YES Termination w.r.t. Q proof of Transformed_CSR_04_Ex1_Zan97_iGM.ari

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 RFCMatchBoundsTRSProof (⇔, 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 transformation)

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, 3, 6, 8, 10, 12, 14, 16, 17, 18, 20

Node 1 is start node and node 3 is final node.

Those nodes are connected through the following edges:

  • 1 to 6 labelled mark_1(0), active_1(0), active_1(1), mark_1(1)
  • 1 to 8 labelled mark_1(0)
  • 1 to 12 labelled active_1(0)
  • 1 to 10 labelled active_1(0)
  • 1 to 3 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 20 labelled active_1(4)
  • 3 to 3 labelled #_1(0)
  • 6 to 3 labelled h_1(0), d(0), h_1(1)
  • 8 to 10 labelled g_1(0)
  • 10 to 3 labelled c(0), c(1)
  • 12 to 3 labelled g_1(0), g_1(1)
  • 14 to 3 labelled h_1(1), d(1)
  • 14 to 10 labelled g_1(1)
  • 16 to 3 labelled h_1(2), d(2), h_1(1)
  • 16 to 10 labelled g_1(2)
  • 17 to 10 labelled h_1(2)
  • 18 to 10 labelled h_1(3)
  • 20 to 10 labelled h_1(4)

(2) YES