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


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


YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) QTRSRRRProof [EQUIVALENT, 52 ms]
(4) QTRS
(5) AAECC Innermost [EQUIVALENT, 0 ms]
(6) QTRS
(7) DependencyPairsProof [EQUIVALENT, 0 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) AND
    (11) QDP
        (12) UsableRulesProof [EQUIVALENT, 0 ms]
        (13) QDP
        (14) QReductionProof [EQUIVALENT, 0 ms]
        (15) QDP
        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (17) YES
    (18) QDP
        (19) UsableRulesProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) QReductionProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (24) YES
    (25) QDP
        (26) UsableRulesProof [EQUIVALENT, 0 ms]
        (27) QDP
        (28) QReductionProof [EQUIVALENT, 0 ms]
        (29) QDP
        (30) RFCMatchBoundsDPProof [EQUIVALENT, 0 ms]
        (31) YES


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

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

   f(g(x)) -> x
   g(b) -> f(g(b))



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(x))) -> up(x)
   down(g(b)) -> up(f(g(b)))
   top(up(x)) -> top(down(x))
   down(f(f(y3))) -> f_flat(down(f(y3)))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(g(f(y6))) -> g_flat(down(f(y6)))
   down(g(g(y7))) -> g_flat(down(g(y7)))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   f_flat(up(x_1)) -> up(f(x_1))
   g_flat(up(x_1)) -> up(g(x_1))

Q is empty.

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

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(b) = 0
   POL(down(x_1)) = 2 + 2*x_1
   POL(f(x_1)) = x_1
   POL(f_flat(x_1)) = x_1
   POL(fresh_constant) = 1
   POL(g(x_1)) = 1 + 2*x_1
   POL(g_flat(x_1)) = 2*x_1
   POL(top(x_1)) = x_1
   POL(up(x_1)) = 2 + 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   down(f(g(x))) -> up(x)




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

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   down(g(b)) -> up(f(g(b)))
   top(up(x)) -> top(down(x))
   down(f(f(y3))) -> f_flat(down(f(y3)))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(g(f(y6))) -> g_flat(down(f(y6)))
   down(g(g(y7))) -> g_flat(down(g(y7)))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   f_flat(up(x_1)) -> up(f(x_1))
   g_flat(up(x_1)) -> up(g(x_1))

Q is empty.

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

(5) AAECC Innermost (EQUIVALENT)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 
   down(f(f(y3))) -> f_flat(down(f(y3)))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(g(f(y6))) -> g_flat(down(f(y6)))
   down(g(g(y7))) -> g_flat(down(g(y7)))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   f_flat(up(x_1)) -> up(f(x_1))
   g_flat(up(x_1)) -> up(g(x_1))
   down(g(b)) -> up(f(g(b)))

The TRS R 2 is 
   top(up(x)) -> top(down(x))

The signature Sigma is {top_1}
----------------------------------------

(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   down(g(b)) -> up(f(g(b)))
   top(up(x)) -> top(down(x))
   down(f(f(y3))) -> f_flat(down(f(y3)))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(g(f(y6))) -> g_flat(down(f(y6)))
   down(g(g(y7))) -> g_flat(down(g(y7)))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   f_flat(up(x_1)) -> up(f(x_1))
   g_flat(up(x_1)) -> up(g(x_1))

The set Q consists of the following terms:

   down(g(b))
   top(up(x0))
   down(f(f(x0)))
   down(f(b))
   down(f(fresh_constant))
   down(g(f(x0)))
   down(g(g(x0)))
   down(g(fresh_constant))
   f_flat(up(x0))
   g_flat(up(x0))


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

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

(8)
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(f(y3))) -> F_FLAT(down(f(y3)))
   DOWN(f(f(y3))) -> DOWN(f(y3))
   DOWN(f(b)) -> F_FLAT(down(b))
   DOWN(f(b)) -> DOWN(b)
   DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant))
   DOWN(f(fresh_constant)) -> DOWN(fresh_constant)
   DOWN(g(f(y6))) -> G_FLAT(down(f(y6)))
   DOWN(g(f(y6))) -> DOWN(f(y6))
   DOWN(g(g(y7))) -> G_FLAT(down(g(y7)))
   DOWN(g(g(y7))) -> DOWN(g(y7))
   DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant))
   DOWN(g(fresh_constant)) -> DOWN(fresh_constant)

The TRS R consists of the following rules:

   down(g(b)) -> up(f(g(b)))
   top(up(x)) -> top(down(x))
   down(f(f(y3))) -> f_flat(down(f(y3)))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(g(f(y6))) -> g_flat(down(f(y6)))
   down(g(g(y7))) -> g_flat(down(g(y7)))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   f_flat(up(x_1)) -> up(f(x_1))
   g_flat(up(x_1)) -> up(g(x_1))

The set Q consists of the following terms:

   down(g(b))
   top(up(x0))
   down(f(f(x0)))
   down(f(b))
   down(f(fresh_constant))
   down(g(f(x0)))
   down(g(g(x0)))
   down(g(fresh_constant))
   f_flat(up(x0))
   g_flat(up(x0))

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

(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 11 less nodes.
----------------------------------------

(10)
Complex Obligation (AND)

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

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

   DOWN(f(f(y3))) -> DOWN(f(y3))

The TRS R consists of the following rules:

   down(g(b)) -> up(f(g(b)))
   top(up(x)) -> top(down(x))
   down(f(f(y3))) -> f_flat(down(f(y3)))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(g(f(y6))) -> g_flat(down(f(y6)))
   down(g(g(y7))) -> g_flat(down(g(y7)))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   f_flat(up(x_1)) -> up(f(x_1))
   g_flat(up(x_1)) -> up(g(x_1))

The set Q consists of the following terms:

   down(g(b))
   top(up(x0))
   down(f(f(x0)))
   down(f(b))
   down(f(fresh_constant))
   down(g(f(x0)))
   down(g(g(x0)))
   down(g(fresh_constant))
   f_flat(up(x0))
   g_flat(up(x0))

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

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

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

   DOWN(f(f(y3))) -> DOWN(f(y3))

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

   down(g(b))
   top(up(x0))
   down(f(f(x0)))
   down(f(b))
   down(f(fresh_constant))
   down(g(f(x0)))
   down(g(g(x0)))
   down(g(fresh_constant))
   f_flat(up(x0))
   g_flat(up(x0))

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

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

   down(g(b))
   top(up(x0))
   down(f(f(x0)))
   down(f(b))
   down(f(fresh_constant))
   down(g(f(x0)))
   down(g(g(x0)))
   down(g(fresh_constant))
   f_flat(up(x0))
   g_flat(up(x0))


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

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   DOWN(f(f(y3))) -> DOWN(f(y3))

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

(16) 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(f(y3))) -> DOWN(f(y3))
The graph contains the following edges 1 > 1


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

(17)
YES

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

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

   DOWN(g(g(y7))) -> DOWN(g(y7))

The TRS R consists of the following rules:

   down(g(b)) -> up(f(g(b)))
   top(up(x)) -> top(down(x))
   down(f(f(y3))) -> f_flat(down(f(y3)))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(g(f(y6))) -> g_flat(down(f(y6)))
   down(g(g(y7))) -> g_flat(down(g(y7)))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   f_flat(up(x_1)) -> up(f(x_1))
   g_flat(up(x_1)) -> up(g(x_1))

The set Q consists of the following terms:

   down(g(b))
   top(up(x0))
   down(f(f(x0)))
   down(f(b))
   down(f(fresh_constant))
   down(g(f(x0)))
   down(g(g(x0)))
   down(g(fresh_constant))
   f_flat(up(x0))
   g_flat(up(x0))

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

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

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

   DOWN(g(g(y7))) -> DOWN(g(y7))

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

   down(g(b))
   top(up(x0))
   down(f(f(x0)))
   down(f(b))
   down(f(fresh_constant))
   down(g(f(x0)))
   down(g(g(x0)))
   down(g(fresh_constant))
   f_flat(up(x0))
   g_flat(up(x0))

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

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

   down(g(b))
   top(up(x0))
   down(f(f(x0)))
   down(f(b))
   down(f(fresh_constant))
   down(g(f(x0)))
   down(g(g(x0)))
   down(g(fresh_constant))
   f_flat(up(x0))
   g_flat(up(x0))


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

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

   DOWN(g(g(y7))) -> DOWN(g(y7))

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

(23) 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(g(g(y7))) -> DOWN(g(y7))
The graph contains the following edges 1 > 1


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

(24)
YES

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

(25)
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(g(b)) -> up(f(g(b)))
   top(up(x)) -> top(down(x))
   down(f(f(y3))) -> f_flat(down(f(y3)))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(g(f(y6))) -> g_flat(down(f(y6)))
   down(g(g(y7))) -> g_flat(down(g(y7)))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   f_flat(up(x_1)) -> up(f(x_1))
   g_flat(up(x_1)) -> up(g(x_1))

The set Q consists of the following terms:

   down(g(b))
   top(up(x0))
   down(f(f(x0)))
   down(f(b))
   down(f(fresh_constant))
   down(g(f(x0)))
   down(g(g(x0)))
   down(g(fresh_constant))
   f_flat(up(x0))
   g_flat(up(x0))

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

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

(27)
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(g(b)) -> up(f(g(b)))
   down(f(f(y3))) -> f_flat(down(f(y3)))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(g(f(y6))) -> g_flat(down(f(y6)))
   down(g(g(y7))) -> g_flat(down(g(y7)))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   g_flat(up(x_1)) -> up(g(x_1))
   f_flat(up(x_1)) -> up(f(x_1))

The set Q consists of the following terms:

   down(g(b))
   top(up(x0))
   down(f(f(x0)))
   down(f(b))
   down(f(fresh_constant))
   down(g(f(x0)))
   down(g(g(x0)))
   down(g(fresh_constant))
   f_flat(up(x0))
   g_flat(up(x0))

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

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

   top(up(x0))


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

(29)
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(g(b)) -> up(f(g(b)))
   down(f(f(y3))) -> f_flat(down(f(y3)))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(g(f(y6))) -> g_flat(down(f(y6)))
   down(g(g(y7))) -> g_flat(down(g(y7)))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   g_flat(up(x_1)) -> up(g(x_1))
   f_flat(up(x_1)) -> up(f(x_1))

The set Q consists of the following terms:

   down(g(b))
   down(f(f(x0)))
   down(f(b))
   down(f(fresh_constant))
   down(g(f(x0)))
   down(g(g(x0)))
   down(g(fresh_constant))
   f_flat(up(x0))
   g_flat(up(x0))

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

(30) RFCMatchBoundsDPProof (EQUIVALENT)
Finiteness of the DP problem can be shown by a matchbound of 3. 
As the DP problem is minimal we only have to initialize the certificate graph by the rules of P:

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

To find matches we regarded all rules of R and P:

   down(g(b)) -> up(f(g(b)))
   down(f(f(y3))) -> f_flat(down(f(y3)))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(g(f(y6))) -> g_flat(down(f(y6)))
   down(g(g(y7))) -> g_flat(down(g(y7)))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   g_flat(up(x_1)) -> up(g(x_1))
   f_flat(up(x_1)) -> up(f(x_1))
   TOP(up(x)) -> TOP(down(x))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
445, 446, 447, 448, 449, 450, 451, 452, 454, 455, 456, 457, 459, 460

Node 445 is start node and node 446 is final node.

Those nodes are connected through the following edges:

* 445 to 447 labelled TOP_1(0)* 445 to 454 labelled TOP_1(1)* 446 to 446 labelled #_1(0)* 447 to 446 labelled down_1(0)* 447 to 448 labelled up_1(1)* 447 to 451 labelled f_flat_1(1), g_flat_1(1)* 447 to 455 labelled up_1(2)* 448 to 449 labelled f_1(1)* 449 to 450 labelled g_1(1)* 450 to 446 labelled b(1)* 451 to 452 labelled down_1(1)* 451 to 451 labelled f_flat_1(1), g_flat_1(1)* 451 to 448 labelled up_1(1)* 451 to 455 labelled up_1(2)* 452 to 446 labelled f_1(1), b(1), fresh_constant(1), g_1(1)* 454 to 448 labelled down_1(1)* 454 to 455 labelled down_1(1)* 454 to 456 labelled f_flat_1(2), g_flat_1(2)* 455 to 448 labelled f_1(2), g_1(2)* 455 to 455 labelled f_1(2), g_1(2)* 456 to 457 labelled down_1(2)* 456 to 456 labelled f_flat_1(2), g_flat_1(2)* 456 to 459 labelled f_flat_1(3), g_flat_1(3)* 457 to 449 labelled f_1(2)* 457 to 448 labelled f_1(2), g_1(2)* 457 to 455 labelled f_1(2), g_1(2)* 459 to 460 labelled down_1(3)* 459 to 456 labelled f_flat_1(2), g_flat_1(2)* 459 to 459 labelled f_flat_1(3), g_flat_1(3)* 460 to 448 labelled f_1(3), g_1(3)* 460 to 455 labelled f_1(3), g_1(3)


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

(31)
YES