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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) Overlay + Local Confluence [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 17 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) TRUE


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

   f(x, y, z) -> g(<=(x, y), x, y, z)
   g(true, x, y, z) -> z
   g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
   p(0) -> 0
   p(s(x)) -> x

Q is empty.

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(1) Overlay + Local Confluence (EQUIVALENT)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(x, y, z) -> g(<=(x, y), x, y, z)
   g(true, x, y, z) -> z
   g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
   p(0) -> 0
   p(s(x)) -> x

The set Q consists of the following terms:

   f(x0, x1, x2)
   g(true, x0, x1, x2)
   g(false, x0, x1, x2)
   p(0)
   p(s(x0))


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(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(x, y, z) -> G(<=(x, y), x, y, z)
   G(false, x, y, z) -> F(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
   G(false, x, y, z) -> F(p(x), y, z)
   G(false, x, y, z) -> P(x)
   G(false, x, y, z) -> F(p(y), z, x)
   G(false, x, y, z) -> P(y)
   G(false, x, y, z) -> F(p(z), x, y)
   G(false, x, y, z) -> P(z)

The TRS R consists of the following rules:

   f(x, y, z) -> g(<=(x, y), x, y, z)
   g(true, x, y, z) -> z
   g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
   p(0) -> 0
   p(s(x)) -> x

The set Q consists of the following terms:

   f(x0, x1, x2)
   g(true, x0, x1, x2)
   g(false, x0, x1, x2)
   p(0)
   p(s(x0))

We have to consider all minimal (P,Q,R)-chains.
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(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 8 less nodes.
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(6)
TRUE