YES
proof of Transformed_CSR_04_Ex18_Luc06_iGM.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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:

   active(f(f(a))) -> mark(f(g(f(a))))
   mark(f(X)) -> active(f(mark(X)))
   mark(a) -> active(a)
   mark(g(X)) -> active(g(X))
   f(mark(X)) -> f(X)
   f(active(X)) -> f(X)
   g(mark(X)) -> g(X)
   g(active(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 4. This implies Q-termination of R.
The following rules were used to construct the certificate:

   active(f(f(a))) -> mark(f(g(f(a))))
   mark(f(X)) -> active(f(mark(X)))
   mark(a) -> active(a)
   mark(g(X)) -> active(g(X))
   f(mark(X)) -> f(X)
   f(active(X)) -> f(X)
   g(mark(X)) -> g(X)
   g(active(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, 7, 8, 11, 12, 13, 14, 23, 24, 25, 26, 27, 33, 34, 41, 42, 43, 44, 46, 47, 48, 51

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

Those nodes are connected through the following edges:

* 1 to 3 labelled mark_1(0)* 1 to 7 labelled active_1(0)* 1 to 6 labelled active_1(0)* 1 to 2 labelled f_1(0), g_1(0), f_1(1), g_1(1)* 1 to 11 labelled active_1(1)* 1 to 24 labelled mark_1(1)* 1 to 33 labelled active_1(2)* 2 to 2 labelled #_1(0)* 3 to 4 labelled f_1(0)* 4 to 5 labelled g_1(0)* 5 to 6 labelled f_1(0)* 6 to 2 labelled a(0)* 7 to 8 labelled f_1(0)* 7 to 2 labelled g_1(0), g_1(1), f_1(1)* 7 to 13 labelled f_1(1)* 7 to 7 labelled f_1(1)* 7 to 24 labelled f_1(1)* 7 to 33 labelled f_1(1)* 7 to 41 labelled f_1(1)* 7 to 46 labelled f_1(1)* 8 to 2 labelled mark_1(0)* 8 to 13 labelled active_1(1)* 8 to 7 labelled active_1(1)* 8 to 24 labelled mark_1(1)* 8 to 33 labelled active_1(2)* 8 to 41 labelled mark_1(2)* 8 to 46 labelled active_1(3)* 11 to 12 labelled f_1(1)* 11 to 4 labelled f_1(2)* 11 to 23 labelled f_1(2)* 12 to 4 labelled mark_1(1)* 12 to 23 labelled active_1(1)* 13 to 14 labelled f_1(1)* 13 to 2 labelled a(1), f_1(2), f_1(1)* 13 to 13 labelled f_1(2)* 13 to 7 labelled f_1(2)* 13 to 24 labelled f_1(2)* 13 to 41 labelled f_1(2)* 13 to 33 labelled f_1(2)* 13 to 46 labelled f_1(2)* 14 to 2 labelled mark_1(1)* 14 to 13 labelled active_1(1)* 14 to 7 labelled active_1(1)* 14 to 24 labelled mark_1(1)* 14 to 41 labelled mark_1(2)* 14 to 33 labelled active_1(2)* 14 to 46 labelled active_1(3)* 23 to 5 labelled g_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 2 labelled a(1)* 33 to 34 labelled f_1(2)* 33 to 25 labelled f_1(3)* 33 to 48 labelled f_1(3)* 34 to 25 labelled mark_1(2)* 34 to 48 labelled active_1(2)* 41 to 42 labelled f_1(2)* 42 to 43 labelled g_1(2)* 43 to 44 labelled f_1(2)* 44 to 2 labelled a(2)* 46 to 47 labelled f_1(3)* 46 to 42 labelled f_1(4)* 46 to 51 labelled f_1(4)* 47 to 42 labelled mark_1(3)* 47 to 51 labelled active_1(3)* 48 to 26 labelled g_1(2)* 51 to 43 labelled g_1(3)


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(2)
YES