YES
proof of Strategy_removed_AG01_#4.21.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


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(1) -> f(g(1))
   f(f(x)) -> f(x)
   g(0) -> g(f(0))
   g(g(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 1. This implies Q-termination of R.
The following rules were used to construct the certificate:

   f(1) -> f(g(1))
   f(f(x)) -> f(x)
   g(0) -> g(f(0))
   g(g(x)) -> g(x)

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 3, 4, 5, 6, 13, 14, 15, 16

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

Those nodes are connected through the following edges:

* 1 to 3 labelled f_1(0)* 1 to 2 labelled f_1(0), g_1(0), f_1(1), g_1(1)* 1 to 5 labelled g_1(0)* 1 to 13 labelled f_1(1)* 1 to 15 labelled g_1(1)* 2 to 2 labelled #_1(0)* 3 to 4 labelled g_1(0)* 4 to 2 labelled 1(0)* 5 to 6 labelled f_1(0)* 6 to 2 labelled 0(0)* 13 to 14 labelled g_1(1)* 14 to 2 labelled 1(1)* 15 to 16 labelled f_1(1)* 16 to 2 labelled 0(1)


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(2)
YES