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(c(b(x1))) → b(a(b(a(x1))))
b(x1) → c(a(c(x1)))
a(a(x1)) → a(b(c(a(x1))))
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(c(b(x1))) → b(a(b(a(x1))))
b(x1) → c(a(c(x1)))
a(a(x1)) → a(b(c(a(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(c(b(x1))) → b(a(b(a(x1))))
b(x1) → c(a(c(x1)))
a(a(x1)) → a(b(c(a(x1))))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 4, 3, 6, 7, 5, 9, 10, 8, 12, 11, 14, 13, 16, 15, 18, 19, 17, 21, 22, 20, 24, 23, 26, 25, 28, 27, 30, 31, 29, 33, 32, 35, 34, 37, 38, 36, 40, 41, 39, 43, 42, 45, 44, 47, 46, 49, 50, 48, 52, 53, 51, 55, 56, 54, 58, 59, 57, 67, 66, 69, 68, 71, 70, 73, 72, 75, 74, 77, 76, 79, 78, 81, 82, 80, 84, 85, 83, 87, 88, 86, 90, 89, 92, 91, 94, 93, 96, 95, 98, 99, 97, 101, 100
Node 1 is start node and node 2 is final node.
Those nodes are connect through the following edges:
- 1 to 3 labelled c_1(0)
- 1 to 5 labelled b_1(0)
- 1 to 8 labelled a_1(0)
- 1 to 15 labelled c_1(1)
- 2 to 2 labelled #_1(0)
- 4 to 2 labelled c_1(0)
- 3 to 4 labelled a_1(0)
- 3 to 17 labelled b_1(1)
- 3 to 27 labelled c_1(2)
- 6 to 7 labelled b_1(0)
- 6 to 13 labelled c_1(1)
- 7 to 2 labelled a_1(0)
- 7 to 17 labelled b_1(1)
- 7 to 20 labelled a_1(1)
- 7 to 27 labelled c_1(2)
- 5 to 6 labelled a_1(0)
- 5 to 39 labelled b_1(1)
- 5 to 46 labelled c_1(2)
- 9 to 10 labelled c_1(0)
- 10 to 2 labelled a_1(0)
- 10 to 17 labelled b_1(1)
- 10 to 20 labelled a_1(1)
- 10 to 27 labelled c_1(2)
- 8 to 9 labelled b_1(0)
- 8 to 11 labelled c_1(1)
- 12 to 9 labelled c_1(1)
- 11 to 12 labelled a_1(1)
- 14 to 7 labelled c_1(1)
- 13 to 14 labelled a_1(1)
- 13 to 29 labelled b_1(2)
- 13 to 34 labelled c_1(3)
- 16 to 5 labelled c_1(1)
- 15 to 16 labelled a_1(1)
- 15 to 57 labelled b_1(2)
- 15 to 78 labelled c_1(3)
- 18 to 19 labelled b_1(1)
- 18 to 25 labelled c_1(2)
- 19 to 2 labelled a_1(1)
- 19 to 17 labelled b_1(1)
- 19 to 20 labelled a_1(1)
- 19 to 27 labelled c_1(2)
- 17 to 18 labelled a_1(1)
- 17 to 51 labelled b_1(2)
- 17 to 76 labelled c_1(3)
- 21 to 22 labelled c_1(1)
- 22 to 2 labelled a_1(1)
- 22 to 17 labelled b_1(1)
- 22 to 20 labelled a_1(1)
- 22 to 27 labelled c_1(2)
- 20 to 21 labelled b_1(1)
- 20 to 23 labelled c_1(2)
- 24 to 21 labelled c_1(2)
- 23 to 24 labelled a_1(2)
- 26 to 19 labelled c_1(2)
- 25 to 26 labelled a_1(2)
- 25 to 29 labelled b_1(2)
- 25 to 34 labelled c_1(3)
- 28 to 17 labelled c_1(2)
- 27 to 28 labelled a_1(2)
- 27 to 86 labelled b_1(3)
- 27 to 93 labelled c_1(4)
- 30 to 31 labelled b_1(2)
- 30 to 32 labelled c_1(3)
- 31 to 17 labelled a_1(2)
- 31 to 36 labelled a_1(2)
- 29 to 30 labelled a_1(2)
- 33 to 31 labelled c_1(3)
- 32 to 33 labelled a_1(3)
- 35 to 29 labelled c_1(3)
- 34 to 35 labelled a_1(3)
- 37 to 38 labelled c_1(2)
- 38 to 18 labelled a_1(2)
- 38 to 54 labelled b_1(3)
- 38 to 68 labelled c_1(4)
- 36 to 37 labelled b_1(2)
- 36 to 42 labelled c_1(3)
- 40 to 41 labelled b_1(1)
- 40 to 44 labelled c_1(2)
- 41 to 29 labelled a_1(1)
- 41 to 48 labelled a_1(2)
- 39 to 40 labelled a_1(1)
- 43 to 37 labelled c_1(3)
- 42 to 43 labelled a_1(3)
- 45 to 41 labelled c_1(2)
- 44 to 45 labelled a_1(2)
- 47 to 39 labelled c_1(2)
- 46 to 47 labelled a_1(2)
- 49 to 50 labelled c_1(2)
- 50 to 30 labelled a_1(2)
- 48 to 49 labelled b_1(2)
- 48 to 74 labelled c_1(3)
- 52 to 53 labelled b_1(2)
- 52 to 72 labelled c_1(3)
- 53 to 29 labelled a_1(2)
- 53 to 83 labelled a_1(3)
- 51 to 52 labelled a_1(2)
- 55 to 56 labelled b_1(3)
- 55 to 66 labelled c_1(4)
- 56 to 29 labelled a_1(3)
- 56 to 83 labelled a_1(3)
- 54 to 55 labelled a_1(3)
- 58 to 59 labelled b_1(2)
- 58 to 70 labelled c_1(3)
- 59 to 39 labelled a_1(2)
- 59 to 80 labelled a_1(2)
- 57 to 58 labelled a_1(2)
- 67 to 56 labelled c_1(4)
- 66 to 67 labelled a_1(4)
- 69 to 54 labelled c_1(4)
- 68 to 69 labelled a_1(4)
- 71 to 59 labelled c_1(3)
- 70 to 71 labelled a_1(3)
- 73 to 53 labelled c_1(3)
- 72 to 73 labelled a_1(3)
- 75 to 49 labelled c_1(3)
- 74 to 75 labelled a_1(3)
- 77 to 51 labelled c_1(3)
- 76 to 77 labelled a_1(3)
- 79 to 57 labelled c_1(3)
- 78 to 79 labelled a_1(3)
- 81 to 82 labelled c_1(2)
- 82 to 40 labelled a_1(2)
- 80 to 81 labelled b_1(2)
- 80 to 95 labelled c_1(3)
- 84 to 85 labelled c_1(3)
- 85 to 30 labelled a_1(3)
- 83 to 84 labelled b_1(3)
- 83 to 91 labelled c_1(4)
- 87 to 88 labelled b_1(3)
- 87 to 89 labelled c_1(4)
- 88 to 51 labelled a_1(3)
- 88 to 97 labelled a_1(3)
- 86 to 87 labelled a_1(3)
- 90 to 88 labelled c_1(4)
- 89 to 90 labelled a_1(4)
- 92 to 84 labelled c_1(4)
- 91 to 92 labelled a_1(4)
- 94 to 86 labelled c_1(4)
- 93 to 94 labelled a_1(4)
- 96 to 81 labelled c_1(3)
- 95 to 96 labelled a_1(3)
- 98 to 99 labelled c_1(3)
- 99 to 52 labelled a_1(3)
- 97 to 98 labelled b_1(3)
- 97 to 100 labelled c_1(4)
- 101 to 98 labelled c_1(4)
- 100 to 101 labelled a_1(4)