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:
h(f(x), y) → f(g(x, y))
g(x, y) → h(x, y)
Q is empty.
↳ QTRS
↳ Overlay + Local Confluence
Q restricted rewrite system:
The TRS R consists of the following rules:
h(f(x), y) → f(g(x, y))
g(x, y) → h(x, y)
Q is empty.
The TRS is overlay and locally confluent. By [15] we can switch to innermost.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
h(f(x), y) → f(g(x, y))
g(x, y) → h(x, y)
The set Q consists of the following terms:
h(f(x0), x1)
g(x0, x1)
Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
G(x, y) → H(x, y)
H(f(x), y) → G(x, y)
The TRS R consists of the following rules:
h(f(x), y) → f(g(x, y))
g(x, y) → h(x, y)
The set Q consists of the following terms:
h(f(x0), x1)
g(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
Q DP problem:
The TRS P consists of the following rules:
G(x, y) → H(x, y)
H(f(x), y) → G(x, y)
The TRS R consists of the following rules:
h(f(x), y) → f(g(x, y))
g(x, y) → h(x, y)
The set Q consists of the following terms:
h(f(x0), x1)
g(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
G(x, y) → H(x, y)
H(f(x), y) → G(x, y)
The TRS R consists of the following rules:
h(f(x), y) → f(g(x, y))
g(x, y) → h(x, y)
The set Q consists of the following terms:
h(f(x0), x1)
g(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].
The following pairs can be oriented strictly and are deleted.
H(f(x), y) → G(x, y)
The remaining pairs can at least be oriented weakly.
G(x, y) → H(x, y)
Used ordering: Combined order from the following AFS and order.
G(x1, x2) = x1
H(x1, x2) = x1
f(x1) = f(x1)
Lexicographic Path Order [19].
Precedence:
trivial
The following usable rules [14] were oriented:
none
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
G(x, y) → H(x, y)
The TRS R consists of the following rules:
h(f(x), y) → f(g(x, y))
g(x, y) → h(x, y)
The set Q consists of the following terms:
h(f(x0), x1)
g(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 1 less node.