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:
f(s(x)) → s(f(f(p(s(x)))))
f(0) → 0
p(s(x)) → x
Q is empty.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(x)) → s(f(f(p(s(x)))))
f(0) → 0
p(s(x)) → x
Q is empty.
The TRS is overlay and locally confluent. By [19] we can switch to innermost.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(x)) → s(f(f(p(s(x)))))
f(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
f(s(x0))
f(0)
p(s(x0))
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
F(s(x)) → F(f(p(s(x))))
F(s(x)) → F(p(s(x)))
F(s(x)) → P(s(x))
The TRS R consists of the following rules:
f(s(x)) → s(f(f(p(s(x)))))
f(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
f(s(x0))
f(0)
p(s(x0))
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
F(s(x)) → F(f(p(s(x))))
F(s(x)) → F(p(s(x)))
F(s(x)) → P(s(x))
The TRS R consists of the following rules:
f(s(x)) → s(f(f(p(s(x)))))
f(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
f(s(x0))
f(0)
p(s(x0))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
F(s(x)) → F(f(p(s(x))))
F(s(x)) → F(p(s(x)))
The TRS R consists of the following rules:
f(s(x)) → s(f(f(p(s(x)))))
f(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
f(s(x0))
f(0)
p(s(x0))
We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [17] to decrease Q to the empty set.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
F(s(x)) → F(f(p(s(x))))
F(s(x)) → F(p(s(x)))
The TRS R consists of the following rules:
f(s(x)) → s(f(f(p(s(x)))))
f(0) → 0
p(s(x)) → x
Q is empty.
We have to consider all (P,Q,R)-chains.
We have reversed the following QTRS:
The set of rules R is
f(s(x)) → s(f(f(p(s(x)))))
f(0) → 0
p(s(x)) → x
The set Q is empty.
We have obtained the following QTRS:
s(f(x)) → s(p(f(f(s(x)))))
0'(f(x)) → 0'(x)
s(p(x)) → x
The set Q is empty.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS Reverse
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
s(f(x)) → s(p(f(f(s(x)))))
0'(f(x)) → 0'(x)
s(p(x)) → x
Q is empty.
Termination of the TRS R could be shown with a Match Bound [6,7] of 2. This implies Q-termination of R.
The following rules were used to construct the certificate:
s(f(x)) → s(p(f(f(s(x)))))
0'(f(x)) → 0'(x)
s(p(x)) → x
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
113, 114, 115, 116, 118, 117, 119, 120, 122, 121
Node 113 is start node and node 114 is final node.
Those nodes are connect through the following edges:
- 113 to 114 labelled p_1(0), 0'_1(0), f_1(0), s_1(0), p_1(1), f_1(1), s_1(1), 0'_1(1)
- 113 to 115 labelled s_1(0)
- 113 to 119 labelled s_1(1)
- 113 to 117 labelled f_1(1)
- 113 to 121 labelled f_1(2)
- 114 to 114 labelled #_1(0)
- 115 to 116 labelled p_1(0)
- 116 to 117 labelled f_1(0)
- 118 to 114 labelled s_1(0), p_1(1), f_1(1), s_1(1), 0'_1(1)
- 118 to 119 labelled s_1(1)
- 118 to 121 labelled f_1(2)
- 117 to 118 labelled f_1(0)
- 119 to 120 labelled p_1(1)
- 120 to 121 labelled f_1(1)
- 122 to 114 labelled s_1(1), p_1(1), f_1(1), 0'_1(1)
- 122 to 119 labelled s_1(1)
- 122 to 121 labelled f_1(2)
- 121 to 122 labelled f_1(1)