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


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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Outermost Termination of the given OTRS could be proven:

(0) OTRS
(1) Raffelsieper-Zantema-Transformation [SOUND, 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) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) TransformationProof [EQUIVALENT, 0 ms]
        (14) QDP
        (15) DependencyGraphProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) QDPOrderProof [EQUIVALENT, 22 ms]
        (18) QDP
        (19) SplitQDPProof [EQUIVALENT, 0 ms]
        (20) AND
            (21) QDP
                (22) SemLabProof [SOUND, 0 ms]
                (23) QDP
                (24) DependencyGraphProof [EQUIVALENT, 0 ms]
                (25) QDP
                (26) QDPOrderProof [EQUIVALENT, 21 ms]
                (27) QDP
                (28) PisEmptyProof [SOUND, 0 ms]
                (29) TRUE
            (30) QDP
                (31) SemLabProof [SOUND, 2990 ms]
                (32) QDP
                (33) DependencyGraphProof [EQUIVALENT, 0 ms]
                (34) QDP
                (35) QDPOrderProof [EQUIVALENT, 0 ms]
                (36) QDP
                (37) PisEmptyProof [EQUIVALENT, 0 ms]
                (38) YES


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

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

   f(g(u, v), y, z) -> c
   f(0, 1, x) -> f(x, x, x)
   g(x, y) -> x
   g(x, y) -> y
   f(f(x, y, z), u, v) -> c
   f(0, u, v) -> c



Outermost Strategy.

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

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

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

   down(f(g(u, v), y, z)) -> up(c)
   down(f(0, 1, x)) -> up(f(x, x, x))
   down(g(x, y)) -> up(x)
   down(g(x, y)) -> up(y)
   down(f(f(x, y, z), u, v)) -> up(c)
   down(f(0, u, v)) -> up(c)
   top(up(x)) -> top(down(x))
   down(f(c, y17, y18)) -> f_flat(down(c), block(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), down(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), block(y17), down(y18))
   down(f(1, y21, y22)) -> f_flat(down(1), block(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), down(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), block(y21), down(y22))
   down(f(fresh_constant, y23, y24)) -> f_flat(down(fresh_constant), block(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), down(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), block(y23), down(y24))
   f_flat(up(x_1), block(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), up(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), block(x_2), up(x_3)) -> up(f(x_1, x_2, x_3))
   g_flat(up(x_1), block(x_2)) -> up(g(x_1, x_2))
   g_flat(block(x_1), up(x_2)) -> up(g(x_1, x_2))

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:

   TOP(up(x)) -> TOP(down(x))
   TOP(up(x)) -> DOWN(x)
   DOWN(f(c, y17, y18)) -> F_FLAT(down(c), block(y17), block(y18))
   DOWN(f(c, y17, y18)) -> DOWN(c)
   DOWN(f(c, y17, y18)) -> F_FLAT(block(c), down(y17), block(y18))
   DOWN(f(c, y17, y18)) -> DOWN(y17)
   DOWN(f(c, y17, y18)) -> F_FLAT(block(c), block(y17), down(y18))
   DOWN(f(c, y17, y18)) -> DOWN(y18)
   DOWN(f(1, y21, y22)) -> F_FLAT(down(1), block(y21), block(y22))
   DOWN(f(1, y21, y22)) -> DOWN(1)
   DOWN(f(1, y21, y22)) -> F_FLAT(block(1), down(y21), block(y22))
   DOWN(f(1, y21, y22)) -> DOWN(y21)
   DOWN(f(1, y21, y22)) -> F_FLAT(block(1), block(y21), down(y22))
   DOWN(f(1, y21, y22)) -> DOWN(y22)
   DOWN(f(fresh_constant, y23, y24)) -> F_FLAT(down(fresh_constant), block(y23), block(y24))
   DOWN(f(fresh_constant, y23, y24)) -> DOWN(fresh_constant)
   DOWN(f(fresh_constant, y23, y24)) -> F_FLAT(block(fresh_constant), down(y23), block(y24))
   DOWN(f(fresh_constant, y23, y24)) -> DOWN(y23)
   DOWN(f(fresh_constant, y23, y24)) -> F_FLAT(block(fresh_constant), block(y23), down(y24))
   DOWN(f(fresh_constant, y23, y24)) -> DOWN(y24)

The TRS R consists of the following rules:

   down(f(g(u, v), y, z)) -> up(c)
   down(f(0, 1, x)) -> up(f(x, x, x))
   down(g(x, y)) -> up(x)
   down(g(x, y)) -> up(y)
   down(f(f(x, y, z), u, v)) -> up(c)
   down(f(0, u, v)) -> up(c)
   top(up(x)) -> top(down(x))
   down(f(c, y17, y18)) -> f_flat(down(c), block(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), down(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), block(y17), down(y18))
   down(f(1, y21, y22)) -> f_flat(down(1), block(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), down(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), block(y21), down(y22))
   down(f(fresh_constant, y23, y24)) -> f_flat(down(fresh_constant), block(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), down(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), block(y23), down(y24))
   f_flat(up(x_1), block(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), up(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), block(x_2), up(x_3)) -> up(f(x_1, x_2, x_3))
   g_flat(up(x_1), block(x_2)) -> up(g(x_1, x_2))
   g_flat(block(x_1), up(x_2)) -> up(g(x_1, x_2))

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 2 SCCs with 13 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

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

   DOWN(f(c, y17, y18)) -> DOWN(y18)
   DOWN(f(c, y17, y18)) -> DOWN(y17)
   DOWN(f(1, y21, y22)) -> DOWN(y21)
   DOWN(f(1, y21, y22)) -> DOWN(y22)
   DOWN(f(fresh_constant, y23, y24)) -> DOWN(y23)
   DOWN(f(fresh_constant, y23, y24)) -> DOWN(y24)

The TRS R consists of the following rules:

   down(f(g(u, v), y, z)) -> up(c)
   down(f(0, 1, x)) -> up(f(x, x, x))
   down(g(x, y)) -> up(x)
   down(g(x, y)) -> up(y)
   down(f(f(x, y, z), u, v)) -> up(c)
   down(f(0, u, v)) -> up(c)
   top(up(x)) -> top(down(x))
   down(f(c, y17, y18)) -> f_flat(down(c), block(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), down(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), block(y17), down(y18))
   down(f(1, y21, y22)) -> f_flat(down(1), block(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), down(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), block(y21), down(y22))
   down(f(fresh_constant, y23, y24)) -> f_flat(down(fresh_constant), block(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), down(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), block(y23), down(y24))
   f_flat(up(x_1), block(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), up(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), block(x_2), up(x_3)) -> up(f(x_1, x_2, x_3))
   g_flat(up(x_1), block(x_2)) -> up(g(x_1, x_2))
   g_flat(block(x_1), up(x_2)) -> up(g(x_1, x_2))

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

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

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

   DOWN(f(c, y17, y18)) -> DOWN(y18)
   DOWN(f(c, y17, y18)) -> DOWN(y17)
   DOWN(f(1, y21, y22)) -> DOWN(y21)
   DOWN(f(1, y21, y22)) -> DOWN(y22)
   DOWN(f(fresh_constant, y23, y24)) -> DOWN(y23)
   DOWN(f(fresh_constant, y23, y24)) -> DOWN(y24)

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

(10) 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(f(c, y17, y18)) -> DOWN(y18)
The graph contains the following edges 1 > 1


*DOWN(f(c, y17, y18)) -> DOWN(y17)
The graph contains the following edges 1 > 1


*DOWN(f(1, y21, y22)) -> DOWN(y21)
The graph contains the following edges 1 > 1


*DOWN(f(1, y21, y22)) -> DOWN(y22)
The graph contains the following edges 1 > 1


*DOWN(f(fresh_constant, y23, y24)) -> DOWN(y23)
The graph contains the following edges 1 > 1


*DOWN(f(fresh_constant, y23, y24)) -> DOWN(y24)
The graph contains the following edges 1 > 1


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

(11)
YES

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

(12)
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(f(g(u, v), y, z)) -> up(c)
   down(f(0, 1, x)) -> up(f(x, x, x))
   down(g(x, y)) -> up(x)
   down(g(x, y)) -> up(y)
   down(f(f(x, y, z), u, v)) -> up(c)
   down(f(0, u, v)) -> up(c)
   top(up(x)) -> top(down(x))
   down(f(c, y17, y18)) -> f_flat(down(c), block(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), down(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), block(y17), down(y18))
   down(f(1, y21, y22)) -> f_flat(down(1), block(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), down(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), block(y21), down(y22))
   down(f(fresh_constant, y23, y24)) -> f_flat(down(fresh_constant), block(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), down(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), block(y23), down(y24))
   f_flat(up(x_1), block(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), up(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), block(x_2), up(x_3)) -> up(f(x_1, x_2, x_3))
   g_flat(up(x_1), block(x_2)) -> up(g(x_1, x_2))
   g_flat(block(x_1), up(x_2)) -> up(g(x_1, x_2))

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

(13) 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(f(g(x0, x1), x2, x3))) -> TOP(up(c)),TOP(up(f(g(x0, x1), x2, x3))) -> TOP(up(c)))
   (TOP(up(f(0, 1, x0))) -> TOP(up(f(x0, x0, x0))),TOP(up(f(0, 1, x0))) -> TOP(up(f(x0, x0, x0))))
   (TOP(up(g(x0, x1))) -> TOP(up(x0)),TOP(up(g(x0, x1))) -> TOP(up(x0)))
   (TOP(up(g(x0, x1))) -> TOP(up(x1)),TOP(up(g(x0, x1))) -> TOP(up(x1)))
   (TOP(up(f(f(x0, x1, x2), x3, x4))) -> TOP(up(c)),TOP(up(f(f(x0, x1, x2), x3, x4))) -> TOP(up(c)))
   (TOP(up(f(0, x0, x1))) -> TOP(up(c)),TOP(up(f(0, x0, x1))) -> TOP(up(c)))
   (TOP(up(f(c, x0, x1))) -> TOP(f_flat(down(c), block(x0), block(x1))),TOP(up(f(c, x0, x1))) -> TOP(f_flat(down(c), block(x0), block(x1))))
   (TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), down(x0), block(x1))),TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), down(x0), block(x1))))
   (TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), block(x0), down(x1))),TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), block(x0), down(x1))))
   (TOP(up(f(1, x0, x1))) -> TOP(f_flat(down(1), block(x0), block(x1))),TOP(up(f(1, x0, x1))) -> TOP(f_flat(down(1), block(x0), block(x1))))
   (TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), down(x0), block(x1))),TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), down(x0), block(x1))))
   (TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), block(x0), down(x1))),TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), block(x0), down(x1))))
   (TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(down(fresh_constant), block(x0), block(x1))),TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(down(fresh_constant), block(x0), block(x1))))
   (TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), down(x0), block(x1))),TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), down(x0), block(x1))))
   (TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), block(x0), down(x1))),TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), block(x0), down(x1))))


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

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(up(f(g(x0, x1), x2, x3))) -> TOP(up(c))
   TOP(up(f(0, 1, x0))) -> TOP(up(f(x0, x0, x0)))
   TOP(up(g(x0, x1))) -> TOP(up(x0))
   TOP(up(g(x0, x1))) -> TOP(up(x1))
   TOP(up(f(f(x0, x1, x2), x3, x4))) -> TOP(up(c))
   TOP(up(f(0, x0, x1))) -> TOP(up(c))
   TOP(up(f(c, x0, x1))) -> TOP(f_flat(down(c), block(x0), block(x1)))
   TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), down(x0), block(x1)))
   TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), block(x0), down(x1)))
   TOP(up(f(1, x0, x1))) -> TOP(f_flat(down(1), block(x0), block(x1)))
   TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), down(x0), block(x1)))
   TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), block(x0), down(x1)))
   TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(down(fresh_constant), block(x0), block(x1)))
   TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), down(x0), block(x1)))
   TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), block(x0), down(x1)))

The TRS R consists of the following rules:

   down(f(g(u, v), y, z)) -> up(c)
   down(f(0, 1, x)) -> up(f(x, x, x))
   down(g(x, y)) -> up(x)
   down(g(x, y)) -> up(y)
   down(f(f(x, y, z), u, v)) -> up(c)
   down(f(0, u, v)) -> up(c)
   top(up(x)) -> top(down(x))
   down(f(c, y17, y18)) -> f_flat(down(c), block(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), down(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), block(y17), down(y18))
   down(f(1, y21, y22)) -> f_flat(down(1), block(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), down(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), block(y21), down(y22))
   down(f(fresh_constant, y23, y24)) -> f_flat(down(fresh_constant), block(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), down(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), block(y23), down(y24))
   f_flat(up(x_1), block(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), up(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), block(x_2), up(x_3)) -> up(f(x_1, x_2, x_3))
   g_flat(up(x_1), block(x_2)) -> up(g(x_1, x_2))
   g_flat(block(x_1), up(x_2)) -> up(g(x_1, x_2))

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

(15) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
----------------------------------------

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

   TOP(up(f(0, 1, x0))) -> TOP(up(f(x0, x0, x0)))
   TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), down(x0), block(x1)))
   TOP(up(g(x0, x1))) -> TOP(up(x0))
   TOP(up(g(x0, x1))) -> TOP(up(x1))
   TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), block(x0), down(x1)))
   TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), down(x0), block(x1)))
   TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), block(x0), down(x1)))
   TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), down(x0), block(x1)))
   TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), block(x0), down(x1)))

The TRS R consists of the following rules:

   down(f(g(u, v), y, z)) -> up(c)
   down(f(0, 1, x)) -> up(f(x, x, x))
   down(g(x, y)) -> up(x)
   down(g(x, y)) -> up(y)
   down(f(f(x, y, z), u, v)) -> up(c)
   down(f(0, u, v)) -> up(c)
   top(up(x)) -> top(down(x))
   down(f(c, y17, y18)) -> f_flat(down(c), block(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), down(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), block(y17), down(y18))
   down(f(1, y21, y22)) -> f_flat(down(1), block(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), down(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), block(y21), down(y22))
   down(f(fresh_constant, y23, y24)) -> f_flat(down(fresh_constant), block(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), down(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), block(y23), down(y24))
   f_flat(up(x_1), block(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), up(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), block(x_2), up(x_3)) -> up(f(x_1, x_2, x_3))
   g_flat(up(x_1), block(x_2)) -> up(g(x_1, x_2))
   g_flat(block(x_1), up(x_2)) -> up(g(x_1, x_2))

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

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


The following pairs can be oriented strictly and are deleted.

   TOP(up(g(x0, x1))) -> TOP(up(x0))
   TOP(up(g(x0, x1))) -> TOP(up(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(1) = 0
   POL(TOP(x_1)) = x_1
   POL(block(x_1)) = 0
   POL(c) = 0
   POL(down(x_1)) = 0
   POL(f(x_1, x_2, x_3)) = 0
   POL(f_flat(x_1, x_2, x_3)) = 0
   POL(fresh_constant) = 0
   POL(g(x_1, x_2)) = 1 + x_1 + x_2
   POL(up(x_1)) = x_1

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

   f_flat(block(x_1), up(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), block(x_2), up(x_3)) -> up(f(x_1, x_2, x_3))


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

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

   TOP(up(f(0, 1, x0))) -> TOP(up(f(x0, x0, x0)))
   TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), down(x0), block(x1)))
   TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), block(x0), down(x1)))
   TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), down(x0), block(x1)))
   TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), block(x0), down(x1)))
   TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), down(x0), block(x1)))
   TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), block(x0), down(x1)))

The TRS R consists of the following rules:

   down(f(g(u, v), y, z)) -> up(c)
   down(f(0, 1, x)) -> up(f(x, x, x))
   down(g(x, y)) -> up(x)
   down(g(x, y)) -> up(y)
   down(f(f(x, y, z), u, v)) -> up(c)
   down(f(0, u, v)) -> up(c)
   top(up(x)) -> top(down(x))
   down(f(c, y17, y18)) -> f_flat(down(c), block(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), down(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), block(y17), down(y18))
   down(f(1, y21, y22)) -> f_flat(down(1), block(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), down(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), block(y21), down(y22))
   down(f(fresh_constant, y23, y24)) -> f_flat(down(fresh_constant), block(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), down(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), block(y23), down(y24))
   f_flat(up(x_1), block(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), up(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), block(x_2), up(x_3)) -> up(f(x_1, x_2, x_3))
   g_flat(up(x_1), block(x_2)) -> up(g(x_1, x_2))
   g_flat(block(x_1), up(x_2)) -> up(g(x_1, x_2))

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

(19) SplitQDPProof (EQUIVALENT)
We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem

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

(20)
Complex Obligation (AND)

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

(21)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP(up(f(0, 1, x0))) -> TOP(up(f(x0, x0, x0)))
   TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), down(x0), block(x1)))
   TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), block(x0), down(x1)))
   TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), down(x0), block(x1)))
   TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), block(x0), down(x1)))
   TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), down(x0), block(x1)))
   TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), block(x0), down(x1)))

The TRS R consists of the following rules:

   down(f(g(u, v), y, z)) -> up(c)
   down(f(0, 1, x)) -> up(f(x, x, x))
   down(g(x, y)) -> up(x)
   down(g(x, y)) -> up(y)
   down(f(f(x, y, z), u, v)) -> up(c)
   down(f(0, u, v)) -> up(c)
   top(up(x)) -> top(down(x))
   down(f(c, y17, y18)) -> f_flat(down(c), block(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), down(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), block(y17), down(y18))
   down(f(1, y21, y22)) -> f_flat(down(1), block(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), down(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), block(y21), down(y22))
   down(f(fresh_constant, y23, y24)) -> f_flat(down(fresh_constant), block(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), down(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), block(y23), down(y24))
   f_flat(up(x_1), block(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), up(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), block(x_2), up(x_3)) -> up(f(x_1, x_2, x_3))
   g_flat(up(x_1), block(x_2)) -> up(g(x_1, x_2))
   g_flat(block(x_1), up(x_2)) -> up(g(x_1, x_2))

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

(22) SemLabProof (SOUND)
We found the following model for the rules of the TRSs R and P.
Interpretation over the domain with elements from 0 to 1.
c: 0
block: 0
TOP: 0
g: 0
top: 0
down: 0
f: 0
0: 0
fresh_constant: 0
up: 0
1: 1
f_flat: 0
g_flat: 0
By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
----------------------------------------

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP.0(up.0(f.0-1-0(0., 1., x0))) -> TOP.0(up.0(f.0-0-0(x0, x0, x0)))
   TOP.0(up.0(f.0-1-1(0., 1., x0))) -> TOP.0(up.0(f.1-1-1(x0, x0, x0)))
   TOP.0(up.0(f.0-0-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.0-0-1(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.0-1-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), down.1(x0), block.0(x1)))
   TOP.0(up.0(f.0-1-1(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), down.1(x0), block.1(x1)))
   TOP.0(up.0(f.0-0-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.0-0-1(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), block.0(x0), down.1(x1)))
   TOP.0(up.0(f.0-1-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.0-1-1(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), block.1(x0), down.1(x1)))
   TOP.0(up.0(f.1-0-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.1-0-1(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.1-1-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.1(x0), block.0(x1)))
   TOP.0(up.0(f.1-1-1(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.1(x0), block.1(x1)))
   TOP.0(up.0(f.1-0-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.1-0-1(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.0(x0), down.1(x1)))
   TOP.0(up.0(f.1-1-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.1-1-1(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.1(x0), down.1(x1)))
   TOP.0(up.0(f.0-0-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.0-0-1(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.0-1-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), down.1(x0), block.0(x1)))
   TOP.0(up.0(f.0-1-1(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), down.1(x0), block.1(x1)))
   TOP.0(up.0(f.0-0-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.0-0-1(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), block.0(x0), down.1(x1)))
   TOP.0(up.0(f.0-1-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.0-1-1(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), block.1(x0), down.1(x1)))

The TRS R consists of the following rules:

   down.0(f.0-0-0(g.0-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-1(g.0-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(g.0-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-1(g.0-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-0(g.0-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-1(g.0-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(g.0-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-1(g.0-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-0(g.1-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-1(g.1-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(g.1-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-1(g.1-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-0(g.1-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-1(g.1-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(g.1-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-1(g.1-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(0., 1., x)) -> up.0(f.0-0-0(x, x, x))
   down.0(f.0-1-1(0., 1., x)) -> up.0(f.1-1-1(x, x, x))
   down.0(g.0-0(x, y)) -> up.0(x)
   down.0(g.0-1(x, y)) -> up.0(x)
   down.0(g.1-0(x, y)) -> up.1(x)
   down.0(g.1-1(x, y)) -> up.1(x)
   down.0(g.0-0(x, y)) -> up.0(y)
   down.0(g.0-1(x, y)) -> up.1(y)
   down.0(g.1-0(x, y)) -> up.0(y)
   down.0(g.1-1(x, y)) -> up.1(y)
   down.0(f.0-0-0(f.0-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.0-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.0-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.0-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.0-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.0-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.0-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.0-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.0-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.0-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.0-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.0-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.0-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.0-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.0-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.0-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.1-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.1-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.1-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.1-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.1-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.1-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.1-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.1-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.1-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.1-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.1-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.1-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.1-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.1-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.1-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.1-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(0., u, v)) -> up.0(c.)
   down.0(f.0-0-1(0., u, v)) -> up.0(c.)
   down.0(f.0-1-0(0., u, v)) -> up.0(c.)
   down.0(f.0-1-1(0., u, v)) -> up.0(c.)
   top.0(up.0(x)) -> top.0(down.0(x))
   top.0(up.1(x)) -> top.0(down.1(x))
   down.0(f.0-0-0(c., y17, y18)) -> f_flat.0-0-0(down.0(c.), block.0(y17), block.0(y18))
   down.0(f.0-0-1(c., y17, y18)) -> f_flat.0-0-0(down.0(c.), block.0(y17), block.1(y18))
   down.0(f.0-1-0(c., y17, y18)) -> f_flat.0-0-0(down.0(c.), block.1(y17), block.0(y18))
   down.0(f.0-1-1(c., y17, y18)) -> f_flat.0-0-0(down.0(c.), block.1(y17), block.1(y18))
   down.0(f.0-0-0(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), down.0(y17), block.0(y18))
   down.0(f.0-0-1(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), down.0(y17), block.1(y18))
   down.0(f.0-1-0(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), down.1(y17), block.0(y18))
   down.0(f.0-1-1(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), down.1(y17), block.1(y18))
   down.0(f.0-0-0(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), block.0(y17), down.0(y18))
   down.0(f.0-0-1(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), block.0(y17), down.1(y18))
   down.0(f.0-1-0(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), block.1(y17), down.0(y18))
   down.0(f.0-1-1(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), block.1(y17), down.1(y18))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.1(y22))
   down.0(f.0-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.0(fresh_constant.), block.0(y23), block.0(y24))
   down.0(f.0-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.0(fresh_constant.), block.0(y23), block.1(y24))
   down.0(f.0-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.0(fresh_constant.), block.1(y23), block.0(y24))
   down.0(f.0-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.0(fresh_constant.), block.1(y23), block.1(y24))
   down.0(f.0-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), down.0(y23), block.0(y24))
   down.0(f.0-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), down.0(y23), block.1(y24))
   down.0(f.0-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), down.1(y23), block.0(y24))
   down.0(f.0-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), down.1(y23), block.1(y24))
   down.0(f.0-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), block.0(y23), down.0(y24))
   down.0(f.0-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), block.0(y23), down.1(y24))
   down.0(f.0-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), block.1(y23), down.0(y24))
   down.0(f.0-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), block.1(y23), down.1(y24))
   f_flat.0-0-0(up.0(x_1), block.0(x_2), block.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.0(x_2), block.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.1(x_2), block.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.1(x_2), block.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   g_flat.0-0(up.0(x_1), block.0(x_2)) -> up.0(g.0-0(x_1, x_2))
   g_flat.0-0(up.0(x_1), block.1(x_2)) -> up.0(g.0-1(x_1, x_2))
   g_flat.0-0(up.1(x_1), block.0(x_2)) -> up.0(g.1-0(x_1, x_2))
   g_flat.0-0(up.1(x_1), block.1(x_2)) -> up.0(g.1-1(x_1, x_2))
   g_flat.0-0(block.0(x_1), up.0(x_2)) -> up.0(g.0-0(x_1, x_2))
   g_flat.0-0(block.0(x_1), up.1(x_2)) -> up.0(g.0-1(x_1, x_2))
   g_flat.0-0(block.1(x_1), up.0(x_2)) -> up.0(g.1-0(x_1, x_2))
   g_flat.0-0(block.1(x_1), up.1(x_2)) -> up.0(g.1-1(x_1, x_2))

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

(24) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 13 less nodes.
----------------------------------------

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP.0(up.0(f.0-0-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.0-1-0(0., 1., x0))) -> TOP.0(up.0(f.0-0-0(x0, x0, x0)))
   TOP.0(up.0(f.0-0-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.0-0-1(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.0-1-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.1-0-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.1-0-1(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.1-0-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.1-1-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.0-0-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.0-0-1(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.0-0-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.0-1-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), block.1(x0), down.0(x1)))

The TRS R consists of the following rules:

   down.0(f.0-0-0(g.0-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-1(g.0-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(g.0-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-1(g.0-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-0(g.0-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-1(g.0-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(g.0-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-1(g.0-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-0(g.1-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-1(g.1-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(g.1-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-1(g.1-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-0(g.1-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-1(g.1-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(g.1-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-1(g.1-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(0., 1., x)) -> up.0(f.0-0-0(x, x, x))
   down.0(f.0-1-1(0., 1., x)) -> up.0(f.1-1-1(x, x, x))
   down.0(g.0-0(x, y)) -> up.0(x)
   down.0(g.0-1(x, y)) -> up.0(x)
   down.0(g.1-0(x, y)) -> up.1(x)
   down.0(g.1-1(x, y)) -> up.1(x)
   down.0(g.0-0(x, y)) -> up.0(y)
   down.0(g.0-1(x, y)) -> up.1(y)
   down.0(g.1-0(x, y)) -> up.0(y)
   down.0(g.1-1(x, y)) -> up.1(y)
   down.0(f.0-0-0(f.0-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.0-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.0-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.0-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.0-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.0-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.0-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.0-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.0-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.0-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.0-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.0-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.0-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.0-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.0-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.0-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.1-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.1-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.1-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.1-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.1-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.1-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.1-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.1-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.1-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.1-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.1-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.1-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.1-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.1-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.1-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.1-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(0., u, v)) -> up.0(c.)
   down.0(f.0-0-1(0., u, v)) -> up.0(c.)
   down.0(f.0-1-0(0., u, v)) -> up.0(c.)
   down.0(f.0-1-1(0., u, v)) -> up.0(c.)
   top.0(up.0(x)) -> top.0(down.0(x))
   top.0(up.1(x)) -> top.0(down.1(x))
   down.0(f.0-0-0(c., y17, y18)) -> f_flat.0-0-0(down.0(c.), block.0(y17), block.0(y18))
   down.0(f.0-0-1(c., y17, y18)) -> f_flat.0-0-0(down.0(c.), block.0(y17), block.1(y18))
   down.0(f.0-1-0(c., y17, y18)) -> f_flat.0-0-0(down.0(c.), block.1(y17), block.0(y18))
   down.0(f.0-1-1(c., y17, y18)) -> f_flat.0-0-0(down.0(c.), block.1(y17), block.1(y18))
   down.0(f.0-0-0(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), down.0(y17), block.0(y18))
   down.0(f.0-0-1(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), down.0(y17), block.1(y18))
   down.0(f.0-1-0(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), down.1(y17), block.0(y18))
   down.0(f.0-1-1(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), down.1(y17), block.1(y18))
   down.0(f.0-0-0(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), block.0(y17), down.0(y18))
   down.0(f.0-0-1(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), block.0(y17), down.1(y18))
   down.0(f.0-1-0(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), block.1(y17), down.0(y18))
   down.0(f.0-1-1(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), block.1(y17), down.1(y18))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.1(y22))
   down.0(f.0-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.0(fresh_constant.), block.0(y23), block.0(y24))
   down.0(f.0-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.0(fresh_constant.), block.0(y23), block.1(y24))
   down.0(f.0-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.0(fresh_constant.), block.1(y23), block.0(y24))
   down.0(f.0-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.0(fresh_constant.), block.1(y23), block.1(y24))
   down.0(f.0-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), down.0(y23), block.0(y24))
   down.0(f.0-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), down.0(y23), block.1(y24))
   down.0(f.0-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), down.1(y23), block.0(y24))
   down.0(f.0-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), down.1(y23), block.1(y24))
   down.0(f.0-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), block.0(y23), down.0(y24))
   down.0(f.0-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), block.0(y23), down.1(y24))
   down.0(f.0-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), block.1(y23), down.0(y24))
   down.0(f.0-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), block.1(y23), down.1(y24))
   f_flat.0-0-0(up.0(x_1), block.0(x_2), block.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.0(x_2), block.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.1(x_2), block.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.1(x_2), block.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   g_flat.0-0(up.0(x_1), block.0(x_2)) -> up.0(g.0-0(x_1, x_2))
   g_flat.0-0(up.0(x_1), block.1(x_2)) -> up.0(g.0-1(x_1, x_2))
   g_flat.0-0(up.1(x_1), block.0(x_2)) -> up.0(g.1-0(x_1, x_2))
   g_flat.0-0(up.1(x_1), block.1(x_2)) -> up.0(g.1-1(x_1, x_2))
   g_flat.0-0(block.0(x_1), up.0(x_2)) -> up.0(g.0-0(x_1, x_2))
   g_flat.0-0(block.0(x_1), up.1(x_2)) -> up.0(g.0-1(x_1, x_2))
   g_flat.0-0(block.1(x_1), up.0(x_2)) -> up.0(g.1-0(x_1, x_2))
   g_flat.0-0(block.1(x_1), up.1(x_2)) -> up.0(g.1-1(x_1, x_2))

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

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


The following pairs can be oriented strictly and are deleted.

   TOP.0(up.0(f.0-1-0(0., 1., x0))) -> TOP.0(up.0(f.0-0-0(x0, x0, x0)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0.) = 1
   POL(1.) = 0
   POL(TOP.0(x_1)) = x_1
   POL(block.0(x_1)) = x_1
   POL(block.1(x_1)) = 0
   POL(c.) = 0
   POL(down.0(x_1)) = 0
   POL(down.1(x_1)) = 0
   POL(f.0-0-0(x_1, x_2, x_3)) = 0
   POL(f.0-0-1(x_1, x_2, x_3)) = 0
   POL(f.0-1-0(x_1, x_2, x_3)) = x_1
   POL(f.0-1-1(x_1, x_2, x_3)) = 0
   POL(f.1-0-0(x_1, x_2, x_3)) = 0
   POL(f.1-0-1(x_1, x_2, x_3)) = 0
   POL(f.1-1-0(x_1, x_2, x_3)) = 0
   POL(f.1-1-1(x_1, x_2, x_3)) = 0
   POL(f_flat.0-0-0(x_1, x_2, x_3)) = x_1
   POL(fresh_constant.) = 0
   POL(g.0-0(x_1, x_2)) = x_1 + x_2
   POL(g.0-1(x_1, x_2)) = x_1 + x_2
   POL(g.1-0(x_1, x_2)) = x_1 + x_2
   POL(g.1-1(x_1, x_2)) = x_1 + x_2
   POL(up.0(x_1)) = x_1
   POL(up.1(x_1)) = x_1

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

   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))


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

(27)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP.0(up.0(f.0-0-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.0-0-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.0-0-1(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.0-1-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(c.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.1-0-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.1-0-1(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.1-0-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.1-1-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.0-0-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.0-0-1(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.0-0-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.0-1-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.0(fresh_constant.), block.1(x0), down.0(x1)))

The TRS R consists of the following rules:

   down.0(f.0-0-0(g.0-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-1(g.0-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(g.0-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-1(g.0-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-0(g.0-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-1(g.0-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(g.0-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-1(g.0-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-0(g.1-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-1(g.1-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(g.1-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-1(g.1-0(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-0(g.1-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-0-1(g.1-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(g.1-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-1(g.1-1(u, v), y, z)) -> up.0(c.)
   down.0(f.0-1-0(0., 1., x)) -> up.0(f.0-0-0(x, x, x))
   down.0(f.0-1-1(0., 1., x)) -> up.0(f.1-1-1(x, x, x))
   down.0(g.0-0(x, y)) -> up.0(x)
   down.0(g.0-1(x, y)) -> up.0(x)
   down.0(g.1-0(x, y)) -> up.1(x)
   down.0(g.1-1(x, y)) -> up.1(x)
   down.0(g.0-0(x, y)) -> up.0(y)
   down.0(g.0-1(x, y)) -> up.1(y)
   down.0(g.1-0(x, y)) -> up.0(y)
   down.0(g.1-1(x, y)) -> up.1(y)
   down.0(f.0-0-0(f.0-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.0-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.0-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.0-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.0-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.0-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.0-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.0-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.0-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.0-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.0-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.0-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.0-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.0-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.0-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.0-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.1-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.1-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.1-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.1-0-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.1-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.1-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.1-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.1-0-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.1-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.1-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.1-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.1-1-0(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(f.1-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-1(f.1-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-0(f.1-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-1-1(f.1-1-1(x, y, z), u, v)) -> up.0(c.)
   down.0(f.0-0-0(0., u, v)) -> up.0(c.)
   down.0(f.0-0-1(0., u, v)) -> up.0(c.)
   down.0(f.0-1-0(0., u, v)) -> up.0(c.)
   down.0(f.0-1-1(0., u, v)) -> up.0(c.)
   top.0(up.0(x)) -> top.0(down.0(x))
   top.0(up.1(x)) -> top.0(down.1(x))
   down.0(f.0-0-0(c., y17, y18)) -> f_flat.0-0-0(down.0(c.), block.0(y17), block.0(y18))
   down.0(f.0-0-1(c., y17, y18)) -> f_flat.0-0-0(down.0(c.), block.0(y17), block.1(y18))
   down.0(f.0-1-0(c., y17, y18)) -> f_flat.0-0-0(down.0(c.), block.1(y17), block.0(y18))
   down.0(f.0-1-1(c., y17, y18)) -> f_flat.0-0-0(down.0(c.), block.1(y17), block.1(y18))
   down.0(f.0-0-0(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), down.0(y17), block.0(y18))
   down.0(f.0-0-1(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), down.0(y17), block.1(y18))
   down.0(f.0-1-0(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), down.1(y17), block.0(y18))
   down.0(f.0-1-1(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), down.1(y17), block.1(y18))
   down.0(f.0-0-0(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), block.0(y17), down.0(y18))
   down.0(f.0-0-1(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), block.0(y17), down.1(y18))
   down.0(f.0-1-0(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), block.1(y17), down.0(y18))
   down.0(f.0-1-1(c., y17, y18)) -> f_flat.0-0-0(block.0(c.), block.1(y17), down.1(y18))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.1(y22))
   down.0(f.0-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.0(fresh_constant.), block.0(y23), block.0(y24))
   down.0(f.0-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.0(fresh_constant.), block.0(y23), block.1(y24))
   down.0(f.0-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.0(fresh_constant.), block.1(y23), block.0(y24))
   down.0(f.0-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.0(fresh_constant.), block.1(y23), block.1(y24))
   down.0(f.0-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), down.0(y23), block.0(y24))
   down.0(f.0-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), down.0(y23), block.1(y24))
   down.0(f.0-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), down.1(y23), block.0(y24))
   down.0(f.0-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), down.1(y23), block.1(y24))
   down.0(f.0-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), block.0(y23), down.0(y24))
   down.0(f.0-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), block.0(y23), down.1(y24))
   down.0(f.0-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), block.1(y23), down.0(y24))
   down.0(f.0-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.0(fresh_constant.), block.1(y23), down.1(y24))
   f_flat.0-0-0(up.0(x_1), block.0(x_2), block.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.0(x_2), block.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.1(x_2), block.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.1(x_2), block.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   g_flat.0-0(up.0(x_1), block.0(x_2)) -> up.0(g.0-0(x_1, x_2))
   g_flat.0-0(up.0(x_1), block.1(x_2)) -> up.0(g.0-1(x_1, x_2))
   g_flat.0-0(up.1(x_1), block.0(x_2)) -> up.0(g.1-0(x_1, x_2))
   g_flat.0-0(up.1(x_1), block.1(x_2)) -> up.0(g.1-1(x_1, x_2))
   g_flat.0-0(block.0(x_1), up.0(x_2)) -> up.0(g.0-0(x_1, x_2))
   g_flat.0-0(block.0(x_1), up.1(x_2)) -> up.0(g.0-1(x_1, x_2))
   g_flat.0-0(block.1(x_1), up.0(x_2)) -> up.0(g.1-0(x_1, x_2))
   g_flat.0-0(block.1(x_1), up.1(x_2)) -> up.0(g.1-1(x_1, x_2))

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

(28) PisEmptyProof (SOUND)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(29)
TRUE

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

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

   TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), down(x0), block(x1)))
   TOP(up(f(c, x0, x1))) -> TOP(f_flat(block(c), block(x0), down(x1)))
   TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), down(x0), block(x1)))
   TOP(up(f(1, x0, x1))) -> TOP(f_flat(block(1), block(x0), down(x1)))
   TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), down(x0), block(x1)))
   TOP(up(f(fresh_constant, x0, x1))) -> TOP(f_flat(block(fresh_constant), block(x0), down(x1)))

The TRS R consists of the following rules:

   down(f(g(u, v), y, z)) -> up(c)
   down(f(0, 1, x)) -> up(f(x, x, x))
   down(g(x, y)) -> up(x)
   down(g(x, y)) -> up(y)
   down(f(f(x, y, z), u, v)) -> up(c)
   down(f(0, u, v)) -> up(c)
   top(up(x)) -> top(down(x))
   down(f(c, y17, y18)) -> f_flat(down(c), block(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), down(y17), block(y18))
   down(f(c, y17, y18)) -> f_flat(block(c), block(y17), down(y18))
   down(f(1, y21, y22)) -> f_flat(down(1), block(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), down(y21), block(y22))
   down(f(1, y21, y22)) -> f_flat(block(1), block(y21), down(y22))
   down(f(fresh_constant, y23, y24)) -> f_flat(down(fresh_constant), block(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), down(y23), block(y24))
   down(f(fresh_constant, y23, y24)) -> f_flat(block(fresh_constant), block(y23), down(y24))
   f_flat(up(x_1), block(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), up(x_2), block(x_3)) -> up(f(x_1, x_2, x_3))
   f_flat(block(x_1), block(x_2), up(x_3)) -> up(f(x_1, x_2, x_3))
   g_flat(up(x_1), block(x_2)) -> up(g(x_1, x_2))
   g_flat(block(x_1), up(x_2)) -> up(g(x_1, x_2))

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

(31) SemLabProof (SOUND)
We found the following model for the rules of the TRSs R and P.
Interpretation over the domain with elements from 0 to 1.
c: 1
block: 0
TOP: 0
g: 0
top: 0
down: 0
f: 0
0: 0
fresh_constant: 1
up: 0
1: 1
f_flat: 0
g_flat: 0
By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
----------------------------------------

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

   TOP.0(up.0(f.1-0-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.1-0-1(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.1-1-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), down.1(x0), block.0(x1)))
   TOP.0(up.0(f.1-1-1(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), down.1(x0), block.1(x1)))
   TOP.0(up.0(f.1-0-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.1-0-1(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), block.0(x0), down.1(x1)))
   TOP.0(up.0(f.1-1-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.1-1-1(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), block.1(x0), down.1(x1)))
   TOP.0(up.0(f.1-0-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.1-0-1(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.1-1-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.1(x0), block.0(x1)))
   TOP.0(up.0(f.1-1-1(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.1(x0), block.1(x1)))
   TOP.0(up.0(f.1-0-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.1-0-1(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.0(x0), down.1(x1)))
   TOP.0(up.0(f.1-1-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.1-1-1(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.1(x0), down.1(x1)))
   TOP.0(up.0(f.1-0-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.1-0-1(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.1-1-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), down.1(x0), block.0(x1)))
   TOP.0(up.0(f.1-1-1(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), down.1(x0), block.1(x1)))
   TOP.0(up.0(f.1-0-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.1-0-1(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), block.0(x0), down.1(x1)))
   TOP.0(up.0(f.1-1-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.1-1-1(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), block.1(x0), down.1(x1)))

The TRS R consists of the following rules:

   down.0(f.0-0-0(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-0(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-0(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-0(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(0., 1., x)) -> up.0(f.0-0-0(x, x, x))
   down.0(f.0-1-1(0., 1., x)) -> up.0(f.1-1-1(x, x, x))
   down.0(g.0-0(x, y)) -> up.0(x)
   down.0(g.0-1(x, y)) -> up.0(x)
   down.0(g.1-0(x, y)) -> up.1(x)
   down.0(g.1-1(x, y)) -> up.1(x)
   down.0(g.0-0(x, y)) -> up.0(y)
   down.0(g.0-1(x, y)) -> up.1(y)
   down.0(g.1-0(x, y)) -> up.0(y)
   down.0(g.1-1(x, y)) -> up.1(y)
   down.0(f.0-0-0(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(0., u, v)) -> up.1(c.)
   down.0(f.0-0-1(0., u, v)) -> up.1(c.)
   down.0(f.0-1-0(0., u, v)) -> up.1(c.)
   down.0(f.0-1-1(0., u, v)) -> up.1(c.)
   top.0(up.0(x)) -> top.0(down.0(x))
   top.0(up.1(x)) -> top.0(down.1(x))
   down.0(f.1-0-0(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.0(y17), block.0(y18))
   down.0(f.1-0-1(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.0(y17), block.1(y18))
   down.0(f.1-1-0(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.1(y17), block.0(y18))
   down.0(f.1-1-1(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.1(y17), block.1(y18))
   down.0(f.1-0-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.0(y17), block.0(y18))
   down.0(f.1-0-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.0(y17), block.1(y18))
   down.0(f.1-1-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.1(y17), block.0(y18))
   down.0(f.1-1-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.1(y17), block.1(y18))
   down.0(f.1-0-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.0(y17), down.0(y18))
   down.0(f.1-0-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.0(y17), down.1(y18))
   down.0(f.1-1-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.1(y17), down.0(y18))
   down.0(f.1-1-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.1(y17), down.1(y18))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.1(y22))
   down.0(f.1-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.0(y23), block.0(y24))
   down.0(f.1-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.0(y23), block.1(y24))
   down.0(f.1-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.1(y23), block.0(y24))
   down.0(f.1-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.1(y23), block.1(y24))
   down.0(f.1-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.0(y23), block.0(y24))
   down.0(f.1-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.0(y23), block.1(y24))
   down.0(f.1-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.1(y23), block.0(y24))
   down.0(f.1-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.1(y23), block.1(y24))
   down.0(f.1-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.0(y23), down.0(y24))
   down.0(f.1-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.0(y23), down.1(y24))
   down.0(f.1-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.1(y23), down.0(y24))
   down.0(f.1-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.1(y23), down.1(y24))
   f_flat.0-0-0(up.0(x_1), block.0(x_2), block.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.0(x_2), block.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.1(x_2), block.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.1(x_2), block.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   g_flat.0-0(up.0(x_1), block.0(x_2)) -> up.0(g.0-0(x_1, x_2))
   g_flat.0-0(up.0(x_1), block.1(x_2)) -> up.0(g.0-1(x_1, x_2))
   g_flat.0-0(up.1(x_1), block.0(x_2)) -> up.0(g.1-0(x_1, x_2))
   g_flat.0-0(up.1(x_1), block.1(x_2)) -> up.0(g.1-1(x_1, x_2))
   g_flat.0-0(block.0(x_1), up.0(x_2)) -> up.0(g.0-0(x_1, x_2))
   g_flat.0-0(block.0(x_1), up.1(x_2)) -> up.0(g.0-1(x_1, x_2))
   g_flat.0-0(block.1(x_1), up.0(x_2)) -> up.0(g.1-0(x_1, x_2))
   g_flat.0-0(block.1(x_1), up.1(x_2)) -> up.0(g.1-1(x_1, x_2))

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

(33) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 12 less nodes.
----------------------------------------

(34)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TOP.0(up.0(f.1-0-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.1-0-1(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.1-0-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.1-1-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.1-0-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.1-0-1(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.1-0-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.1-1-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.1-0-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.1-0-1(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.1-0-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.1-1-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), block.1(x0), down.0(x1)))

The TRS R consists of the following rules:

   down.0(f.0-0-0(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-0(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-0(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-0(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(0., 1., x)) -> up.0(f.0-0-0(x, x, x))
   down.0(f.0-1-1(0., 1., x)) -> up.0(f.1-1-1(x, x, x))
   down.0(g.0-0(x, y)) -> up.0(x)
   down.0(g.0-1(x, y)) -> up.0(x)
   down.0(g.1-0(x, y)) -> up.1(x)
   down.0(g.1-1(x, y)) -> up.1(x)
   down.0(g.0-0(x, y)) -> up.0(y)
   down.0(g.0-1(x, y)) -> up.1(y)
   down.0(g.1-0(x, y)) -> up.0(y)
   down.0(g.1-1(x, y)) -> up.1(y)
   down.0(f.0-0-0(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(0., u, v)) -> up.1(c.)
   down.0(f.0-0-1(0., u, v)) -> up.1(c.)
   down.0(f.0-1-0(0., u, v)) -> up.1(c.)
   down.0(f.0-1-1(0., u, v)) -> up.1(c.)
   top.0(up.0(x)) -> top.0(down.0(x))
   top.0(up.1(x)) -> top.0(down.1(x))
   down.0(f.1-0-0(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.0(y17), block.0(y18))
   down.0(f.1-0-1(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.0(y17), block.1(y18))
   down.0(f.1-1-0(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.1(y17), block.0(y18))
   down.0(f.1-1-1(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.1(y17), block.1(y18))
   down.0(f.1-0-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.0(y17), block.0(y18))
   down.0(f.1-0-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.0(y17), block.1(y18))
   down.0(f.1-1-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.1(y17), block.0(y18))
   down.0(f.1-1-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.1(y17), block.1(y18))
   down.0(f.1-0-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.0(y17), down.0(y18))
   down.0(f.1-0-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.0(y17), down.1(y18))
   down.0(f.1-1-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.1(y17), down.0(y18))
   down.0(f.1-1-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.1(y17), down.1(y18))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.1(y22))
   down.0(f.1-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.0(y23), block.0(y24))
   down.0(f.1-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.0(y23), block.1(y24))
   down.0(f.1-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.1(y23), block.0(y24))
   down.0(f.1-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.1(y23), block.1(y24))
   down.0(f.1-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.0(y23), block.0(y24))
   down.0(f.1-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.0(y23), block.1(y24))
   down.0(f.1-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.1(y23), block.0(y24))
   down.0(f.1-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.1(y23), block.1(y24))
   down.0(f.1-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.0(y23), down.0(y24))
   down.0(f.1-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.0(y23), down.1(y24))
   down.0(f.1-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.1(y23), down.0(y24))
   down.0(f.1-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.1(y23), down.1(y24))
   f_flat.0-0-0(up.0(x_1), block.0(x_2), block.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.0(x_2), block.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.1(x_2), block.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.1(x_2), block.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   g_flat.0-0(up.0(x_1), block.0(x_2)) -> up.0(g.0-0(x_1, x_2))
   g_flat.0-0(up.0(x_1), block.1(x_2)) -> up.0(g.0-1(x_1, x_2))
   g_flat.0-0(up.1(x_1), block.0(x_2)) -> up.0(g.1-0(x_1, x_2))
   g_flat.0-0(up.1(x_1), block.1(x_2)) -> up.0(g.1-1(x_1, x_2))
   g_flat.0-0(block.0(x_1), up.0(x_2)) -> up.0(g.0-0(x_1, x_2))
   g_flat.0-0(block.0(x_1), up.1(x_2)) -> up.0(g.0-1(x_1, x_2))
   g_flat.0-0(block.1(x_1), up.0(x_2)) -> up.0(g.1-0(x_1, x_2))
   g_flat.0-0(block.1(x_1), up.1(x_2)) -> up.0(g.1-1(x_1, x_2))

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

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


The following pairs can be oriented strictly and are deleted.

   TOP.0(up.0(f.1-0-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.1-0-1(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.1-0-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.1-1-0(c., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(c.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.1-0-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.1-0-1(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.1-0-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.1-1-0(1., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(1.), block.1(x0), down.0(x1)))
   TOP.0(up.0(f.1-0-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), down.0(x0), block.0(x1)))
   TOP.0(up.0(f.1-0-1(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), down.0(x0), block.1(x1)))
   TOP.0(up.0(f.1-0-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), block.0(x0), down.0(x1)))
   TOP.0(up.0(f.1-1-0(fresh_constant., x0, x1))) -> TOP.0(f_flat.0-0-0(block.1(fresh_constant.), block.1(x0), down.0(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0.) = 1
   POL(1.) = 0
   POL(TOP.0(x_1)) = x_1
   POL(block.0(x_1)) = x_1
   POL(block.1(x_1)) = 0
   POL(c.) = 0
   POL(down.0(x_1)) = x_1
   POL(down.1(x_1)) = 0
   POL(f.0-0-0(x_1, x_2, x_3)) = 1 + x_2
   POL(f.0-0-1(x_1, x_2, x_3)) = 1
   POL(f.0-1-0(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(f.0-1-1(x_1, x_2, x_3)) = 1 + x_2 + x_3
   POL(f.1-0-0(x_1, x_2, x_3)) = x_2 + x_3
   POL(f.1-0-1(x_1, x_2, x_3)) = x_2
   POL(f.1-1-0(x_1, x_2, x_3)) = x_3
   POL(f.1-1-1(x_1, x_2, x_3)) = 0
   POL(f_flat.0-0-0(x_1, x_2, x_3)) = x_2 + x_3
   POL(fresh_constant.) = 0
   POL(g.0-0(x_1, x_2)) = 1 + x_1 + x_2
   POL(g.0-1(x_1, x_2)) = 1 + x_1 + x_2
   POL(g.1-0(x_1, x_2)) = 1 + x_1 + x_2
   POL(g.1-1(x_1, x_2)) = 1 + x_1 + x_2
   POL(up.0(x_1)) = 1 + x_1
   POL(up.1(x_1)) = 1

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

   down.0(f.0-0-0(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-0(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-0(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-0(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(0., 1., x)) -> up.0(f.0-0-0(x, x, x))
   down.0(f.0-1-1(0., 1., x)) -> up.0(f.1-1-1(x, x, x))
   down.0(g.0-0(x, y)) -> up.0(x)
   down.0(g.0-1(x, y)) -> up.0(x)
   down.0(g.1-0(x, y)) -> up.1(x)
   down.0(g.1-1(x, y)) -> up.1(x)
   down.0(g.0-0(x, y)) -> up.0(y)
   down.0(g.0-1(x, y)) -> up.1(y)
   down.0(g.1-0(x, y)) -> up.0(y)
   down.0(g.1-1(x, y)) -> up.1(y)
   down.0(f.0-0-0(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(0., u, v)) -> up.1(c.)
   down.0(f.0-0-1(0., u, v)) -> up.1(c.)
   down.0(f.0-1-0(0., u, v)) -> up.1(c.)
   down.0(f.0-1-1(0., u, v)) -> up.1(c.)
   down.0(f.1-0-0(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.0(y17), block.0(y18))
   down.0(f.1-0-1(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.0(y17), block.1(y18))
   down.0(f.1-1-0(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.1(y17), block.0(y18))
   down.0(f.1-1-1(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.1(y17), block.1(y18))
   down.0(f.1-0-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.0(y17), block.0(y18))
   down.0(f.1-0-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.0(y17), block.1(y18))
   down.0(f.1-1-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.1(y17), block.0(y18))
   down.0(f.1-1-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.1(y17), block.1(y18))
   down.0(f.1-0-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.0(y17), down.0(y18))
   down.0(f.1-0-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.0(y17), down.1(y18))
   down.0(f.1-1-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.1(y17), down.0(y18))
   down.0(f.1-1-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.1(y17), down.1(y18))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.1(y22))
   down.0(f.1-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.0(y23), block.0(y24))
   down.0(f.1-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.0(y23), block.1(y24))
   down.0(f.1-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.1(y23), block.0(y24))
   down.0(f.1-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.1(y23), block.1(y24))
   down.0(f.1-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.0(y23), block.0(y24))
   down.0(f.1-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.0(y23), block.1(y24))
   down.0(f.1-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.1(y23), block.0(y24))
   down.0(f.1-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.1(y23), block.1(y24))
   down.0(f.1-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.0(y23), down.0(y24))
   down.0(f.1-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.0(y23), down.1(y24))
   down.0(f.1-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.1(y23), down.0(y24))
   down.0(f.1-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.1(y23), down.1(y24))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))


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

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

   down.0(f.0-0-0(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.0-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-0(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.0-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-0(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.1-0(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-0(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-0-1(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-1(g.1-1(u, v), y, z)) -> up.1(c.)
   down.0(f.0-1-0(0., 1., x)) -> up.0(f.0-0-0(x, x, x))
   down.0(f.0-1-1(0., 1., x)) -> up.0(f.1-1-1(x, x, x))
   down.0(g.0-0(x, y)) -> up.0(x)
   down.0(g.0-1(x, y)) -> up.0(x)
   down.0(g.1-0(x, y)) -> up.1(x)
   down.0(g.1-1(x, y)) -> up.1(x)
   down.0(g.0-0(x, y)) -> up.0(y)
   down.0(g.0-1(x, y)) -> up.1(y)
   down.0(g.1-0(x, y)) -> up.0(y)
   down.0(g.1-1(x, y)) -> up.1(y)
   down.0(f.0-0-0(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.0-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-0-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-0-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-1-0(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-1(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-0(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-1-1(f.1-1-1(x, y, z), u, v)) -> up.1(c.)
   down.0(f.0-0-0(0., u, v)) -> up.1(c.)
   down.0(f.0-0-1(0., u, v)) -> up.1(c.)
   down.0(f.0-1-0(0., u, v)) -> up.1(c.)
   down.0(f.0-1-1(0., u, v)) -> up.1(c.)
   top.0(up.0(x)) -> top.0(down.0(x))
   top.0(up.1(x)) -> top.0(down.1(x))
   down.0(f.1-0-0(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.0(y17), block.0(y18))
   down.0(f.1-0-1(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.0(y17), block.1(y18))
   down.0(f.1-1-0(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.1(y17), block.0(y18))
   down.0(f.1-1-1(c., y17, y18)) -> f_flat.0-0-0(down.1(c.), block.1(y17), block.1(y18))
   down.0(f.1-0-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.0(y17), block.0(y18))
   down.0(f.1-0-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.0(y17), block.1(y18))
   down.0(f.1-1-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.1(y17), block.0(y18))
   down.0(f.1-1-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), down.1(y17), block.1(y18))
   down.0(f.1-0-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.0(y17), down.0(y18))
   down.0(f.1-0-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.0(y17), down.1(y18))
   down.0(f.1-1-0(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.1(y17), down.0(y18))
   down.0(f.1-1-1(c., y17, y18)) -> f_flat.0-0-0(block.1(c.), block.1(y17), down.1(y18))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(down.1(1.), block.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.0(y21), block.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), down.1(y21), block.1(y22))
   down.0(f.1-0-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.0(y22))
   down.0(f.1-0-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.0(y21), down.1(y22))
   down.0(f.1-1-0(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.0(y22))
   down.0(f.1-1-1(1., y21, y22)) -> f_flat.0-0-0(block.1(1.), block.1(y21), down.1(y22))
   down.0(f.1-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.0(y23), block.0(y24))
   down.0(f.1-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.0(y23), block.1(y24))
   down.0(f.1-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.1(y23), block.0(y24))
   down.0(f.1-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(down.1(fresh_constant.), block.1(y23), block.1(y24))
   down.0(f.1-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.0(y23), block.0(y24))
   down.0(f.1-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.0(y23), block.1(y24))
   down.0(f.1-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.1(y23), block.0(y24))
   down.0(f.1-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), down.1(y23), block.1(y24))
   down.0(f.1-0-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.0(y23), down.0(y24))
   down.0(f.1-0-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.0(y23), down.1(y24))
   down.0(f.1-1-0(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.1(y23), down.0(y24))
   down.0(f.1-1-1(fresh_constant., y23, y24)) -> f_flat.0-0-0(block.1(fresh_constant.), block.1(y23), down.1(y24))
   f_flat.0-0-0(up.0(x_1), block.0(x_2), block.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.0(x_2), block.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.1(x_2), block.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.0(x_1), block.1(x_2), block.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(up.1(x_1), block.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.0(x_2), block.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), up.1(x_2), block.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.0-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.0-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.0-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.0(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.0-1-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.0(x_3)) -> up.0(f.1-0-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.0(x_2), up.1(x_3)) -> up.0(f.1-0-1(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.0(x_3)) -> up.0(f.1-1-0(x_1, x_2, x_3))
   f_flat.0-0-0(block.1(x_1), block.1(x_2), up.1(x_3)) -> up.0(f.1-1-1(x_1, x_2, x_3))
   g_flat.0-0(up.0(x_1), block.0(x_2)) -> up.0(g.0-0(x_1, x_2))
   g_flat.0-0(up.0(x_1), block.1(x_2)) -> up.0(g.0-1(x_1, x_2))
   g_flat.0-0(up.1(x_1), block.0(x_2)) -> up.0(g.1-0(x_1, x_2))
   g_flat.0-0(up.1(x_1), block.1(x_2)) -> up.0(g.1-1(x_1, x_2))
   g_flat.0-0(block.0(x_1), up.0(x_2)) -> up.0(g.0-0(x_1, x_2))
   g_flat.0-0(block.0(x_1), up.1(x_2)) -> up.0(g.0-1(x_1, x_2))
   g_flat.0-0(block.1(x_1), up.0(x_2)) -> up.0(g.1-0(x_1, x_2))
   g_flat.0-0(block.1(x_1), up.1(x_2)) -> up.0(g.1-1(x_1, x_2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(37) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(38)
YES