YES
proof of Transformed_CSR_04_Ex23_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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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(c(f(g(f(a)))))
   mark(f(X)) -> active(f(mark(X)))
   mark(a) -> active(a)
   mark(c(X)) -> active(c(X))
   mark(g(X)) -> active(g(mark(X)))
   f(mark(X)) -> f(X)
   f(active(X)) -> f(X)
   c(mark(X)) -> c(X)
   c(active(X)) -> c(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 3. This implies Q-termination of R.
The following rules were used to construct the certificate:

   active(f(f(a))) -> mark(c(f(g(f(a)))))
   mark(f(X)) -> active(f(mark(X)))
   mark(a) -> active(a)
   mark(c(X)) -> active(c(X))
   mark(g(X)) -> active(g(mark(X)))
   f(mark(X)) -> f(X)
   f(active(X)) -> f(X)
   c(mark(X)) -> c(X)
   c(active(X)) -> c(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, 9, 10, 11, 12, 13, 14, 15, 16, 32, 33, 34, 35, 36, 42, 52, 53, 54, 55, 56, 57

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 8 labelled active_1(0)* 1 to 7 labelled active_1(0)* 1 to 10 labelled active_1(0)* 1 to 2 labelled f_1(0), c_1(0), g_1(0), f_1(1), c_1(1), g_1(1)* 1 to 12 labelled active_1(1)* 1 to 32 labelled mark_1(1)* 1 to 42 labelled active_1(2)* 2 to 2 labelled #_1(0)* 3 to 4 labelled c_1(0)* 4 to 5 labelled f_1(0)* 5 to 6 labelled g_1(0)* 6 to 7 labelled f_1(0)* 7 to 2 labelled a(0)* 8 to 9 labelled f_1(0)* 8 to 2 labelled c_1(0), c_1(1), f_1(1)* 8 to 13 labelled f_1(1)* 8 to 8 labelled f_1(1)* 8 to 15 labelled f_1(1)* 8 to 32 labelled f_1(1)* 8 to 42 labelled f_1(1)* 8 to 52 labelled f_1(1)* 8 to 57 labelled f_1(1)* 9 to 2 labelled mark_1(0)* 9 to 13 labelled active_1(1)* 9 to 8 labelled active_1(1)* 9 to 15 labelled active_1(1)* 9 to 32 labelled mark_1(1)* 9 to 42 labelled active_1(2)* 9 to 52 labelled mark_1(2)* 9 to 57 labelled active_1(3)* 10 to 11 labelled g_1(0)* 10 to 2 labelled g_1(1)* 10 to 13 labelled g_1(1)* 10 to 8 labelled g_1(1)* 10 to 15 labelled g_1(1)* 10 to 32 labelled g_1(1)* 10 to 42 labelled g_1(1)* 10 to 52 labelled g_1(1)* 10 to 57 labelled g_1(1)* 11 to 2 labelled mark_1(0)* 11 to 13 labelled active_1(1)* 11 to 8 labelled active_1(1)* 11 to 15 labelled active_1(1)* 11 to 32 labelled mark_1(1)* 11 to 42 labelled active_1(2)* 11 to 52 labelled mark_1(2)* 11 to 57 labelled active_1(3)* 12 to 4 labelled c_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 8 labelled f_1(2)* 13 to 15 labelled f_1(2)* 13 to 32 labelled f_1(2)* 13 to 52 labelled f_1(2)* 13 to 42 labelled f_1(2)* 13 to 57 labelled f_1(2)* 14 to 2 labelled mark_1(1)* 14 to 13 labelled active_1(1)* 14 to 8 labelled active_1(1)* 14 to 15 labelled active_1(1)* 14 to 32 labelled mark_1(1)* 14 to 52 labelled mark_1(2)* 14 to 42 labelled active_1(2)* 14 to 57 labelled active_1(3)* 15 to 16 labelled g_1(1)* 15 to 2 labelled g_1(2), g_1(1)* 15 to 13 labelled g_1(2)* 15 to 8 labelled g_1(2)* 15 to 15 labelled g_1(2)* 15 to 32 labelled g_1(2)* 15 to 52 labelled g_1(2)* 15 to 42 labelled g_1(2)* 15 to 57 labelled g_1(2)* 16 to 2 labelled mark_1(1)* 16 to 13 labelled active_1(1)* 16 to 8 labelled active_1(1)* 16 to 15 labelled active_1(1)* 16 to 32 labelled mark_1(1)* 16 to 52 labelled mark_1(2)* 16 to 42 labelled active_1(2)* 16 to 57 labelled active_1(3)* 32 to 33 labelled c_1(1)* 33 to 34 labelled f_1(1)* 34 to 35 labelled g_1(1)* 35 to 36 labelled f_1(1)* 36 to 2 labelled a(1)* 42 to 33 labelled c_1(2)* 52 to 53 labelled c_1(2)* 53 to 54 labelled f_1(2)* 54 to 55 labelled g_1(2)* 55 to 56 labelled f_1(2)* 56 to 2 labelled a(2)* 57 to 53 labelled c_1(3)


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(2)
YES