/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) QDPOrderProof [EQUIVALENT, 36 ms]
(4) QDP
(5) PisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

   f(f(X, Y), Z) -> f(X, f(Y, Z))
   f(X, f(Y, Z)) -> f(Y, Y)

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(f(X, Y), Z) -> F(X, f(Y, Z))
   F(f(X, Y), Z) -> F(Y, Z)
   F(X, f(Y, Z)) -> F(Y, Y)

The TRS R consists of the following rules:

   f(f(X, Y), Z) -> f(X, f(Y, Z))
   f(X, f(Y, Z)) -> f(Y, Y)

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


The following pairs can be oriented strictly and are deleted.

   F(f(X, Y), Z) -> F(X, f(Y, Z))
   F(f(X, Y), Z) -> F(Y, Z)
   F(X, f(Y, Z)) -> F(Y, Y)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(F(x_1, x_2)) = x_1*x_2 + 2*x_1^2 + 2*x_2
   POL(f(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_1^2 + 2*x_2

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

   f(X, f(Y, Z)) -> f(Y, Y)
   f(f(X, Y), Z) -> f(X, f(Y, Z))


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(4)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   f(f(X, Y), Z) -> f(X, f(Y, Z))
   f(X, f(Y, Z)) -> f(Y, Y)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(5) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(6)
YES