YES
proof of Transformed_CSR_04_ExConc_Zan97_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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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, 36 ms]
(2) YES


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

   active(f(X)) -> mark(g(h(f(X))))
   mark(f(X)) -> active(f(mark(X)))
   mark(g(X)) -> active(g(X))
   mark(h(X)) -> active(h(mark(X)))
   f(mark(X)) -> f(X)
   f(active(X)) -> f(X)
   g(mark(X)) -> g(X)
   g(active(X)) -> g(X)
   h(mark(X)) -> h(X)
   h(active(X)) -> h(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(X)) -> mark(g(h(f(X))))
   mark(f(X)) -> active(f(mark(X)))
   mark(g(X)) -> active(g(X))
   mark(h(X)) -> active(h(mark(X)))
   f(mark(X)) -> f(X)
   f(active(X)) -> f(X)
   g(mark(X)) -> g(X)
   g(active(X)) -> g(X)
   h(mark(X)) -> h(X)
   h(active(X)) -> h(X)

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24

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

Those nodes are connected through the following edges:

* 1 to 5 labelled mark_1(0)* 1 to 8 labelled active_1(0)* 1 to 10 labelled active_1(0)* 1 to 2 labelled f_1(0), g_1(0), h_1(0), f_1(1), g_1(1), h_1(1)* 1 to 12 labelled active_1(1)* 1 to 13 labelled mark_1(1)* 1 to 23 labelled active_1(2)* 2 to 2 labelled #_1(0)* 5 to 6 labelled g_1(0)* 6 to 7 labelled h_1(0)* 7 to 2 labelled f_1(0), f_1(1)* 8 to 9 labelled f_1(0)* 8 to 2 labelled g_1(0), g_1(1), f_1(1)* 8 to 16 labelled f_1(1)* 8 to 8 labelled f_1(1)* 8 to 18 labelled f_1(1)* 8 to 13 labelled f_1(1)* 8 to 20 labelled f_1(1)* 8 to 23 labelled f_1(1)* 8 to 24 labelled f_1(1)* 9 to 2 labelled mark_1(0)* 9 to 16 labelled active_1(1)* 9 to 8 labelled active_1(1)* 9 to 18 labelled active_1(1)* 9 to 13 labelled mark_1(1)* 9 to 20 labelled mark_1(2)* 9 to 23 labelled active_1(2)* 9 to 24 labelled active_1(3)* 10 to 11 labelled h_1(0)* 10 to 2 labelled h_1(1)* 10 to 16 labelled h_1(1)* 10 to 8 labelled h_1(1)* 10 to 18 labelled h_1(1)* 10 to 13 labelled h_1(1)* 10 to 20 labelled h_1(1)* 10 to 23 labelled h_1(1)* 10 to 24 labelled h_1(1)* 11 to 2 labelled mark_1(0)* 11 to 16 labelled active_1(1)* 11 to 8 labelled active_1(1)* 11 to 18 labelled active_1(1)* 11 to 13 labelled mark_1(1)* 11 to 20 labelled mark_1(2)* 11 to 23 labelled active_1(2)* 11 to 24 labelled active_1(3)* 12 to 6 labelled g_1(1)* 13 to 14 labelled g_1(1)* 14 to 15 labelled h_1(1)* 15 to 9 labelled f_1(1)* 15 to 2 labelled f_1(1)* 15 to 16 labelled f_1(2), f_1(1)* 15 to 8 labelled f_1(2), f_1(1)* 15 to 18 labelled f_1(2), f_1(1)* 15 to 13 labelled f_1(2), f_1(1)* 15 to 20 labelled f_1(2), f_1(1)* 15 to 23 labelled f_1(2), f_1(1)* 15 to 24 labelled f_1(2), f_1(1)* 16 to 17 labelled f_1(1)* 16 to 2 labelled f_1(2), f_1(1)* 16 to 16 labelled f_1(2)* 16 to 8 labelled f_1(2)* 16 to 18 labelled f_1(2)* 16 to 13 labelled f_1(2)* 16 to 20 labelled f_1(2)* 16 to 23 labelled f_1(2)* 16 to 24 labelled f_1(2)* 17 to 2 labelled mark_1(1)* 17 to 16 labelled active_1(1)* 17 to 8 labelled active_1(1)* 17 to 18 labelled active_1(1)* 17 to 13 labelled mark_1(1)* 17 to 20 labelled mark_1(2)* 17 to 23 labelled active_1(2)* 17 to 24 labelled active_1(3)* 18 to 19 labelled h_1(1)* 18 to 2 labelled h_1(2), h_1(1)* 18 to 16 labelled h_1(2)* 18 to 8 labelled h_1(2)* 18 to 18 labelled h_1(2)* 18 to 13 labelled h_1(2)* 18 to 20 labelled h_1(2)* 18 to 23 labelled h_1(2)* 18 to 24 labelled h_1(2)* 19 to 2 labelled mark_1(1)* 19 to 16 labelled active_1(1)* 19 to 8 labelled active_1(1)* 19 to 18 labelled active_1(1)* 19 to 13 labelled mark_1(1)* 19 to 20 labelled mark_1(2)* 19 to 23 labelled active_1(2)* 19 to 24 labelled active_1(3)* 20 to 21 labelled g_1(2)* 21 to 22 labelled h_1(2)* 22 to 2 labelled f_1(2), f_1(1)* 22 to 17 labelled f_1(2)* 22 to 16 labelled f_1(2)* 22 to 8 labelled f_1(2)* 22 to 18 labelled f_1(2)* 22 to 13 labelled f_1(2)* 22 to 20 labelled f_1(2), f_1(3)* 22 to 23 labelled f_1(2), f_1(3)* 22 to 24 labelled f_1(2), f_1(3)* 23 to 14 labelled g_1(2)* 24 to 21 labelled g_1(3)


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(2)
YES