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


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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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) QTRSRRRProof [EQUIVALENT, 45 ms]
(4) QTRS
(5) AAECC Innermost [EQUIVALENT, 0 ms]
(6) QTRS
(7) DependencyPairsProof [EQUIVALENT, 0 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) QDP
(11) UsableRulesProof [EQUIVALENT, 0 ms]
(12) QDP
(13) QReductionProof [EQUIVALENT, 0 ms]
(14) QDP
(15) RFCMatchBoundsDPProof [EQUIVALENT, 0 ms]
(16) YES


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

   f(x) -> g(f(x))
   g(g(x)) -> c



Outermost Strategy.

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(1) Raffelsieper-Zantema-Transformation (SOUND)
We applied the Raffelsieper-Zantema transformation  to transform the outermost TRS to a standard TRS.
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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   down(f(x)) -> up(g(f(x)))
   down(g(g(x))) -> up(c)
   top(up(x)) -> top(down(x))
   down(g(f(y3))) -> g_flat(down(f(y3)))
   down(g(c)) -> g_flat(down(c))
   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.

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

   POL(c) = 0
   POL(down(x_1)) = 2*x_1
   POL(f(x_1)) = x_1
   POL(f_flat(x_1)) = 2 + x_1
   POL(fresh_constant) = 0
   POL(g(x_1)) = x_1
   POL(g_flat(x_1)) = 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:

   f_flat(up(x_1)) -> up(f(x_1))




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

   down(f(x)) -> up(g(f(x)))
   down(g(g(x))) -> up(c)
   top(up(x)) -> top(down(x))
   down(g(f(y3))) -> g_flat(down(f(y3)))
   down(g(c)) -> g_flat(down(c))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   g_flat(up(x_1)) -> up(g(x_1))

Q is empty.

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

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(f(x)) -> up(g(f(x)))
   down(g(g(x))) -> up(c)
   top(up(x)) -> top(down(x))
   down(g(f(y3))) -> g_flat(down(f(y3)))
   down(g(c)) -> g_flat(down(c))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   g_flat(up(x_1)) -> up(g(x_1))

The set Q consists of the following terms:

   down(f(x0))
   down(g(g(x0)))
   top(up(x0))
   down(g(f(x0)))
   down(g(c))
   down(g(fresh_constant))
   g_flat(up(x0))


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(7) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(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(g(f(y3))) -> G_FLAT(down(f(y3)))
   DOWN(g(f(y3))) -> DOWN(f(y3))
   DOWN(g(c)) -> G_FLAT(down(c))
   DOWN(g(c)) -> DOWN(c)
   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(f(x)) -> up(g(f(x)))
   down(g(g(x))) -> up(c)
   top(up(x)) -> top(down(x))
   down(g(f(y3))) -> g_flat(down(f(y3)))
   down(g(c)) -> g_flat(down(c))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   g_flat(up(x_1)) -> up(g(x_1))

The set Q consists of the following terms:

   down(f(x0))
   down(g(g(x0)))
   top(up(x0))
   down(g(f(x0)))
   down(g(c))
   down(g(fresh_constant))
   g_flat(up(x0))

We have to consider all minimal (P,Q,R)-chains.
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(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.
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(10)
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(x)) -> up(g(f(x)))
   down(g(g(x))) -> up(c)
   top(up(x)) -> top(down(x))
   down(g(f(y3))) -> g_flat(down(f(y3)))
   down(g(c)) -> g_flat(down(c))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   g_flat(up(x_1)) -> up(g(x_1))

The set Q consists of the following terms:

   down(f(x0))
   down(g(g(x0)))
   top(up(x0))
   down(g(f(x0)))
   down(g(c))
   down(g(fresh_constant))
   g_flat(up(x0))

We have to consider all minimal (P,Q,R)-chains.
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(11) 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.
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(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(x)) -> up(g(f(x)))
   down(g(g(x))) -> up(c)
   down(g(f(y3))) -> g_flat(down(f(y3)))
   down(g(c)) -> g_flat(down(c))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   g_flat(up(x_1)) -> up(g(x_1))

The set Q consists of the following terms:

   down(f(x0))
   down(g(g(x0)))
   top(up(x0))
   down(g(f(x0)))
   down(g(c))
   down(g(fresh_constant))
   g_flat(up(x0))

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


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(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(x)) -> up(g(f(x)))
   down(g(g(x))) -> up(c)
   down(g(f(y3))) -> g_flat(down(f(y3)))
   down(g(c)) -> g_flat(down(c))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   g_flat(up(x_1)) -> up(g(x_1))

The set Q consists of the following terms:

   down(f(x0))
   down(g(g(x0)))
   down(g(f(x0)))
   down(g(c))
   down(g(fresh_constant))
   g_flat(up(x0))

We have to consider all minimal (P,Q,R)-chains.
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(15) 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(x)) -> up(g(f(x)))
   down(g(g(x))) -> up(c)
   down(g(f(y3))) -> g_flat(down(f(y3)))
   down(g(c)) -> g_flat(down(c))
   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
   g_flat(up(x_1)) -> up(g(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:
293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 309, 310, 311, 312, 313

Node 293 is start node and node 294 is final node.

Those nodes are connected through the following edges:

* 293 to 295 labelled TOP_1(0)* 293 to 300 labelled TOP_1(1)* 293 to 311 labelled TOP_1(2)* 293 to 313 labelled TOP_1(3)* 294 to 294 labelled #_1(0)* 295 to 294 labelled down_1(0)* 295 to 296 labelled up_1(1)* 295 to 298 labelled g_flat_1(1)* 295 to 303 labelled up_1(2)* 296 to 297 labelled g_1(1)* 296 to 294 labelled c(1)* 297 to 294 labelled f_1(1)* 298 to 299 labelled down_1(1)* 298 to 301 labelled up_1(2)* 299 to 294 labelled f_1(1), c(1), fresh_constant(1)* 300 to 296 labelled down_1(1)* 300 to 304 labelled g_flat_1(2)* 300 to 303 labelled down_1(1)* 300 to 309 labelled up_1(3)* 300 to 310 labelled up_1(2)* 301 to 302 labelled g_1(2)* 302 to 294 labelled f_1(2)* 303 to 301 labelled g_1(2)* 304 to 305 labelled down_1(2)* 304 to 306 labelled up_1(3)* 305 to 294 labelled f_1(2)* 306 to 307 labelled g_1(3)* 307 to 294 labelled f_1(3)* 309 to 306 labelled g_1(3)* 310 to 302 labelled c(2)* 311 to 309 labelled down_1(2)* 311 to 310 labelled down_1(2)* 311 to 312 labelled up_1(3)* 312 to 307 labelled c(3)* 313 to 312 labelled down_1(3)


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(16)
YES