
   down(cons(y48, s(y49))) -> cons_flat(block(y48), down(s(y49)))
   down(cons(y59, fresh_constant)) -> cons_flat(down(y59), block(fresh_constant))
   down(cons(y59, fresh_constant)) -> cons_flat(block(y59), down(fresh_constant))
   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
   inf_flat(up(x_1)) -> up(inf(x_1))
   s_flat(up(x_1)) -> up(s(x_1))

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

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


The following pairs can be oriented strictly and are deleted.

   TOP(up(cons(x0, inf(x1)))) -> TOP(cons_flat(block(x0), down(inf(x1))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO]:

Non-tuple symbols: 
<<<
 M( s_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
>>>

<<<
 M( s_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
>>>

<<<
 M( block_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [0, 0]] 	* 	x_1
>>>

<<<
 M( down_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1
>>>

<<<
 M( fresh_constant ) =  	[[0], [0]]
>>>

<<<
 M( up_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [1, 0]] 	* 	x_1
>>>

<<<
 M( cons_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 1], [0, 0]] 	* 	x_2
>>>

<<<
 M( big ) =  	[[0], [0]]
>>>

<<<
 M( inf_1(x_1) ) =  	[[0], [1]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
>>>

<<<
 M( cons_flat_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 0], [1, 0]] 	* 	x_2
>>>

Tuple symbols: 
<<<
 M( TOP_1(x_1) ) =  	[[0]] 	+ 	[[0, 1]] 	* 	x_1
>>>



Matrix type: 

We used a basic matrix type which is not further parametrizeable.





As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   down(inf(x)) -> up(cons(x, inf(s(x))))
   down(s(y3)) -> s_flat(down(y3))
   down(cons(y32, big)) -> cons_flat(down(y32), block(big))
   down(cons(y37, inf(y38))) -> cons_flat(down(y37), block(inf(y38)))
   down(cons(y48, s(y49))) -> cons_flat(down(y48), block(s(y49)))
   down(cons(y48, s(y49))) -> cons_flat(block(y48), down(s(y49)))
   down(cons(y59, fresh_constant)) -> cons_flat(down(y59), block(fresh_constant))
   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
   s_flat(up(x_1)) -> up(s(x_1))
   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))


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

(66)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(up(cons(x0, inf(x1)))) -> TOP(cons_flat(down(x0), block(inf(x1))))
   TOP(up(s(x0))) -> TOP(s_flat(down(x0)))
   TOP(up(cons(x0, big))) -> TOP(cons_flat(down(x0), block(big)))
   TOP(up(cons(x0, s(x1)))) -> TOP(cons_flat(down(x0), block(s(x1))))
   TOP(up(cons(x0, s(x1)))) -> TOP(cons_flat(block(x0), down(s(x1))))
   TOP(up(cons(x0, fresh_constant))) -> TOP(cons_flat(down(x0), block(fresh_constant)))

The TRS R consists of the following rules:

   down(cons(x, cons(y, z))) -> up(big)
   down(inf(x)) -> up(cons(x, inf(s(x))))
   top(up(x)) -> top(down(x))
   down(s(y3)) -> s_flat(down(y3))
   down(cons(y32, big)) -> cons_flat(down(y32), block(big))
   down(cons(y32, big)) -> cons_flat(block(y32), down(big))
   down(cons(y37, inf(y38))) -> cons_flat(down(y37), block(inf(y38)))
   down(cons(y37, inf(y38))) -> cons_flat(block(y37), down(inf(y38)))
   down(cons(y48, s(y49))) -> cons_flat(down(y48), block(s(y49)))
   down(cons(y48, s(y49))) -> cons_flat(block(y48), down(s(y49)))
   down(cons(y59, fresh_constant)) -> cons_flat(down(y59), block(fresh_constant))
   down(cons(y59, fresh_constant)) -> cons_flat(block(y59), down(fresh_constant))
   cons_flat(up(x_1), block(x_2)) -> up(cons(x_1, x_2))
   cons_flat(block(x_1), up(x_2)) -> up(cons(x_1, x_2))
   inf_flat(up(x_1)) -> up(inf(x_1))
   s_flat(up(x_1)) -> up(s(x_1))

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