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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(2) YES


----------------------------------------

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

   0(1(2(3(4(x1))))) -> 0(2(1(3(4(x1)))))
   0(5(1(2(4(3(x1)))))) -> 0(5(2(1(4(3(x1))))))
   0(5(2(4(1(3(x1)))))) -> 0(1(5(2(4(3(x1))))))
   0(5(3(1(2(4(x1)))))) -> 0(1(5(3(2(4(x1))))))
   0(5(4(1(3(2(x1)))))) -> 0(5(4(3(1(2(x1))))))

Q is empty.

----------------------------------------

(1) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 0. This implies Q-termination of R.
The following rules were used to construct the certificate:

   0(1(2(3(4(x1))))) -> 0(2(1(3(4(x1)))))
   0(5(1(2(4(3(x1)))))) -> 0(5(2(1(4(3(x1))))))
   0(5(2(4(1(3(x1)))))) -> 0(1(5(2(4(3(x1))))))
   0(5(3(1(2(4(x1)))))) -> 0(1(5(3(2(4(x1))))))
   0(5(4(1(3(2(x1)))))) -> 0(5(4(3(1(2(x1))))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26

Node 1 is start node and node 2 is final node.

Those nodes are connected through the following edges:

* 1 to 3 labelled 0_1(0)* 1 to 7 labelled 0_1(0)* 1 to 12 labelled 0_1(0)* 1 to 17 labelled 0_1(0)* 1 to 22 labelled 0_1(0)* 2 to 2 labelled #_1(0)* 3 to 4 labelled 2_1(0)* 4 to 5 labelled 1_1(0)* 5 to 6 labelled 3_1(0)* 6 to 2 labelled 4_1(0)* 7 to 8 labelled 5_1(0)* 8 to 9 labelled 2_1(0)* 9 to 10 labelled 1_1(0)* 10 to 11 labelled 4_1(0)* 11 to 2 labelled 3_1(0)* 12 to 13 labelled 1_1(0)* 13 to 14 labelled 5_1(0)* 14 to 15 labelled 2_1(0)* 15 to 16 labelled 4_1(0)* 16 to 2 labelled 3_1(0)* 17 to 18 labelled 1_1(0)* 18 to 19 labelled 5_1(0)* 19 to 20 labelled 3_1(0)* 20 to 21 labelled 2_1(0)* 21 to 2 labelled 4_1(0)* 22 to 23 labelled 5_1(0)* 23 to 24 labelled 4_1(0)* 24 to 25 labelled 3_1(0)* 25 to 26 labelled 1_1(0)* 26 to 2 labelled 2_1(0)


----------------------------------------

(2)
YES