YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

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


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

   :(:(x, y), z) -> :(x, :(y, z))
   :(+(x, y), z) -> +(:(x, z), :(y, z))
   :(z, +(x, f(y))) -> :(g(z, y), +(x, a))

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:

   :^1(:(x, y), z) -> :^1(x, :(y, z))
   :^1(:(x, y), z) -> :^1(y, z)
   :^1(+(x, y), z) -> :^1(x, z)
   :^1(+(x, y), z) -> :^1(y, z)
   :^1(z, +(x, f(y))) -> :^1(g(z, y), +(x, a))

The TRS R consists of the following rules:

   :(:(x, y), z) -> :(x, :(y, z))
   :(+(x, y), z) -> +(:(x, z), :(y, z))
   :(z, +(x, f(y))) -> :(g(z, y), +(x, a))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
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(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   :^1(:(x, y), z) -> :^1(y, z)
   :^1(:(x, y), z) -> :^1(x, :(y, z))
   :^1(+(x, y), z) -> :^1(x, z)
   :^1(+(x, y), z) -> :^1(y, z)

The TRS R consists of the following rules:

   :(:(x, y), z) -> :(x, :(y, z))
   :(+(x, y), z) -> +(:(x, z), :(y, z))
   :(z, +(x, f(y))) -> :(g(z, y), +(x, a))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(5) 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:
*:^1(:(x, y), z) -> :^1(y, z)
The graph contains the following edges 1 > 1, 2 >= 2


*:^1(:(x, y), z) -> :^1(x, :(y, z))
The graph contains the following edges 1 > 1


*:^1(+(x, y), z) -> :^1(x, z)
The graph contains the following edges 1 > 1, 2 >= 2


*:^1(+(x, y), z) -> :^1(y, z)
The graph contains the following edges 1 > 1, 2 >= 2


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(6)
YES