/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(4) YES


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

   g(x, y) -> x
   g(x, y) -> y
   f(0, 1, x) -> f(s(x), x, x)
   f(x, y, s(z)) -> s(f(0, 1, z))

Q is empty.

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(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(0, 1, x) -> F(s(x), x, x)
   F(x, y, s(z)) -> F(0, 1, z)

The TRS R consists of the following rules:

   g(x, y) -> x
   g(x, y) -> y
   f(0, 1, x) -> f(s(x), x, x)
   f(x, y, s(z)) -> s(f(0, 1, z))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(3) 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:
*F(x, y, s(z)) -> F(0, 1, z)
The graph contains the following edges 3 > 3


*F(0, 1, x) -> F(s(x), x, x)
The graph contains the following edges 3 >= 2, 3 >= 3


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(4)
YES