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:
*2(i1(x), x) -> 1
*2(1, y) -> y
*2(x, 0) -> 0
*2(*2(x, y), z) -> *2(x, *2(y, z))
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
*2(i1(x), x) -> 1
*2(1, y) -> y
*2(x, 0) -> 0
*2(*2(x, y), z) -> *2(x, *2(y, z))
Q is empty.
Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
*12(*2(x, y), z) -> *12(y, z)
*12(*2(x, y), z) -> *12(x, *2(y, z))
The TRS R consists of the following rules:
*2(i1(x), x) -> 1
*2(1, y) -> y
*2(x, 0) -> 0
*2(*2(x, y), z) -> *2(x, *2(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
*12(*2(x, y), z) -> *12(y, z)
*12(*2(x, y), z) -> *12(x, *2(y, z))
The TRS R consists of the following rules:
*2(i1(x), x) -> 1
*2(1, y) -> y
*2(x, 0) -> 0
*2(*2(x, y), z) -> *2(x, *2(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].
The following pairs can be strictly oriented and are deleted.
*12(*2(x, y), z) -> *12(y, z)
*12(*2(x, y), z) -> *12(x, *2(y, z))
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
*12(x1, x2) = *12(x1, x2)
*2(x1, x2) = *2(x1, x2)
i1(x1) = i1(x1)
1 = 1
0 = 0
Lexicographic Path Order [19].
Precedence:
[*^12, *2]
i1 > 1
The following usable rules [14] were oriented:
*2(i1(x), x) -> 1
*2(1, y) -> y
*2(x, 0) -> 0
*2(*2(x, y), z) -> *2(x, *2(y, z))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
Q DP problem:
P is empty.
The TRS R consists of the following rules:
*2(i1(x), x) -> 1
*2(1, y) -> y
*2(x, 0) -> 0
*2(*2(x, y), z) -> *2(x, *2(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.