YES
proof of /TPDB/AG01_#3.51.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# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty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(f(x)) -> f(c(f(x)))
   f(f(x)) -> f(d(f(x)))
   g(c(x)) -> x
   g(d(x)) -> x
   g(c(h(0))) -> g(d(1))
   g(c(1)) -> g(d(h(0)))
   g(h(x)) -> g(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(f(x)) -> f(c(f(x)))
   f(f(x)) -> f(d(f(x)))
   g(c(x)) -> x
   g(d(x)) -> x
   g(c(h(0))) -> g(d(1))
   g(c(1)) -> g(d(h(0)))
   g(h(x)) -> g(x)

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
2, 3, 5, 8, 11, 12, 18, 19, 20, 21, 22, 23, 24, 27, 28, 31, 32, 33, 34, 35

Node 2 is start node and node 3 is final node.

Those nodes are connected through the following edges:

* 2 to 5 labelled f_1(0)* 2 to 11 labelled f_1(0)* 2 to 3 labelled f_1(0), c_1(0), d_1(0), g_1(0), h_1(0), 0(0), 1(0), f_1(1), c_1(1), d_1(1), g_1(1), h_1(1), 0(1), 1(1), 1(2)* 2 to 18 labelled g_1(0)* 2 to 20 labelled g_1(0)* 2 to 23 labelled f_1(1)* 2 to 27 labelled f_1(1)* 2 to 31 labelled g_1(1)* 2 to 33 labelled g_1(1)* 2 to 22 labelled h_1(1)* 2 to 35 labelled h_1(2)* 3 to 3 labelled #_1(0)* 5 to 8 labelled c_1(0)* 8 to 3 labelled f_1(0)* 8 to 23 labelled f_1(1)* 8 to 27 labelled f_1(1)* 11 to 12 labelled d_1(0)* 12 to 3 labelled f_1(0)* 12 to 23 labelled f_1(1)* 12 to 27 labelled f_1(1)* 18 to 19 labelled d_1(0)* 19 to 3 labelled 1(0)* 20 to 21 labelled d_1(0)* 21 to 22 labelled h_1(0)* 22 to 3 labelled 0(0)* 23 to 24 labelled c_1(1)* 24 to 3 labelled f_1(1)* 24 to 23 labelled f_1(1)* 24 to 27 labelled f_1(1)* 27 to 28 labelled d_1(1)* 28 to 3 labelled f_1(1)* 28 to 23 labelled f_1(1)* 28 to 27 labelled f_1(1)* 31 to 32 labelled d_1(1)* 32 to 3 labelled 1(1)* 33 to 34 labelled d_1(1)* 34 to 35 labelled h_1(1)* 35 to 3 labelled 0(1)


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(2)
YES