YES
proof of Transformed_CSR_04_Ex6_GM04_C.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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:

   active(c) -> mark(f(g(c)))
   active(f(g(X))) -> mark(g(X))
   proper(c) -> ok(c)
   proper(f(X)) -> f(proper(X))
   proper(g(X)) -> g(proper(X))
   f(ok(X)) -> ok(f(X))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(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:

   c'(active(x)) -> c'(g(f(mark(x))))
   g(f(active(X))) -> g(mark(X))
   c'(proper(x)) -> c'(ok(x))
   f(proper(X)) -> proper(f(X))
   g(proper(X)) -> proper(g(X))
   ok(f(X)) -> f(ok(X))
   ok(g(X)) -> g(ok(X))
   mark(top(X)) -> proper(top(X))
   ok(top(X)) -> active(top(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 5. This implies Q-termination of R.
The following rules were used to construct the certificate:

   c'(active(x)) -> c'(g(f(mark(x))))
   g(f(active(X))) -> g(mark(X))
   c'(proper(x)) -> c'(ok(x))
   f(proper(X)) -> proper(f(X))
   g(proper(X)) -> proper(g(X))
   ok(f(X)) -> f(ok(X))
   ok(g(X)) -> g(ok(X))
   mark(top(X)) -> proper(top(X))
   ok(top(X)) -> active(top(X))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 36, 37, 38, 39, 41, 42, 43, 44, 45, 48, 49, 53, 54, 55, 59, 60, 61, 64, 65, 66, 67, 68, 73, 74, 75, 76, 77, 78, 80, 81, 84

Node 24 is start node and node 25 is final node.

Those nodes are connected through the following edges:

* 24 to 26 labelled c'_1(0), f_1(0), g_1(0)* 24 to 28 labelled g_1(0)* 24 to 29 labelled proper_1(0), active_1(0)* 24 to 36 labelled c'_1(1)* 24 to 39 labelled proper_1(1)* 24 to 38 labelled g_1(1)* 24 to 48 labelled proper_1(2)* 24 to 59 labelled c'_1(2)* 24 to 80 labelled c'_1(3)* 25 to 25 labelled #_1(0)* 26 to 27 labelled g_1(0)* 26 to 25 labelled ok_1(0)* 26 to 30 labelled f_1(1), g_1(1)* 26 to 31 labelled active_1(1)* 26 to 42 labelled proper_1(1)* 26 to 45 labelled g_1(2)* 26 to 53 labelled proper_1(3)* 26 to 60 labelled proper_1(2)* 27 to 28 labelled f_1(0)* 27 to 41 labelled proper_1(1)* 28 to 25 labelled mark_1(0)* 28 to 31 labelled proper_1(1)* 29 to 25 labelled f_1(0), g_1(0), top_1(0)* 29 to 32 labelled proper_1(1)* 29 to 33 labelled g_1(1)* 29 to 43 labelled proper_1(2)* 30 to 25 labelled ok_1(1)* 30 to 30 labelled f_1(1), g_1(1)* 30 to 31 labelled active_1(1)* 30 to 45 labelled g_1(2)* 30 to 53 labelled proper_1(3)* 30 to 60 labelled proper_1(2)* 31 to 25 labelled top_1(1)* 32 to 25 labelled f_1(1), g_1(1)* 32 to 32 labelled proper_1(1)* 32 to 33 labelled g_1(1)* 32 to 43 labelled proper_1(2)* 33 to 25 labelled mark_1(1)* 33 to 31 labelled proper_1(1)* 36 to 37 labelled g_1(1)* 36 to 42 labelled ok_1(1)* 36 to 54 labelled proper_1(2)* 36 to 55 labelled g_1(2)* 36 to 53 labelled ok_1(1)* 36 to 60 labelled ok_1(1)* 36 to 65 labelled f_1(2), g_1(2)* 36 to 68 labelled g_1(3)* 36 to 75 labelled proper_1(4)* 37 to 38 labelled f_1(1)* 37 to 49 labelled proper_1(2)* 38 to 31 labelled mark_1(1)* 38 to 44 labelled proper_1(2)* 39 to 31 labelled g_1(1)* 39 to 42 labelled f_1(1), g_1(1)* 39 to 53 labelled f_1(1), g_1(1)* 39 to 60 labelled f_1(1), g_1(1)* 41 to 31 labelled f_1(1)* 42 to 41 labelled g_1(1)* 43 to 31 labelled g_1(2)* 44 to 25 labelled top_1(2)* 45 to 31 labelled mark_1(2)* 45 to 44 labelled proper_1(2)* 48 to 44 labelled g_1(2)* 49 to 44 labelled f_1(2)* 53 to 44 labelled g_1(3)* 54 to 49 labelled g_1(2)* 55 to 41 labelled ok_1(2)* 55 to 61 labelled f_1(2)* 55 to 44 labelled ok_1(2)* 55 to 67 labelled active_1(3)* 59 to 54 labelled ok_1(2)* 59 to 64 labelled g_1(3)* 59 to 76 labelled g_1(4)* 59 to 75 labelled ok_1(2)* 59 to 78 labelled proper_1(5)* 60 to 53 labelled f_1(2), g_1(2)* 60 to 60 labelled f_1(2), g_1(2)* 61 to 31 labelled ok_1(2)* 61 to 44 labelled active_1(2)* 64 to 49 labelled ok_1(3)* 64 to 66 labelled f_1(3)* 64 to 67 labelled ok_1(3)* 64 to 77 labelled active_1(4)* 65 to 53 labelled ok_1(2)* 65 to 60 labelled ok_1(2)* 65 to 66 labelled g_1(3)* 65 to 73 labelled f_1(3), g_1(3)* 66 to 44 labelled ok_1(3)* 66 to 67 labelled active_1(3)* 67 to 25 labelled top_1(3)* 68 to 44 labelled mark_1(3)* 68 to 67 labelled proper_1(3)* 73 to 53 labelled ok_1(3)* 73 to 60 labelled ok_1(3)* 73 to 74 labelled g_1(4)* 73 to 73 labelled f_1(3), g_1(3)* 74 to 44 labelled ok_1(4)* 74 to 67 labelled active_1(3)* 75 to 67 labelled g_1(4)* 76 to 67 labelled mark_1(4)* 76 to 77 labelled proper_1(4)* 77 to 25 labelled top_1(4)* 78 to 77 labelled g_1(5)* 80 to 78 labelled ok_1(3)* 80 to 81 labelled g_1(4)* 81 to 77 labelled ok_1(4)* 81 to 84 labelled active_1(5)* 84 to 25 labelled top_1(5)


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(4)
YES