Termination w.r.t. Q of the following Term Rewriting System could be proven:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(c(x1))) → a(a(b(x1)))
a(b(c(x1))) → b(c(b(c(x1))))
a(b(c(x1))) → c(b(c(a(x1))))
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(c(x1))) → a(a(b(x1)))
a(b(c(x1))) → b(c(b(c(x1))))
a(b(c(x1))) → c(b(c(a(x1))))
Q is empty.
Termination of the TRS R could be shown with a Match Bound [6,7] of 3. This implies Q-termination of R.
The following rules were used to construct the certificate:
a(b(c(x1))) → a(a(b(x1)))
a(b(c(x1))) → b(c(b(c(x1))))
a(b(c(x1))) → c(b(c(a(x1))))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 5, 3, 4, 7, 6, 9, 10, 8, 13, 11, 12, 15, 14, 17, 18, 16, 20, 19, 22, 23, 21, 26, 24, 25, 28, 27, 30, 31, 29, 34, 32, 33, 36, 35, 38, 39, 37, 48, 46, 47, 50, 49, 52, 53, 51
Node 1 is start node and node 2 is final node.
Those nodes are connect through the following edges:
- 1 to 3 labelled b_1(0)
- 1 to 6 labelled a_1(0)
- 1 to 8 labelled c_1(0)
- 1 to 17 labelled b_1(1)
- 1 to 19 labelled a_1(1)
- 1 to 21 labelled c_1(1)
- 2 to 2 labelled #_1(0)
- 5 to 2 labelled c_1(0)
- 3 to 4 labelled c_1(0)
- 4 to 5 labelled b_1(0)
- 7 to 2 labelled b_1(0)
- 6 to 7 labelled a_1(0)
- 6 to 11 labelled b_1(1)
- 6 to 14 labelled a_1(1)
- 6 to 16 labelled c_1(1)
- 6 to 32 labelled b_1(2)
- 6 to 35 labelled a_1(2)
- 6 to 37 labelled c_1(2)
- 9 to 10 labelled c_1(0)
- 10 to 2 labelled a_1(0)
- 10 to 11 labelled b_1(1)
- 10 to 14 labelled a_1(1)
- 10 to 16 labelled c_1(1)
- 10 to 32 labelled b_1(2)
- 10 to 35 labelled a_1(2)
- 10 to 37 labelled c_1(2)
- 8 to 9 labelled b_1(0)
- 13 to 2 labelled c_1(1)
- 11 to 12 labelled c_1(1)
- 11 to 33 labelled c_1(1)
- 12 to 13 labelled b_1(1)
- 15 to 2 labelled b_1(1)
- 14 to 15 labelled a_1(1)
- 14 to 11 labelled b_1(1)
- 14 to 14 labelled a_1(1)
- 14 to 16 labelled c_1(1)
- 14 to 32 labelled b_1(2)
- 14 to 35 labelled a_1(2)
- 14 to 37 labelled c_1(2)
- 17 to 18 labelled c_1(1)
- 18 to 2 labelled a_1(1)
- 18 to 11 labelled b_1(1)
- 18 to 14 labelled a_1(1)
- 18 to 16 labelled c_1(1)
- 18 to 32 labelled b_1(2)
- 18 to 35 labelled a_1(2)
- 18 to 37 labelled c_1(2)
- 16 to 17 labelled b_1(1)
- 20 to 12 labelled b_1(1)
- 20 to 33 labelled b_1(1)
- 19 to 20 labelled a_1(1)
- 22 to 23 labelled c_1(1)
- 23 to 12 labelled a_1(1)
- 23 to 24 labelled b_1(2)
- 23 to 27 labelled a_1(2)
- 23 to 29 labelled c_1(2)
- 23 to 33 labelled a_1(1)
- 23 to 32 labelled b_1(2)
- 23 to 35 labelled a_1(2)
- 23 to 37 labelled c_1(2)
- 23 to 46 labelled b_1(3)
- 23 to 49 labelled a_1(3)
- 23 to 51 labelled c_1(3)
- 21 to 22 labelled b_1(1)
- 26 to 2 labelled c_1(2)
- 24 to 25 labelled c_1(2)
- 25 to 26 labelled b_1(2)
- 28 to 2 labelled b_1(2)
- 27 to 28 labelled a_1(2)
- 27 to 11 labelled b_1(1)
- 27 to 14 labelled a_1(1)
- 27 to 16 labelled c_1(1)
- 27 to 32 labelled b_1(2)
- 27 to 35 labelled a_1(2)
- 27 to 37 labelled c_1(2)
- 30 to 31 labelled c_1(2)
- 31 to 2 labelled a_1(2)
- 31 to 11 labelled b_1(1)
- 31 to 14 labelled a_1(1)
- 31 to 16 labelled c_1(1)
- 31 to 32 labelled b_1(2)
- 31 to 35 labelled a_1(2)
- 31 to 37 labelled c_1(2)
- 29 to 30 labelled b_1(2)
- 34 to 12 labelled c_1(2)
- 34 to 33 labelled c_1(2)
- 32 to 33 labelled c_1(2)
- 33 to 34 labelled b_1(2)
- 36 to 12 labelled b_1(2)
- 36 to 33 labelled b_1(2)
- 35 to 36 labelled a_1(2)
- 38 to 39 labelled c_1(2)
- 39 to 12 labelled a_1(2)
- 39 to 33 labelled a_1(2)
- 39 to 24 labelled b_1(2)
- 39 to 27 labelled a_1(2)
- 39 to 29 labelled c_1(2)
- 39 to 46 labelled b_1(3)
- 39 to 49 labelled a_1(3)
- 39 to 51 labelled c_1(3)
- 39 to 32 labelled b_1(2)
- 39 to 35 labelled a_1(2)
- 39 to 37 labelled c_1(2)
- 37 to 38 labelled b_1(2)
- 48 to 33 labelled c_1(3)
- 48 to 12 labelled c_1(3)
- 46 to 47 labelled c_1(3)
- 47 to 48 labelled b_1(3)
- 50 to 33 labelled b_1(3)
- 50 to 12 labelled b_1(3)
- 49 to 50 labelled a_1(3)
- 52 to 53 labelled c_1(3)
- 53 to 33 labelled a_1(3)
- 53 to 12 labelled a_1(3)
- 53 to 46 labelled b_1(3)
- 53 to 49 labelled a_1(3)
- 53 to 51 labelled c_1(3)
- 53 to 24 labelled b_1(2)
- 53 to 27 labelled a_1(2)
- 53 to 29 labelled c_1(2)
- 53 to 32 labelled b_1(2)
- 53 to 35 labelled a_1(2)
- 53 to 37 labelled c_1(2)
- 51 to 52 labelled b_1(3)