YES
proof of Transformed_CSR_04_Ex1_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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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, 0 ms]
(2) YES


----------------------------------------

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

   active(g(X)) -> mark(h(X))
   active(c) -> mark(d)
   active(h(d)) -> mark(g(c))
   mark(g(X)) -> active(g(X))
   mark(h(X)) -> active(h(X))
   mark(c) -> active(c)
   mark(d) -> active(d)
   g(mark(X)) -> g(X)
   g(active(X)) -> g(X)
   h(mark(X)) -> h(X)
   h(active(X)) -> h(X)

Q is empty.

----------------------------------------

(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(g(X)) -> mark(h(X))
   active(c) -> mark(d)
   active(h(d)) -> mark(g(c))
   mark(g(X)) -> active(g(X))
   mark(h(X)) -> active(h(X))
   mark(c) -> active(c)
   mark(d) -> active(d)
   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, 14, 16, 17, 18, 19

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), active_1(0), active_1(1), mark_1(1)* 1 to 6 labelled mark_1(0)* 1 to 8 labelled active_1(0)* 1 to 7 labelled active_1(0)* 1 to 2 labelled g_1(0), h_1(0), g_1(1), h_1(1)* 1 to 14 labelled active_1(1), mark_1(1)* 1 to 16 labelled active_1(2)* 1 to 17 labelled mark_1(2)* 1 to 18 labelled mark_1(3), active_1(3)* 1 to 19 labelled active_1(4)* 2 to 2 labelled #_1(0)* 5 to 2 labelled h_1(0), d(0), h_1(1)* 6 to 7 labelled g_1(0)* 7 to 2 labelled c(0), c(1)* 8 to 2 labelled g_1(0), g_1(1)* 14 to 2 labelled h_1(1), d(1)* 14 to 7 labelled g_1(1)* 16 to 2 labelled h_1(2), d(2), h_1(1)* 16 to 7 labelled g_1(2)* 17 to 7 labelled h_1(2)* 18 to 7 labelled h_1(3)* 19 to 7 labelled h_1(4)


----------------------------------------

(2)
YES