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:
double1(0) -> 0
double1(s1(x)) -> s1(s1(double1(x)))
half1(0) -> 0
half1(s1(0)) -> 0
half1(s1(s1(x))) -> s1(half1(x))
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
if3(0, y, z) -> y
if3(s1(x), y, z) -> z
half1(double1(x)) -> x
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
double1(0) -> 0
double1(s1(x)) -> s1(s1(double1(x)))
half1(0) -> 0
half1(s1(0)) -> 0
half1(s1(s1(x))) -> s1(half1(x))
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
if3(0, y, z) -> y
if3(s1(x), y, z) -> z
half1(double1(x)) -> x
Q is empty.
Q DP problem:
The TRS P consists of the following rules:
DOUBLE1(s1(x)) -> DOUBLE1(x)
-12(s1(x), s1(y)) -> -12(x, y)
HALF1(s1(s1(x))) -> HALF1(x)
The TRS R consists of the following rules:
double1(0) -> 0
double1(s1(x)) -> s1(s1(double1(x)))
half1(0) -> 0
half1(s1(0)) -> 0
half1(s1(s1(x))) -> s1(half1(x))
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
if3(0, y, z) -> y
if3(s1(x), y, z) -> z
half1(double1(x)) -> x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
DOUBLE1(s1(x)) -> DOUBLE1(x)
-12(s1(x), s1(y)) -> -12(x, y)
HALF1(s1(s1(x))) -> HALF1(x)
The TRS R consists of the following rules:
double1(0) -> 0
double1(s1(x)) -> s1(s1(double1(x)))
half1(0) -> 0
half1(s1(0)) -> 0
half1(s1(s1(x))) -> s1(half1(x))
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
if3(0, y, z) -> y
if3(s1(x), y, z) -> z
half1(double1(x)) -> x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 3 SCCs.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
-12(s1(x), s1(y)) -> -12(x, y)
The TRS R consists of the following rules:
double1(0) -> 0
double1(s1(x)) -> s1(s1(double1(x)))
half1(0) -> 0
half1(s1(0)) -> 0
half1(s1(s1(x))) -> s1(half1(x))
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
if3(0, y, z) -> y
if3(s1(x), y, z) -> z
half1(double1(x)) -> x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.
-12(s1(x), s1(y)) -> -12(x, y)
Used argument filtering: -12(x1, x2) = x2
s1(x1) = s1(x1)
Used ordering: Quasi Precedence:
trivial
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
Q DP problem:
P is empty.
The TRS R consists of the following rules:
double1(0) -> 0
double1(s1(x)) -> s1(s1(double1(x)))
half1(0) -> 0
half1(s1(0)) -> 0
half1(s1(s1(x))) -> s1(half1(x))
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
if3(0, y, z) -> y
if3(s1(x), y, z) -> z
half1(double1(x)) -> x
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.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
HALF1(s1(s1(x))) -> HALF1(x)
The TRS R consists of the following rules:
double1(0) -> 0
double1(s1(x)) -> s1(s1(double1(x)))
half1(0) -> 0
half1(s1(0)) -> 0
half1(s1(s1(x))) -> s1(half1(x))
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
if3(0, y, z) -> y
if3(s1(x), y, z) -> z
half1(double1(x)) -> x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.
HALF1(s1(s1(x))) -> HALF1(x)
Used argument filtering: HALF1(x1) = x1
s1(x1) = s1(x1)
Used ordering: Quasi Precedence:
trivial
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
↳ QDP
Q DP problem:
P is empty.
The TRS R consists of the following rules:
double1(0) -> 0
double1(s1(x)) -> s1(s1(double1(x)))
half1(0) -> 0
half1(s1(0)) -> 0
half1(s1(s1(x))) -> s1(half1(x))
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
if3(0, y, z) -> y
if3(s1(x), y, z) -> z
half1(double1(x)) -> x
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.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
Q DP problem:
The TRS P consists of the following rules:
DOUBLE1(s1(x)) -> DOUBLE1(x)
The TRS R consists of the following rules:
double1(0) -> 0
double1(s1(x)) -> s1(s1(double1(x)))
half1(0) -> 0
half1(s1(0)) -> 0
half1(s1(s1(x))) -> s1(half1(x))
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
if3(0, y, z) -> y
if3(s1(x), y, z) -> z
half1(double1(x)) -> x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.
DOUBLE1(s1(x)) -> DOUBLE1(x)
Used argument filtering: DOUBLE1(x1) = x1
s1(x1) = s1(x1)
Used ordering: Quasi Precedence:
trivial
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
Q DP problem:
P is empty.
The TRS R consists of the following rules:
double1(0) -> 0
double1(s1(x)) -> s1(s1(double1(x)))
half1(0) -> 0
half1(s1(0)) -> 0
half1(s1(s1(x))) -> s1(half1(x))
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
if3(0, y, z) -> y
if3(s1(x), y, z) -> z
half1(double1(x)) -> x
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.