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(a(x1)) → a(b(b(b(x1))))
b(a(x1)) → b(b(c(x1)))
a(b(b(c(x1)))) → a(a(a(b(x1))))
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x1)) → a(b(b(b(x1))))
b(a(x1)) → b(b(c(x1)))
a(b(b(c(x1)))) → a(a(a(b(x1))))
Q is empty.
Termination of the TRS R could be shown with a Match Bound [6,7] of 4. This implies Q-termination of R.
The following rules were used to construct the certificate:
a(a(x1)) → a(b(b(b(x1))))
b(a(x1)) → b(b(c(x1)))
a(b(b(c(x1)))) → a(a(a(b(x1))))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 4, 3, 7, 6, 5, 10, 9, 8, 12, 11, 15, 14, 13, 18, 17, 16, 21, 20, 19, 23, 22, 25, 24, 28, 27, 26, 31, 30, 29, 33, 32, 35, 34, 38, 37, 36, 41, 40, 39, 44, 43, 42, 46, 45, 48, 47, 50, 49, 56, 55, 54, 61, 60, 63, 62
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 5 labelled a_1(0)
- 1 to 8 labelled a_1(0)
- 1 to 13 labelled a_1(1)
- 2 to 2 labelled #_1(0)
- 4 to 2 labelled c_1(0)
- 3 to 4 labelled b_1(0)
- 7 to 2 labelled b_1(0)
- 7 to 11 labelled b_1(1)
- 6 to 7 labelled a_1(0)
- 6 to 19 labelled a_1(1)
- 6 to 26 labelled a_1(2)
- 5 to 6 labelled a_1(0)
- 5 to 16 labelled a_1(1)
- 10 to 2 labelled b_1(0)
- 10 to 11 labelled b_1(1)
- 9 to 10 labelled b_1(0)
- 8 to 9 labelled b_1(0)
- 12 to 2 labelled c_1(1)
- 11 to 12 labelled b_1(1)
- 15 to 6 labelled b_1(1)
- 15 to 22 labelled b_1(1)
- 15 to 24 labelled b_1(2)
- 15 to 16 labelled b_1(1)
- 14 to 15 labelled b_1(1)
- 13 to 14 labelled b_1(1)
- 18 to 7 labelled b_1(1)
- 18 to 19 labelled b_1(1)
- 18 to 34 labelled b_1(2)
- 18 to 26 labelled b_1(1)
- 17 to 18 labelled b_1(1)
- 16 to 17 labelled b_1(1)
- 21 to 2 labelled b_1(1)
- 21 to 11 labelled b_1(1)
- 20 to 21 labelled a_1(1)
- 20 to 19 labelled a_1(1)
- 20 to 26 labelled a_1(2)
- 20 to 36 labelled a_1(2)
- 20 to 39 labelled a_1(3)
- 20 to 54 labelled a_1(3)
- 19 to 20 labelled a_1(1)
- 19 to 29 labelled a_1(2)
- 23 to 7 labelled c_1(1)
- 22 to 23 labelled b_1(1)
- 25 to 19 labelled c_1(2)
- 25 to 26 labelled c_1(2)
- 24 to 25 labelled b_1(2)
- 28 to 20 labelled b_1(2)
- 28 to 32 labelled b_1(2)
- 28 to 24 labelled b_1(2)
- 28 to 29 labelled b_1(2)
- 28 to 45 labelled b_1(3)
- 27 to 28 labelled b_1(2)
- 26 to 27 labelled b_1(2)
- 31 to 21 labelled b_1(2)
- 31 to 19 labelled b_1(2)
- 31 to 34 labelled b_1(2)
- 31 to 26 labelled b_1(2)
- 31 to 36 labelled b_1(2)
- 31 to 47 labelled b_1(3)
- 31 to 39 labelled b_1(2)
- 31 to 54 labelled b_1(2)
- 30 to 31 labelled b_1(2)
- 29 to 30 labelled b_1(2)
- 33 to 21 labelled c_1(2)
- 32 to 33 labelled b_1(2)
- 35 to 20 labelled c_1(2)
- 35 to 29 labelled c_1(2)
- 34 to 35 labelled b_1(2)
- 38 to 2 labelled b_1(2)
- 38 to 11 labelled b_1(1)
- 37 to 38 labelled a_1(2)
- 37 to 19 labelled a_1(1)
- 37 to 26 labelled a_1(2)
- 37 to 36 labelled a_1(2)
- 37 to 54 labelled a_1(3)
- 37 to 39 labelled a_1(3)
- 36 to 37 labelled a_1(2)
- 36 to 42 labelled a_1(3)
- 36 to 29 labelled a_1(2)
- 41 to 37 labelled b_1(3)
- 41 to 24 labelled b_1(2)
- 41 to 49 labelled b_1(3)
- 41 to 42 labelled b_1(3)
- 41 to 45 labelled b_1(3)
- 41 to 60 labelled b_1(4)
- 40 to 41 labelled b_1(3)
- 39 to 40 labelled b_1(3)
- 44 to 38 labelled b_1(3)
- 44 to 36 labelled b_1(3)
- 44 to 26 labelled b_1(3)
- 44 to 47 labelled b_1(3)
- 44 to 39 labelled b_1(3)
- 44 to 54 labelled b_1(3)
- 44 to 62 labelled b_1(4)
- 43 to 44 labelled b_1(3)
- 42 to 43 labelled b_1(3)
- 46 to 26 labelled c_1(3)
- 46 to 36 labelled c_1(3)
- 46 to 39 labelled c_1(3)
- 46 to 54 labelled c_1(3)
- 45 to 46 labelled b_1(3)
- 48 to 29 labelled c_1(3)
- 48 to 37 labelled c_1(3)
- 48 to 42 labelled c_1(3)
- 47 to 48 labelled b_1(3)
- 50 to 38 labelled c_1(3)
- 49 to 50 labelled b_1(3)
- 56 to 29 labelled b_1(3)
- 55 to 56 labelled b_1(3)
- 54 to 55 labelled b_1(3)
- 61 to 39 labelled c_1(4)
- 61 to 54 labelled c_1(4)
- 60 to 61 labelled b_1(4)
- 63 to 42 labelled c_1(4)
- 62 to 63 labelled b_1(4)