YES Termination w.r.t. Q proof of SK90_2.48.ari

(0) Obligation:

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

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

Q is empty.

(1) 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:

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

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 4, 8, 9, 10, 11

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

Those nodes are connected through the following edges:

  • 1 to 8 labelled e_1(0)
  • 1 to 4 labelled c_1(0), b_1(0), d_1(0), e_1(0), u_1(0), v_1(0), a_1(0), b_1(1), c_1(1), d_1(1), e_1(1), u_1(1), v_1(1), a_1(1)
  • 1 to 1 labelled a_1(0), a_1(1)
  • 1 to 9 labelled e_1(1)
  • 1 to 10 labelled a_1(1)
  • 1 to 11 labelled e_1(2)
  • 4 to 4 labelled #_1(0)
  • 8 to 4 labelled u_1(0)
  • 9 to 4 labelled u_1(1)
  • 10 to 4 labelled e_1(1)
  • 11 to 4 labelled u_1(2)

(2) YES