YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema_04/z082.srs

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

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 RFCMatchBoundsTRSProof (⇔, 38 ms)
↳4 YES

(0) Obligation:

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

a(c(a(x))) → c(a(c(x)))
a(a(b(x))) → a(d(b(x)))
a(b(x)) → b(a(a(x)))
d(d(x)) → a(d(b(x)))
b(b(x)) → b(c(x))
a(d(c(x))) → c(a(x))
b(c(x)) → a(a(a(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(c(a(x))) → c(a(c(x)))
b(a(a(x))) → b(d(a(x)))
b(a(x)) → a(a(b(x)))
d(d(x)) → b(d(a(x)))
b(b(x)) → c(b(x))
c(d(a(x))) → a(c(x))
c(b(x)) → a(a(a(x)))

Q is empty.

(3) 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:

a(c(a(x))) → c(a(c(x)))
b(a(a(x))) → b(d(a(x)))
b(a(x)) → a(a(b(x)))
d(d(x)) → b(d(a(x)))
b(b(x)) → c(b(x))
c(d(a(x))) → a(c(x))
c(b(x)) → a(a(a(x)))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 43, 44, 45, 46, 47, 48, 50, 51, 82, 83, 84, 85, 86, 87, 140, 141

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

Those nodes are connected through the following edges:

  • 1 to 5 labelled c_1(0)
  • 1 to 7 labelled b_1(0)
  • 1 to 9 labelled a_1(0)
  • 1 to 10 labelled c_1(0)
  • 1 to 6 labelled a_1(0), c_1(1)
  • 1 to 11 labelled a_1(1)
  • 1 to 20 labelled c_1(1)
  • 1 to 47 labelled c_1(2)
  • 3 to 3 labelled #_1(0)
  • 5 to 6 labelled a_1(0), c_1(1)
  • 5 to 20 labelled c_1(1)
  • 6 to 3 labelled c_1(0)
  • 6 to 13 labelled a_1(1)
  • 6 to 11 labelled a_1(1)
  • 6 to 6 labelled c_1(1)
  • 6 to 43 labelled c_1(2)
  • 6 to 47 labelled c_1(2)
  • 6 to 45 labelled c_1(2)
  • 6 to 50 labelled c_1(2)
  • 7 to 8 labelled d_1(0)
  • 8 to 3 labelled a_1(0)
  • 8 to 6 labelled c_1(1)
  • 9 to 10 labelled a_1(0)
  • 9 to 20 labelled c_1(1)
  • 10 to 3 labelled b_1(0), a_1(0)
  • 10 to 14 labelled b_1(1)
  • 10 to 16 labelled a_1(1)
  • 10 to 17 labelled c_1(1)
  • 10 to 6 labelled c_1(1)
  • 10 to 18 labelled a_1(2)
  • 10 to 50 labelled c_1(2)
  • 10 to 82 labelled c_1(3)
  • 11 to 12 labelled a_1(1)
  • 11 to 43 labelled c_1(2)
  • 12 to 3 labelled a_1(1)
  • 12 to 14 labelled a_1(1)
  • 12 to 6 labelled c_1(1)
  • 13 to 3 labelled c_1(1)
  • 13 to 13 labelled a_1(1)
  • 13 to 11 labelled a_1(1)
  • 13 to 21 labelled c_1(1)
  • 13 to 6 labelled c_1(1)
  • 13 to 43 labelled c_1(2)
  • 13 to 47 labelled c_1(2)
  • 13 to 44 labelled c_1(1)
  • 13 to 48 labelled c_1(1)
  • 13 to 45 labelled c_1(2)
  • 13 to 46 labelled c_1(1)
  • 13 to 50 labelled c_1(2)
  • 14 to 15 labelled d_1(1)
  • 15 to 3 labelled a_1(1)
  • 15 to 6 labelled c_1(1)
  • 16 to 17 labelled a_1(1)
  • 16 to 45 labelled c_1(2)
  • 17 to 3 labelled b_1(1)
  • 17 to 14 labelled b_1(1)
  • 17 to 16 labelled a_1(1)
  • 17 to 17 labelled c_1(1)
  • 17 to 18 labelled a_1(2)
  • 17 to 50 labelled c_1(2)
  • 17 to 82 labelled c_1(3)
  • 18 to 19 labelled a_1(2)
  • 18 to 43 labelled c_1(2)
  • 19 to 3 labelled a_1(2)
  • 19 to 6 labelled c_1(1)
  • 19 to 14 labelled a_1(2)
  • 20 to 21 labelled a_1(1)
  • 20 to 43 labelled c_1(2)
  • 21 to 13 labelled c_1(1)
  • 21 to 11 labelled c_1(1)
  • 21 to 16 labelled c_1(1)
  • 21 to 18 labelled c_1(1)
  • 21 to 51 labelled c_1(1)
  • 21 to 83 labelled c_1(1)
  • 43 to 44 labelled a_1(2)
  • 43 to 43 labelled c_1(2)
  • 43 to 6 labelled c_1(1)
  • 43 to 45 labelled c_1(2)
  • 43 to 84 labelled c_1(3)
  • 43 to 82 labelled c_1(3)
  • 44 to 13 labelled c_1(2), a_1(1)
  • 44 to 11 labelled c_1(2), a_1(1)
  • 44 to 12 labelled c_1(2)
  • 44 to 17 labelled c_1(2)
  • 44 to 3 labelled c_1(2)
  • 44 to 14 labelled c_1(2)
  • 44 to 18 labelled a_1(2)
  • 44 to 48 labelled c_1(2)
  • 44 to 19 labelled c_1(2)
  • 44 to 44 labelled a_1(2)
  • 44 to 6 labelled c_1(1)
  • 44 to 43 labelled c_1(2)
  • 44 to 47 labelled c_1(2)
  • 44 to 82 labelled c_1(3)
  • 44 to 45 labelled c_1(2)
  • 44 to 84 labelled c_1(3)
  • 44 to 50 labelled c_1(2)
  • 45 to 46 labelled a_1(2)
  • 45 to 43 labelled c_1(2)
  • 45 to 86 labelled c_1(3)
  • 46 to 16 labelled c_1(2)
  • 46 to 18 labelled c_1(2)
  • 46 to 51 labelled c_1(2)
  • 46 to 83 labelled c_1(2)
  • 47 to 48 labelled a_1(2)
  • 47 to 43 labelled c_1(2)
  • 47 to 84 labelled c_1(3)
  • 47 to 82 labelled c_1(3)
  • 48 to 44 labelled c_1(2)
  • 50 to 51 labelled a_1(2)
  • 51 to 46 labelled c_1(2)
  • 82 to 83 labelled a_1(3)
  • 82 to 43 labelled c_1(2)
  • 82 to 84 labelled c_1(3)
  • 82 to 82 labelled c_1(3)
  • 83 to 44 labelled c_1(3)
  • 84 to 85 labelled a_1(3)
  • 84 to 86 labelled c_1(3)
  • 84 to 6 labelled c_1(1)
  • 84 to 43 labelled c_1(2)
  • 84 to 82 labelled c_1(3)
  • 84 to 84 labelled c_1(3)
  • 84 to 140 labelled c_1(4)
  • 85 to 18 labelled c_1(3)
  • 85 to 3 labelled c_1(3)
  • 85 to 14 labelled c_1(3)
  • 85 to 13 labelled a_1(1)
  • 85 to 11 labelled a_1(1)
  • 85 to 48 labelled c_1(3)
  • 85 to 46 labelled c_1(3)
  • 85 to 83 labelled c_1(3)
  • 85 to 85 labelled c_1(3)
  • 85 to 44 labelled a_1(2)
  • 85 to 6 labelled c_1(1)
  • 85 to 43 labelled c_1(2)
  • 85 to 47 labelled c_1(2)
  • 85 to 82 labelled c_1(3)
  • 85 to 84 labelled c_1(3)
  • 85 to 50 labelled c_1(2)
  • 85 to 45 labelled c_1(2)
  • 85 to 51 labelled c_1(3)
  • 86 to 87 labelled a_1(3)
  • 86 to 84 labelled c_1(3)
  • 87 to 19 labelled c_1(3)
  • 140 to 141 labelled a_1(4)
  • 140 to 43 labelled c_1(2)
  • 140 to 82 labelled c_1(3)
  • 141 to 83 labelled c_1(4)
  • 141 to 85 labelled c_1(4)

(4) YES