YES
proof of /TPDB/AG01_#3.24.ari
# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty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:

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(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:

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
2, 4, 6, 9, 10, 13, 15, 16, 17

Node 2 is start node and node 4 is final node.

Those nodes are connected through the following edges:

* 2 to 6 labelled s_1(0)* 2 to 9 labelled f_1(0)* 2 to 13 labelled s_1(1)* 4 to 4 labelled #_1(0)* 6 to 4 labelled 0(0)* 9 to 10 labelled f_1(0)* 9 to 13 labelled s_1(1)* 9 to 15 labelled f_1(1)* 9 to 17 labelled s_1(2)* 10 to 4 labelled s_1(0)* 13 to 4 labelled 0(1)* 15 to 16 labelled f_1(1)* 15 to 13 labelled s_1(1)* 15 to 15 labelled f_1(1)* 15 to 17 labelled s_1(2)* 16 to 4 labelled s_1(1)* 17 to 4 labelled 0(2)


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(2)
YES