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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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


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

   d(x) -> e(u(x))
   d(u(x)) -> c(x)
   c(u(x)) -> b(x)
   v(e(x)) -> x
   b(u(x)) -> a(e(x))

Q is empty.

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(1) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R.
The following rules were used to construct the certificate:

   d(x) -> e(u(x))
   d(u(x)) -> c(x)
   c(u(x)) -> b(x)
   v(e(x)) -> x
   b(u(x)) -> a(e(x))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 3, 4, 5, 6

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

Those nodes are connected through the following edges:

* 1 to 3 labelled e_1(0)* 1 to 2 labelled c_1(0), b_1(0), d_1(0), e_1(0), u_1(0), v_1(0), a_1(0), b_1(1), c_1(1), d_1(1), e_1(1), u_1(1), v_1(1), a_1(1)* 1 to 1 labelled a_1(0), a_1(1)* 1 to 4 labelled e_1(1)* 1 to 5 labelled a_1(1)* 1 to 6 labelled e_1(2)* 2 to 2 labelled #_1(0)* 3 to 2 labelled u_1(0)* 4 to 2 labelled u_1(1)* 5 to 2 labelled e_1(1)* 6 to 2 labelled u_1(2)


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(2)
YES