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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) Strip Symbols Proof [SOUND, 0 ms]
(4) QTRS
(5) RFCMatchBoundsTRSProof [EQUIVALENT, 8 ms]
(6) YES


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

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

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

Q is empty.

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

(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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

(3) Strip Symbols Proof (SOUND)
We were given the following TRS:

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

By stripping symbols from the only rule of the system, we obtained the following TRS [ENDRULLIS]: 

   a(a(a(a(c(a(b(x))))))) -> c(a(b(a(a(a(c(a(b(a(a(a(a(x)))))))))))))

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

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(a(a(a(c(a(b(x))))))) -> c(a(b(a(a(a(c(a(b(a(a(a(a(x)))))))))))))

Q is empty.

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

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

   a(a(a(a(c(a(b(x))))))) -> c(a(b(a(a(a(c(a(b(a(a(a(a(x)))))))))))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
83, 84, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145

Node 83 is start node and node 84 is final node.

Those nodes are connected through the following edges:

* 83 to 110 labelled c_1(0)* 84 to 84 labelled #_1(0)* 110 to 111 labelled a_1(0)* 111 to 112 labelled b_1(0)* 112 to 113 labelled a_1(0)* 113 to 114 labelled a_1(0)* 114 to 115 labelled a_1(0)* 115 to 116 labelled c_1(0)* 116 to 117 labelled a_1(0)* 117 to 118 labelled b_1(0)* 118 to 119 labelled a_1(0)* 118 to 134 labelled c_1(1)* 119 to 120 labelled a_1(0)* 119 to 134 labelled c_1(1)* 120 to 121 labelled a_1(0)* 120 to 134 labelled c_1(1)* 121 to 84 labelled a_1(0)* 121 to 134 labelled c_1(1)* 134 to 135 labelled a_1(1)* 135 to 136 labelled b_1(1)* 136 to 137 labelled a_1(1)* 137 to 138 labelled a_1(1)* 138 to 139 labelled a_1(1)* 139 to 140 labelled c_1(1)* 140 to 141 labelled a_1(1)* 141 to 142 labelled b_1(1)* 142 to 143 labelled a_1(1)* 142 to 134 labelled c_1(1)* 143 to 144 labelled a_1(1)* 143 to 134 labelled c_1(1)* 144 to 145 labelled a_1(1)* 144 to 134 labelled c_1(1)* 145 to 84 labelled a_1(1)* 145 to 134 labelled c_1(1)


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

(6)
YES