YES Termination w.r.t. Q proof of SK90_2.49.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:

a(c(d(x))) → c(x)
u(b(d(d(x)))) → b(x)
v(a(a(x))) → u(v(x))
v(a(c(x))) → u(b(d(x)))
v(c(x)) → b(x)
w(a(a(x))) → u(w(x))
w(a(c(x))) → u(b(d(x)))
w(c(x)) → b(x)

Q is empty.

(1) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
The following rules were used to construct the certificate:

a(c(d(x))) → c(x)
u(b(d(d(x)))) → b(x)
v(a(a(x))) → u(v(x))
v(a(c(x))) → u(b(d(x)))
v(c(x)) → b(x)
w(a(a(x))) → u(w(x))
w(a(c(x))) → u(b(d(x)))
w(c(x)) → b(x)

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

2, 4, 6, 9, 10, 11, 12, 13

Node 2 is start node and node 4 is final node.

Those nodes are connected through the following edges:

  • 2 to 4 labelled c_1(0), b_1(0), b_1(1)
  • 2 to 6 labelled u_1(0)
  • 2 to 9 labelled u_1(0)
  • 4 to 4 labelled #_1(0)
  • 6 to 4 labelled v_1(0), w_1(0), b_1(1)
  • 6 to 11 labelled u_1(1)
  • 6 to 12 labelled u_1(1)
  • 9 to 10 labelled b_1(0)
  • 10 to 4 labelled d_1(0)
  • 11 to 4 labelled v_1(1), w_1(1), b_1(1)
  • 11 to 11 labelled u_1(1)
  • 11 to 12 labelled u_1(1)
  • 12 to 13 labelled b_1(1)
  • 13 to 4 labelled d_1(1)

(2) YES