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


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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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 40 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 26 ms]
(6) QTRS
(7) AAECC Innermost [EQUIVALENT, 0 ms]
(8) QTRS
(9) DependencyPairsProof [EQUIVALENT, 0 ms]
(10) QDP
(11) DependencyGraphProof [EQUIVALENT, 0 ms]
(12) QDP
(13) UsableRulesProof [EQUIVALENT, 0 ms]
(14) QDP
(15) QReductionProof [EQUIVALENT, 0 ms]
(16) QDP
(17) RFCMatchBoundsDPProof [EQUIVALENT, 0 ms]
(18) YES


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

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

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



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(a))) -> up(a)
   down(f(f(x))) -> up(b)
   down(g(x)) -> up(f(g(x)))
   top(up(x)) -> top(down(x))
   down(f(a)) -> f_flat(down(a))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(f(g(f(y6)))) -> f_flat(down(g(f(y6))))
   down(f(g(g(y7)))) -> f_flat(down(g(g(y7))))
   down(f(g(b))) -> f_flat(down(g(b)))
   down(f(g(fresh_constant))) -> f_flat(down(g(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(a) = 0
   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) = 0
   POL(g(x_1)) = x_1
   POL(g_flat(x_1)) = 1 + 2*x_1
   POL(top(x_1)) = 2*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:

   g_flat(up(x_1)) -> up(g(x_1))




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

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

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

Q is empty.

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

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

   POL(a) = 1
   POL(b) = 0
   POL(down(x_1)) = 2*x_1
   POL(f(x_1)) = x_1
   POL(f_flat(x_1)) = x_1
   POL(fresh_constant) = 0
   POL(g(x_1)) = 2*x_1
   POL(top(x_1)) = 2*x_1
   POL(up(x_1)) = 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(a))) -> up(a)




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

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

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

Q is empty.

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

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

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

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

(8)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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


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

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

(10)
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(a)) -> F_FLAT(down(a))
   DOWN(f(a)) -> DOWN(a)
   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(f(g(f(y6)))) -> F_FLAT(down(g(f(y6))))
   DOWN(f(g(f(y6)))) -> DOWN(g(f(y6)))
   DOWN(f(g(g(y7)))) -> F_FLAT(down(g(g(y7))))
   DOWN(f(g(g(y7)))) -> DOWN(g(g(y7)))
   DOWN(f(g(b))) -> F_FLAT(down(g(b)))
   DOWN(f(g(b))) -> DOWN(g(b))
   DOWN(f(g(fresh_constant))) -> F_FLAT(down(g(fresh_constant)))
   DOWN(f(g(fresh_constant))) -> DOWN(g(fresh_constant))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
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(13) 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.
----------------------------------------

(14)
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(f(x))) -> up(b)
   down(g(x)) -> up(f(g(x)))
   down(f(a)) -> f_flat(down(a))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(f(g(f(y6)))) -> f_flat(down(g(f(y6))))
   down(f(g(g(y7)))) -> f_flat(down(g(g(y7))))
   down(f(g(b))) -> f_flat(down(g(b)))
   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
   f_flat(up(x_1)) -> up(f(x_1))

The set Q consists of the following terms:

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

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

(15) 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))


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

(16)
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(f(x))) -> up(b)
   down(g(x)) -> up(f(g(x)))
   down(f(a)) -> f_flat(down(a))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(f(g(f(y6)))) -> f_flat(down(g(f(y6))))
   down(f(g(g(y7)))) -> f_flat(down(g(g(y7))))
   down(f(g(b))) -> f_flat(down(g(b)))
   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
   f_flat(up(x_1)) -> up(f(x_1))

The set Q consists of the following terms:

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

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

(17) 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(f(f(x))) -> up(b)
   down(g(x)) -> up(f(g(x)))
   down(f(a)) -> f_flat(down(a))
   down(f(b)) -> f_flat(down(b))
   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
   down(f(g(f(y6)))) -> f_flat(down(g(f(y6))))
   down(f(g(g(y7)))) -> f_flat(down(g(g(y7))))
   down(f(g(b))) -> f_flat(down(g(b)))
   down(f(g(fresh_constant))) -> f_flat(down(g(fresh_constant)))
   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:
355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372

Node 355 is start node and node 356 is final node.

Those nodes are connected through the following edges:

* 355 to 357 labelled TOP_1(0)* 355 to 366 labelled TOP_1(1)* 355 to 370 labelled TOP_1(2)* 355 to 372 labelled TOP_1(3)* 356 to 356 labelled #_1(0)* 357 to 356 labelled down_1(0)* 357 to 358 labelled up_1(1)* 357 to 359 labelled up_1(1)* 357 to 361 labelled f_flat_1(1)* 357 to 363 labelled f_flat_1(1)* 357 to 369 labelled up_1(2)* 358 to 356 labelled b(1)* 359 to 360 labelled f_1(1)* 360 to 356 labelled g_1(1)* 361 to 362 labelled down_1(1)* 362 to 356 labelled a(1), b(1), fresh_constant(1)* 363 to 364 labelled down_1(1)* 363 to 367 labelled up_1(2)* 364 to 365 labelled g_1(1)* 365 to 356 labelled f_1(1), g_1(1), b(1), fresh_constant(1)* 366 to 358 labelled down_1(1)* 366 to 359 labelled down_1(1)* 366 to 363 labelled f_flat_1(1)* 366 to 369 labelled down_1(1), up_1(2)* 367 to 368 labelled f_1(2)* 368 to 365 labelled g_1(2)* 369 to 367 labelled f_1(2)* 369 to 368 labelled b(2)* 370 to 369 labelled down_1(2)* 370 to 371 labelled up_1(3)* 371 to 368 labelled b(3)* 372 to 371 labelled down_1(3)


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(18)
YES