YES
proof of Transformed_CSR_04_Ex23_Luc06_GM.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


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, 0 ms]
(4) YES


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

   a__f(f(a)) -> c(f(g(f(a))))
   mark(f(X)) -> a__f(mark(X))
   mark(a) -> a
   mark(c(X)) -> c(X)
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

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:

   a'(f(a__f(x))) -> a'(f(g(f(c(x)))))
   f(mark(X)) -> mark(a__f(X))
   a'(mark(x)) -> a'(x)
   c(mark(X)) -> c(X)
   g(mark(X)) -> mark(g(X))
   a__f(X) -> f(X)

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 2. This implies Q-termination of R.
The following rules were used to construct the certificate:

   a'(f(a__f(x))) -> a'(f(g(f(c(x)))))
   f(mark(X)) -> mark(a__f(X))
   a'(mark(x)) -> a'(x)
   c(mark(X)) -> c(X)
   g(mark(X)) -> mark(g(X))
   a__f(X) -> f(X)

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29

Node 17 is start node and node 18 is final node.

Those nodes are connected through the following edges:

* 17 to 19 labelled a'_1(0)* 17 to 23 labelled mark_1(0)* 17 to 18 labelled a'_1(0), c_1(0), f_1(0), a'_1(1), c_1(1)* 17 to 24 labelled a'_1(1)* 17 to 28 labelled mark_1(1)* 18 to 18 labelled #_1(0)* 19 to 20 labelled f_1(0)* 20 to 21 labelled g_1(0)* 21 to 22 labelled f_1(0)* 22 to 18 labelled c_1(0), c_1(1)* 23 to 18 labelled a__f_1(0), g_1(0), f_1(1)* 23 to 29 labelled mark_1(1)* 23 to 28 labelled mark_1(1)* 24 to 25 labelled f_1(1)* 25 to 26 labelled g_1(1)* 26 to 27 labelled f_1(1)* 27 to 18 labelled c_1(1)* 28 to 18 labelled a__f_1(1), f_1(2)* 28 to 28 labelled mark_1(1)* 29 to 18 labelled g_1(1)* 29 to 29 labelled mark_1(1)


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(4)
YES