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:
and2(x, or2(y, z)) -> or2(and2(x, y), and2(x, z))
and2(x, and2(y, y)) -> and2(x, y)
or2(or2(x, y), and2(y, z)) -> or2(x, y)
or2(x, and2(x, y)) -> x
or2(true, y) -> true
or2(x, false) -> x
or2(x, x) -> x
or2(x, or2(y, y)) -> or2(x, y)
and2(x, true) -> x
and2(false, y) -> false
and2(x, x) -> x
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
and2(x, or2(y, z)) -> or2(and2(x, y), and2(x, z))
and2(x, and2(y, y)) -> and2(x, y)
or2(or2(x, y), and2(y, z)) -> or2(x, y)
or2(x, and2(x, y)) -> x
or2(true, y) -> true
or2(x, false) -> x
or2(x, x) -> x
or2(x, or2(y, y)) -> or2(x, y)
and2(x, true) -> x
and2(false, y) -> false
and2(x, x) -> x
Q is empty.
Q DP problem:
The TRS P consists of the following rules:
AND2(x, or2(y, z)) -> OR2(and2(x, y), and2(x, z))
AND2(x, or2(y, z)) -> AND2(x, z)
AND2(x, or2(y, z)) -> AND2(x, y)
OR2(x, or2(y, y)) -> OR2(x, y)
AND2(x, and2(y, y)) -> AND2(x, y)
The TRS R consists of the following rules:
and2(x, or2(y, z)) -> or2(and2(x, y), and2(x, z))
and2(x, and2(y, y)) -> and2(x, y)
or2(or2(x, y), and2(y, z)) -> or2(x, y)
or2(x, and2(x, y)) -> x
or2(true, y) -> true
or2(x, false) -> x
or2(x, x) -> x
or2(x, or2(y, y)) -> or2(x, y)
and2(x, true) -> x
and2(false, y) -> false
and2(x, 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:
AND2(x, or2(y, z)) -> OR2(and2(x, y), and2(x, z))
AND2(x, or2(y, z)) -> AND2(x, z)
AND2(x, or2(y, z)) -> AND2(x, y)
OR2(x, or2(y, y)) -> OR2(x, y)
AND2(x, and2(y, y)) -> AND2(x, y)
The TRS R consists of the following rules:
and2(x, or2(y, z)) -> or2(and2(x, y), and2(x, z))
and2(x, and2(y, y)) -> and2(x, y)
or2(or2(x, y), and2(y, z)) -> or2(x, y)
or2(x, and2(x, y)) -> x
or2(true, y) -> true
or2(x, false) -> x
or2(x, x) -> x
or2(x, or2(y, y)) -> or2(x, y)
and2(x, true) -> x
and2(false, y) -> false
and2(x, x) -> x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 2 SCCs with 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
OR2(x, or2(y, y)) -> OR2(x, y)
The TRS R consists of the following rules:
and2(x, or2(y, z)) -> or2(and2(x, y), and2(x, z))
and2(x, and2(y, y)) -> and2(x, y)
or2(or2(x, y), and2(y, z)) -> or2(x, y)
or2(x, and2(x, y)) -> x
or2(true, y) -> true
or2(x, false) -> x
or2(x, x) -> x
or2(x, or2(y, y)) -> or2(x, y)
and2(x, true) -> x
and2(false, y) -> false
and2(x, 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.
OR2(x, or2(y, y)) -> OR2(x, y)
Used argument filtering: OR2(x1, x2) = x2
or2(x1, x2) = or1(x2)
Used ordering: Quasi Precedence:
trivial
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
↳ QDP
Q DP problem:
P is empty.
The TRS R consists of the following rules:
and2(x, or2(y, z)) -> or2(and2(x, y), and2(x, z))
and2(x, and2(y, y)) -> and2(x, y)
or2(or2(x, y), and2(y, z)) -> or2(x, y)
or2(x, and2(x, y)) -> x
or2(true, y) -> true
or2(x, false) -> x
or2(x, x) -> x
or2(x, or2(y, y)) -> or2(x, y)
and2(x, true) -> x
and2(false, y) -> false
and2(x, 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
Q DP problem:
The TRS P consists of the following rules:
AND2(x, or2(y, z)) -> AND2(x, z)
AND2(x, or2(y, z)) -> AND2(x, y)
AND2(x, and2(y, y)) -> AND2(x, y)
The TRS R consists of the following rules:
and2(x, or2(y, z)) -> or2(and2(x, y), and2(x, z))
and2(x, and2(y, y)) -> and2(x, y)
or2(or2(x, y), and2(y, z)) -> or2(x, y)
or2(x, and2(x, y)) -> x
or2(true, y) -> true
or2(x, false) -> x
or2(x, x) -> x
or2(x, or2(y, y)) -> or2(x, y)
and2(x, true) -> x
and2(false, y) -> false
and2(x, 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.
AND2(x, or2(y, z)) -> AND2(x, z)
AND2(x, or2(y, z)) -> AND2(x, y)
Used argument filtering: AND2(x1, x2) = x2
or2(x1, x2) = or2(x1, x2)
and2(x1, x2) = x2
Used ordering: Quasi Precedence:
trivial
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ QDPAfsSolverProof
Q DP problem:
The TRS P consists of the following rules:
AND2(x, and2(y, y)) -> AND2(x, y)
The TRS R consists of the following rules:
and2(x, or2(y, z)) -> or2(and2(x, y), and2(x, z))
and2(x, and2(y, y)) -> and2(x, y)
or2(or2(x, y), and2(y, z)) -> or2(x, y)
or2(x, and2(x, y)) -> x
or2(true, y) -> true
or2(x, false) -> x
or2(x, x) -> x
or2(x, or2(y, y)) -> or2(x, y)
and2(x, true) -> x
and2(false, y) -> false
and2(x, 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.
AND2(x, and2(y, y)) -> AND2(x, y)
Used argument filtering: AND2(x1, x2) = x2
and2(x1, x2) = and1(x2)
Used ordering: Quasi Precedence:
trivial
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
Q DP problem:
P is empty.
The TRS R consists of the following rules:
and2(x, or2(y, z)) -> or2(and2(x, y), and2(x, z))
and2(x, and2(y, y)) -> and2(x, y)
or2(or2(x, y), and2(y, z)) -> or2(x, y)
or2(x, and2(x, y)) -> x
or2(true, y) -> true
or2(x, false) -> x
or2(x, x) -> x
or2(x, or2(y, y)) -> or2(x, y)
and2(x, true) -> x
and2(false, y) -> false
and2(x, 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.