/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) QTRS Reverse [SOUND, 0 ms]
(2) QTRS
(3) RFCMatchBoundsTRSProof [EQUIVALENT, 14 ms]
(4) YES


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

   a__g(X) -> a__h(X)
   a__c -> d
   a__h(d) -> a__g(c)
   mark(g(X)) -> a__g(X)
   mark(h(X)) -> a__h(X)
   mark(c) -> a__c
   mark(d) -> d
   a__g(X) -> g(X)
   a__h(X) -> h(X)
   a__c -> c

The set Q consists of the following terms:

   a__g(x0)
   a__c
   mark(g(x0))
   mark(h(x0))
   mark(c)
   mark(d)
   a__h(x0)


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(1) QTRS Reverse (SOUND)
We applied the QTRS Reverse Processor [REVERSE].
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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a__g(X) -> a__h(X)
   a__c'(x) -> d'(x)
   d'(a__h(x)) -> c'(a__g(x))
   g(mark(X)) -> a__g(X)
   h(mark(X)) -> a__h(X)
   c'(mark(x)) -> a__c'(x)
   d'(mark(x)) -> d'(x)
   a__g(X) -> g(X)
   a__h(X) -> h(X)
   a__c'(x) -> c'(x)

Q is empty.

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(3) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 5. This implies Q-termination of R.
The following rules were used to construct the certificate:

   a__g(X) -> a__h(X)
   a__c'(x) -> d'(x)
   d'(a__h(x)) -> c'(a__g(x))
   g(mark(X)) -> a__g(X)
   h(mark(X)) -> a__h(X)
   c'(mark(x)) -> a__c'(x)
   d'(mark(x)) -> d'(x)
   a__g(X) -> g(X)
   a__h(X) -> h(X)
   a__c'(x) -> c'(x)

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
3, 4, 5, 7

Node 3 is start node and node 4 is final node.

Those nodes are connected through the following edges:

* 3 to 4 labelled a__h_1(0), d'_1(0), a__g_1(0), a__c'_1(0), g_1(0), h_1(0), c'_1(0), h_1(1), a__h_1(1), g_1(1), d'_1(1), c'_1(1), a__g_1(1), a__c'_1(1), h_1(2), a__h_1(2), g_1(2), d'_1(2), c'_1(2), h_1(3)* 3 to 5 labelled c'_1(0)* 3 to 7 labelled c'_1(3)* 4 to 4 labelled #_1(0), a__g_1(1), a__h_1(2), g_1(2), h_1(3), a__g_1(2), a__h_1(3), g_1(3), a__h_1(1), h_1(4), h_1(2)* 5 to 4 labelled a__g_1(0), a__h_1(1), g_1(1), h_1(2), a__g_1(1), a__h_1(2), g_1(2), h_1(3)* 7 to 4 labelled a__g_1(3), a__h_1(4), g_1(4), h_1(5), a__g_1(1), a__h_1(2), g_1(2), a__h_1(1), h_1(3), h_1(2)


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(4)
YES