YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(4) YES


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

   b(c(a(x1))) -> a(b(a(b(c(x1)))))
   b(x1) -> c(c(x1))
   c(d(x1)) -> a(b(c(a(x1))))
   a(a(x1)) -> a(c(b(a(x1))))

Q is empty.

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(1) QTRS Reverse (EQUIVALENT)
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(c(b(x1))) -> c(b(a(b(a(x1)))))
   b(x1) -> c(c(x1))
   d(c(x1)) -> a(c(b(a(x1))))
   a(a(x1)) -> a(b(c(a(x1))))

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 3. This implies Q-termination of R.
The following rules were used to construct the certificate:

   a(c(b(x1))) -> c(b(a(b(a(x1)))))
   b(x1) -> c(c(x1))
   d(c(x1)) -> a(c(b(a(x1))))
   a(a(x1)) -> a(b(c(a(x1))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
3, 4, 6, 8, 10, 12, 16, 17, 18, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 50, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 77, 78, 79, 80, 81, 82, 83

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

Those nodes are connected through the following edges:

* 3 to 6 labelled c_1(0)* 3 to 16 labelled a_1(0)* 3 to 22 labelled a_1(0)* 3 to 36 labelled c_1(1)* 4 to 4 labelled #_1(0)* 6 to 8 labelled b_1(0)* 6 to 4 labelled c_1(0)* 6 to 25 labelled c_1(1)* 8 to 10 labelled a_1(0)* 10 to 12 labelled b_1(0)* 10 to 26 labelled c_1(1)* 12 to 4 labelled a_1(0)* 12 to 29 labelled c_1(1)* 12 to 33 labelled a_1(1)* 16 to 17 labelled c_1(0)* 17 to 18 labelled b_1(0)* 17 to 27 labelled c_1(1)* 18 to 4 labelled a_1(0)* 18 to 29 labelled c_1(1)* 18 to 33 labelled a_1(1)* 22 to 23 labelled b_1(0)* 22 to 28 labelled c_1(1)* 23 to 24 labelled c_1(0)* 24 to 4 labelled a_1(0)* 24 to 29 labelled c_1(1)* 24 to 33 labelled a_1(1)* 25 to 8 labelled c_1(1)* 26 to 12 labelled c_1(1)* 27 to 18 labelled c_1(1)* 28 to 23 labelled c_1(1)* 29 to 30 labelled b_1(1)* 29 to 50 labelled c_1(2)* 30 to 31 labelled a_1(1)* 31 to 32 labelled b_1(1)* 31 to 55 labelled c_1(2)* 32 to 4 labelled a_1(1)* 32 to 29 labelled c_1(1)* 32 to 33 labelled a_1(1)* 33 to 34 labelled b_1(1)* 33 to 56 labelled c_1(2)* 34 to 35 labelled c_1(1)* 35 to 4 labelled a_1(1)* 35 to 29 labelled c_1(1)* 35 to 33 labelled a_1(1)* 36 to 37 labelled b_1(1)* 36 to 57 labelled c_1(2)* 37 to 38 labelled a_1(1)* 38 to 39 labelled b_1(1)* 38 to 58 labelled c_1(2)* 39 to 18 labelled a_1(1)* 39 to 33 labelled a_1(1)* 39 to 59 labelled a_1(2)* 39 to 62 labelled c_1(2)* 50 to 30 labelled c_1(2)* 55 to 32 labelled c_1(2)* 56 to 34 labelled c_1(2)* 57 to 37 labelled c_1(2)* 58 to 39 labelled c_1(2)* 59 to 60 labelled b_1(2)* 59 to 77 labelled c_1(3)* 60 to 61 labelled c_1(2)* 61 to 33 labelled a_1(2)* 62 to 63 labelled b_1(2)* 62 to 78 labelled c_1(3)* 63 to 64 labelled a_1(2)* 64 to 65 labelled b_1(2)* 64 to 79 labelled c_1(3)* 65 to 30 labelled a_1(2)* 65 to 80 labelled a_1(2)* 77 to 60 labelled c_1(3)* 78 to 63 labelled c_1(3)* 79 to 65 labelled c_1(3)* 80 to 81 labelled b_1(2)* 80 to 83 labelled c_1(3)* 81 to 82 labelled c_1(2)* 82 to 31 labelled a_1(2)* 83 to 81 labelled c_1(3)


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(4)
YES