YES Termination w.r.t. Q proof of Transformed_CSR_04_Ex23_Luc06_GM.ari

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

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 RFCMatchBoundsTRSProof (⇔, 0 ms)
↳4 YES

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a__f(f(a)) → c(f(g(f(a))))
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c(X)) → c(X)
mark(g(X)) → g(mark(X))
a__f(X) → f(X)

Q is empty.

(1) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a'(f(a__f(x))) → a'(f(g(f(c(x)))))
f(mark(X)) → mark(a__f(X))
a'(mark(x)) → a'(x)
c(mark(X)) → c(X)
g(mark(X)) → mark(g(X))
a__f(X) → f(X)

Q is empty.

(3) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R.
The following rules were used to construct the certificate:

a'(f(a__f(x))) → a'(f(g(f(c(x)))))
f(mark(X)) → mark(a__f(X))
a'(mark(x)) → a'(x)
c(mark(X)) → c(X)
g(mark(X)) → mark(g(X))
a__f(X) → f(X)

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 4, 5, 7, 8, 9, 10, 16, 17, 18, 19, 20, 21

Node 1 is start node and node 4 is final node.

Those nodes are connected through the following edges:

  • 1 to 5 labelled a'_1(0)
  • 1 to 10 labelled mark_1(0)
  • 1 to 4 labelled a'_1(0), c_1(0), f_1(0), a'_1(1), c_1(1)
  • 1 to 16 labelled a'_1(1)
  • 1 to 20 labelled mark_1(1)
  • 4 to 4 labelled #_1(0)
  • 5 to 7 labelled f_1(0)
  • 7 to 8 labelled g_1(0)
  • 8 to 9 labelled f_1(0)
  • 9 to 4 labelled c_1(0), c_1(1)
  • 10 to 4 labelled a__f_1(0), g_1(0), f_1(1)
  • 10 to 21 labelled mark_1(1)
  • 10 to 20 labelled mark_1(1)
  • 16 to 17 labelled f_1(1)
  • 17 to 18 labelled g_1(1)
  • 18 to 19 labelled f_1(1)
  • 19 to 4 labelled c_1(1)
  • 20 to 4 labelled a__f_1(1), f_1(2)
  • 20 to 20 labelled mark_1(1)
  • 21 to 4 labelled g_1(1)
  • 21 to 21 labelled mark_1(1)

(4) YES