/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 5 ms]
(4) YES


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

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

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

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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
117, 118, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158

Node 117 is start node and node 118 is final node.

Those nodes are connected through the following edges:

* 117 to 137 labelled a_1(0)* 117 to 139 labelled b_1(0)* 117 to 140 labelled a_1(0)* 117 to 142 labelled a_1(1)* 118 to 118 labelled #_1(0)* 137 to 138 labelled a_1(0)* 137 to 149 labelled a_1(1)* 138 to 118 labelled b_1(0)* 138 to 144 labelled a_1(1)* 138 to 153 labelled a_1(2)* 139 to 118 labelled c_1(0)* 139 to 146 labelled b_1(1)* 140 to 141 labelled c_1(0)* 141 to 118 labelled a_1(0)* 141 to 147 labelled a_1(1)* 142 to 143 labelled c_1(1)* 143 to 138 labelled a_1(1)* 143 to 151 labelled a_1(2)* 143 to 149 labelled a_1(1)* 144 to 145 labelled a_1(1)* 144 to 151 labelled a_1(2)* 145 to 118 labelled b_1(1)* 145 to 144 labelled a_1(1)* 145 to 153 labelled a_1(2)* 146 to 118 labelled c_1(1)* 146 to 146 labelled b_1(1)* 147 to 148 labelled c_1(1)* 148 to 118 labelled a_1(1)* 148 to 147 labelled a_1(1)* 149 to 150 labelled c_1(1)* 150 to 144 labelled a_1(1)* 150 to 153 labelled a_1(1), a_1(2)* 151 to 152 labelled c_1(2)* 152 to 144 labelled a_1(2)* 152 to 153 labelled a_1(2)* 152 to 155 labelled a_1(3)* 153 to 154 labelled c_1(2)* 154 to 145 labelled a_1(2)* 154 to 151 labelled a_1(2)* 154 to 157 labelled a_1(3)* 155 to 156 labelled c_1(3)* 156 to 151 labelled a_1(3)* 157 to 158 labelled c_1(3)* 158 to 153 labelled a_1(3)


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(4)
YES