/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) QTRSRRRProof [EQUIVALENT, 58 ms]
(2) QTRS
(3) Overlay + Local Confluence [EQUIVALENT, 8 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 0 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 0 ms]
(8) QDP
(9) PisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

   f(g(x)) -> g(f(f(x)))
   f(h(x)) -> h(g(x))
   f'(s(x), y, y) -> f'(y, x, s(x))

Q is empty.

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

   POL(f(x_1)) = x_1
   POL(f'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3
   POL(g(x_1)) = x_1
   POL(h(x_1)) = 2*x_1
   POL(s(x_1)) = 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'(s(x), y, y) -> f'(y, x, s(x))




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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(g(x)) -> g(f(f(x)))
   f(h(x)) -> h(g(x))

Q is empty.

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(3) Overlay + Local Confluence (EQUIVALENT)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
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(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(g(x)) -> g(f(f(x)))
   f(h(x)) -> h(g(x))

The set Q consists of the following terms:

   f(g(x0))
   f(h(x0))


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(5) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(g(x)) -> F(f(x))
   F(g(x)) -> F(x)

The TRS R consists of the following rules:

   f(g(x)) -> g(f(f(x)))
   f(h(x)) -> h(g(x))

The set Q consists of the following terms:

   f(g(x0))
   f(h(x0))

We have to consider all minimal (P,Q,R)-chains.
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(7) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   F(g(x)) -> F(f(x))
   F(g(x)) -> F(x)
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
F(x1)  =  F(x1)

g(x1)  =  g(x1)

f(x1)  =  x1

h(x1)  =  h


Recursive path order with status [RPO].
Quasi-Precedence: h > g_1

Status: F_1: [1]
g_1: multiset status
h: multiset status


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   f(g(x)) -> g(f(f(x)))
   f(h(x)) -> h(g(x))


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(8)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   f(g(x)) -> g(f(f(x)))
   f(h(x)) -> h(g(x))

The set Q consists of the following terms:

   f(g(x0))
   f(h(x0))

We have to consider all minimal (P,Q,R)-chains.
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(9) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(10)
YES