YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Mixed_SRS/3.srs

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

Q is empty.

(3) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 5. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 2, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 29, 30, 31, 32, 37, 38, 39, 40, 43, 44, 51, 52, 53, 54, 55, 56, 59, 60

Node 1 is start node and node 2 is final node.

Those nodes are connected through the following edges:

  • 1 to 13 labelled a_1(0)
  • 1 to 15 labelled a_1(0)
  • 1 to 23 labelled a_1(1)
  • 2 to 2 labelled #_1(0)
  • 13 to 14 labelled b_1(0)
  • 13 to 21 labelled a_1(1)
  • 14 to 2 labelled a_1(0)
  • 14 to 17 labelled a_1(1)
  • 14 to 29 labelled a_1(2)
  • 14 to 37 labelled a_1(3)
  • 15 to 16 labelled c_1(0)
  • 16 to 2 labelled a_1(0)
  • 16 to 17 labelled a_1(1)
  • 16 to 29 labelled a_1(2)
  • 16 to 37 labelled a_1(3)
  • 17 to 18 labelled b_1(1)
  • 17 to 21 labelled a_1(1)
  • 17 to 31 labelled a_1(2)
  • 18 to 2 labelled a_1(1)
  • 18 to 17 labelled a_1(1)
  • 18 to 29 labelled a_1(2)
  • 18 to 37 labelled a_1(3)
  • 21 to 22 labelled c_1(1)
  • 22 to 2 labelled a_1(1)
  • 22 to 17 labelled a_1(1)
  • 22 to 18 labelled a_1(1)
  • 22 to 29 labelled a_1(2)
  • 22 to 30 labelled a_1(1)
  • 22 to 37 labelled a_1(3)
  • 22 to 38 labelled a_1(1)
  • 23 to 24 labelled b_1(1)
  • 24 to 21 labelled a_1(1)
  • 29 to 30 labelled b_1(2)
  • 29 to 21 labelled a_1(1)
  • 29 to 31 labelled a_1(2)
  • 29 to 43 labelled a_1(3)
  • 30 to 21 labelled a_1(2)
  • 30 to 2 labelled a_1(2)
  • 30 to 17 labelled a_1(2), a_1(1)
  • 30 to 29 labelled a_1(2)
  • 30 to 31 labelled a_1(2)
  • 30 to 37 labelled a_1(3), a_1(2)
  • 30 to 43 labelled a_1(2)
  • 31 to 32 labelled c_1(2)
  • 32 to 18 labelled a_1(2)
  • 32 to 29 labelled a_1(2)
  • 32 to 39 labelled a_1(3)
  • 32 to 30 labelled a_1(2)
  • 32 to 37 labelled a_1(3)
  • 32 to 51 labelled a_1(4)
  • 32 to 38 labelled a_1(2)
  • 37 to 38 labelled b_1(3)
  • 38 to 31 labelled a_1(3)
  • 38 to 43 labelled a_1(3)
  • 39 to 40 labelled b_1(3)
  • 39 to 43 labelled a_1(3)
  • 39 to 21 labelled a_1(1)
  • 39 to 31 labelled a_1(2)
  • 39 to 55 labelled a_1(4)
  • 40 to 29 labelled a_1(3), a_1(2)
  • 40 to 37 labelled a_1(3)
  • 40 to 21 labelled a_1(3)
  • 40 to 2 labelled a_1(3)
  • 40 to 17 labelled a_1(3), a_1(1)
  • 40 to 51 labelled a_1(4)
  • 43 to 44 labelled c_1(3)
  • 44 to 30 labelled a_1(3)
  • 44 to 38 labelled a_1(3)
  • 44 to 39 labelled a_1(3)
  • 44 to 29 labelled a_1(2)
  • 44 to 37 labelled a_1(3)
  • 44 to 53 labelled a_1(4)
  • 44 to 51 labelled a_1(4)
  • 44 to 59 labelled a_1(5)
  • 51 to 52 labelled b_1(4)
  • 52 to 43 labelled a_1(4)
  • 52 to 55 labelled a_1(4)
  • 53 to 54 labelled b_1(4)
  • 53 to 55 labelled a_1(4)
  • 54 to 37 labelled a_1(4)
  • 54 to 31 labelled a_1(4)
  • 55 to 56 labelled c_1(4)
  • 56 to 38 labelled a_1(4)
  • 56 to 53 labelled a_1(4)
  • 56 to 51 labelled a_1(4)
  • 56 to 52 labelled a_1(4)
  • 56 to 59 labelled a_1(5)
  • 59 to 60 labelled b_1(5)
  • 60 to 55 labelled a_1(5)
  • 60 to 43 labelled a_1(5)

(4) YES