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(d(x1)) → d(b(c(b(d(x1)))))
a(x1) → b(b(f(b(b(x1)))))
b(d(b(x1))) → a(d(x1))
d(f(x1)) → b(d(x1))
Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(d(x1)) → d(b(c(b(d(x1)))))
a(x1) → b(b(f(b(b(x1)))))
b(d(b(x1))) → a(d(x1))
d(f(x1)) → b(d(x1))
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
A(x1) → B(f(b(b(x1))))
A(x1) → B(b(x1))
B(d(b(x1))) → A(d(x1))
B(d(b(x1))) → D(x1)
A(d(x1)) → B(d(x1))
A(x1) → B(x1)
A(x1) → B(b(f(b(b(x1)))))
D(f(x1)) → D(x1)
D(f(x1)) → B(d(x1))
A(d(x1)) → B(c(b(d(x1))))
A(d(x1)) → D(b(c(b(d(x1)))))
The TRS R consists of the following rules:
a(d(x1)) → d(b(c(b(d(x1)))))
a(x1) → b(b(f(b(b(x1)))))
b(d(b(x1))) → a(d(x1))
d(f(x1)) → b(d(x1))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ RFCMatchBoundsTRSProof
Q DP problem:
The TRS P consists of the following rules:
A(x1) → B(f(b(b(x1))))
A(x1) → B(b(x1))
B(d(b(x1))) → A(d(x1))
B(d(b(x1))) → D(x1)
A(d(x1)) → B(d(x1))
A(x1) → B(x1)
A(x1) → B(b(f(b(b(x1)))))
D(f(x1)) → D(x1)
D(f(x1)) → B(d(x1))
A(d(x1)) → B(c(b(d(x1))))
A(d(x1)) → D(b(c(b(d(x1)))))
The TRS R consists of the following rules:
a(d(x1)) → d(b(c(b(d(x1)))))
a(x1) → b(b(f(b(b(x1)))))
b(d(b(x1))) → a(d(x1))
d(f(x1)) → b(d(x1))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RFCMatchBoundsTRSProof
Q DP problem:
The TRS P consists of the following rules:
A(x1) → B(b(x1))
B(d(b(x1))) → A(d(x1))
B(d(b(x1))) → D(x1)
A(d(x1)) → B(d(x1))
A(x1) → B(x1)
D(f(x1)) → D(x1)
D(f(x1)) → B(d(x1))
The TRS R consists of the following rules:
a(d(x1)) → d(b(c(b(d(x1)))))
a(x1) → b(b(f(b(b(x1)))))
b(d(b(x1))) → a(d(x1))
d(f(x1)) → b(d(x1))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(d(x1)) → d(b(c(b(d(x1)))))
a(x1) → b(b(f(b(b(x1)))))
b(d(b(x1))) → a(d(x1))
d(f(x1)) → b(d(x1))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 6, 3, 4, 5, 10, 9, 7, 8, 14, 13, 11, 12, 18, 15, 16, 17, 22, 21, 19, 20, 24, 25, 26, 23, 27, 31, 30, 28, 29, 33, 34, 35, 32, 42, 41, 39, 40, 48, 49, 50, 47, 53, 54, 55, 52, 57, 58, 59, 56, 63, 60, 61, 62, 67, 64, 65, 66
Node 1 is start node and node 2 is final node.
Those nodes are connect through the following edges:
- 1 to 3 labelled d_1(0)
- 1 to 7 labelled b_1(0)
- 1 to 6 labelled b_1(0), a_1(0)
- 1 to 11 labelled b_1(1)
- 1 to 15 labelled d_1(1)
- 1 to 18 labelled a_1(1)
- 1 to 19 labelled b_1(2)
- 1 to 23 labelled d_1(2)
- 1 to 27 labelled a_1(1)
- 1 to 28 labelled b_1(2)
- 1 to 47 labelled d_1(2)
- 1 to 32 labelled d_1(2)
- 2 to 2 labelled #_1(0)
- 6 to 2 labelled d_1(0)
- 6 to 18 labelled b_1(1), a_1(1)
- 6 to 19 labelled b_1(2)
- 6 to 23 labelled d_1(2)
- 6 to 47 labelled d_1(2)
- 6 to 35 labelled a_1(2)
- 6 to 39 labelled b_1(3)
- 6 to 52 labelled d_1(3)
- 6 to 32 labelled d_1(2)
- 3 to 4 labelled b_1(0)
- 4 to 5 labelled c_1(0)
- 5 to 6 labelled b_1(0)
- 5 to 18 labelled a_1(1)
- 5 to 19 labelled b_1(2)
- 5 to 23 labelled d_1(2)
- 5 to 27 labelled a_1(1)
- 5 to 28 labelled b_1(2)
- 5 to 47 labelled d_1(2)
- 5 to 32 labelled d_1(2)
- 10 to 2 labelled b_1(0)
- 10 to 18 labelled a_1(1)
- 10 to 19 labelled b_1(2)
- 10 to 23 labelled d_1(2)
- 10 to 47 labelled d_1(2)
- 10 to 32 labelled d_1(2)
- 9 to 10 labelled b_1(0)
- 9 to 27 labelled a_1(1)
- 9 to 28 labelled b_1(2)
- 9 to 32 labelled d_1(2)
- 9 to 18 labelled a_1(1)
- 9 to 19 labelled b_1(2)
- 9 to 23 labelled d_1(2)
- 9 to 47 labelled d_1(2)
- 7 to 8 labelled b_1(0)
- 8 to 9 labelled f_1(0)
- 14 to 6 labelled b_1(1)
- 14 to 18 labelled a_1(1)
- 14 to 19 labelled b_1(2)
- 14 to 23 labelled d_1(2)
- 14 to 35 labelled a_1(2)
- 14 to 39 labelled b_1(3)
- 14 to 47 labelled d_1(2)
- 14 to 52 labelled d_1(3)
- 14 to 32 labelled d_1(2)
- 13 to 14 labelled b_1(1)
- 13 to 35 labelled a_1(2)
- 13 to 39 labelled b_1(3)
- 13 to 52 labelled d_1(3)
- 11 to 12 labelled b_1(1)
- 12 to 13 labelled f_1(1)
- 18 to 2 labelled d_1(1)
- 18 to 18 labelled b_1(1), a_1(1)
- 18 to 19 labelled b_1(2)
- 18 to 23 labelled d_1(2), d_1(1)
- 18 to 47 labelled d_1(2), d_1(1)
- 18 to 35 labelled a_1(2)
- 18 to 39 labelled b_1(3)
- 18 to 52 labelled d_1(3), d_1(1)
- 18 to 53 labelled d_1(1)
- 18 to 32 labelled d_1(2), d_1(1)
- 18 to 33 labelled d_1(1)
- 15 to 16 labelled b_1(1)
- 16 to 17 labelled c_1(1)
- 17 to 18 labelled b_1(1), a_1(1)
- 17 to 19 labelled b_1(2)
- 17 to 23 labelled d_1(2)
- 17 to 47 labelled d_1(2)
- 17 to 35 labelled a_1(2)
- 17 to 39 labelled b_1(3)
- 17 to 52 labelled d_1(3)
- 17 to 32 labelled d_1(2)
- 22 to 18 labelled b_1(2), a_1(1)
- 22 to 19 labelled b_1(2)
- 22 to 23 labelled d_1(2)
- 22 to 47 labelled d_1(2)
- 22 to 35 labelled a_1(2)
- 22 to 55 labelled a_1(3)
- 22 to 39 labelled b_1(3)
- 22 to 56 labelled b_1(4)
- 22 to 52 labelled d_1(3)
- 22 to 60 labelled d_1(4)
- 22 to 64 labelled d_1(4)
- 22 to 32 labelled d_1(2)
- 21 to 22 labelled b_1(2)
- 21 to 55 labelled a_1(3)
- 21 to 56 labelled b_1(4)
- 21 to 60 labelled d_1(4)
- 21 to 64 labelled d_1(4)
- 19 to 20 labelled b_1(2)
- 20 to 21 labelled f_1(2)
- 24 to 25 labelled c_1(2)
- 25 to 26 labelled b_1(2)
- 25 to 18 labelled a_1(1)
- 25 to 19 labelled b_1(2)
- 25 to 23 labelled d_1(2)
- 25 to 47 labelled d_1(2)
- 25 to 55 labelled a_1(3)
- 25 to 56 labelled b_1(4)
- 25 to 64 labelled d_1(4)
- 25 to 60 labelled d_1(4)
- 25 to 32 labelled d_1(2)
- 26 to 2 labelled d_1(2)
- 26 to 18 labelled b_1(1), a_1(1)
- 26 to 19 labelled b_1(2)
- 26 to 47 labelled d_1(2)
- 26 to 23 labelled d_1(2)
- 26 to 35 labelled a_1(2)
- 26 to 39 labelled b_1(3)
- 26 to 52 labelled d_1(3)
- 26 to 32 labelled d_1(2)
- 23 to 24 labelled b_1(2)
- 27 to 24 labelled d_1(1)
- 27 to 48 labelled d_1(1)
- 31 to 27 labelled b_1(2)
- 30 to 31 labelled b_1(2)
- 28 to 29 labelled b_1(2)
- 29 to 30 labelled f_1(2)
- 33 to 34 labelled c_1(2)
- 34 to 35 labelled b_1(2)
- 35 to 24 labelled d_1(2)
- 35 to 53 labelled d_1(2)
- 35 to 48 labelled d_1(2)
- 35 to 33 labelled d_1(2)
- 32 to 33 labelled b_1(2)
- 42 to 35 labelled b_1(3)
- 41 to 42 labelled b_1(3)
- 39 to 40 labelled b_1(3)
- 40 to 41 labelled f_1(3)
- 48 to 49 labelled c_1(2)
- 49 to 50 labelled b_1(2)
- 49 to 55 labelled a_1(3)
- 49 to 56 labelled b_1(4)
- 49 to 64 labelled d_1(4)
- 49 to 60 labelled d_1(4)
- 50 to 23 labelled d_1(2)
- 50 to 52 labelled d_1(2)
- 47 to 48 labelled b_1(2)
- 53 to 54 labelled c_1(3)
- 54 to 55 labelled b_1(3)
- 55 to 24 labelled d_1(3)
- 55 to 48 labelled d_1(3)
- 55 to 53 labelled d_1(3)
- 55 to 61 labelled d_1(3)
- 55 to 65 labelled d_1(3)
- 55 to 33 labelled d_1(3)
- 52 to 53 labelled b_1(3)
- 57 to 58 labelled f_1(4)
- 58 to 59 labelled b_1(4)
- 59 to 55 labelled b_1(4)
- 56 to 57 labelled b_1(4)
- 63 to 24 labelled d_1(4)
- 60 to 61 labelled b_1(4)
- 61 to 62 labelled c_1(4)
- 62 to 63 labelled b_1(4)
- 67 to 48 labelled d_1(4)
- 67 to 61 labelled d_1(4)
- 67 to 53 labelled d_1(4)
- 67 to 65 labelled d_1(4)
- 67 to 33 labelled d_1(4)
- 64 to 65 labelled b_1(4)
- 65 to 66 labelled c_1(4)
- 66 to 67 labelled b_1(4)