Termination w.r.t. Q of the following Term Rewriting System could not be shown:
Q restricted rewrite system:
The TRS R consists of the following rules:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
Q is empty.
↳ QTRS
↳ Non-Overlap Check
Q restricted rewrite system:
The TRS R consists of the following rules:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
Q is empty.
The TRS is non-overlapping. Hence, we can switch to innermost.
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
The set Q consists of the following terms:
+2(x0, 0)
+2(x0, s1(x1))
*2(x0, 0)
*2(x0, s1(x1))
ge2(x0, 0)
ge2(0, s1(x0))
ge2(s1(x0), s1(x1))
-2(x0, 0)
-2(s1(x0), s1(x1))
fact1(x0)
iffact2(x0, true)
iffact2(x0, false)
Q DP problem:
The TRS P consists of the following rules:
IFFACT2(x, true) -> *12(x, fact1(-2(x, s1(0))))
*12(x, s1(y)) -> +12(*2(x, y), x)
+12(x, s1(y)) -> +12(x, y)
*12(x, s1(y)) -> *12(x, y)
FACT1(x) -> IFFACT2(x, ge2(x, s1(s1(0))))
-12(s1(x), s1(y)) -> -12(x, y)
IFFACT2(x, true) -> FACT1(-2(x, s1(0)))
IFFACT2(x, true) -> -12(x, s1(0))
FACT1(x) -> GE2(x, s1(s1(0)))
GE2(s1(x), s1(y)) -> GE2(x, y)
The TRS R consists of the following rules:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
The set Q consists of the following terms:
+2(x0, 0)
+2(x0, s1(x1))
*2(x0, 0)
*2(x0, s1(x1))
ge2(x0, 0)
ge2(0, s1(x0))
ge2(s1(x0), s1(x1))
-2(x0, 0)
-2(s1(x0), s1(x1))
fact1(x0)
iffact2(x0, true)
iffact2(x0, false)
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
IFFACT2(x, true) -> *12(x, fact1(-2(x, s1(0))))
*12(x, s1(y)) -> +12(*2(x, y), x)
+12(x, s1(y)) -> +12(x, y)
*12(x, s1(y)) -> *12(x, y)
FACT1(x) -> IFFACT2(x, ge2(x, s1(s1(0))))
-12(s1(x), s1(y)) -> -12(x, y)
IFFACT2(x, true) -> FACT1(-2(x, s1(0)))
IFFACT2(x, true) -> -12(x, s1(0))
FACT1(x) -> GE2(x, s1(s1(0)))
GE2(s1(x), s1(y)) -> GE2(x, y)
The TRS R consists of the following rules:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
The set Q consists of the following terms:
+2(x0, 0)
+2(x0, s1(x1))
*2(x0, 0)
*2(x0, s1(x1))
ge2(x0, 0)
ge2(0, s1(x0))
ge2(s1(x0), s1(x1))
-2(x0, 0)
-2(s1(x0), s1(x1))
fact1(x0)
iffact2(x0, true)
iffact2(x0, false)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 5 SCCs with 4 less nodes.
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ QDP
↳ 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:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
The set Q consists of the following terms:
+2(x0, 0)
+2(x0, s1(x1))
*2(x0, 0)
*2(x0, s1(x1))
ge2(x0, 0)
ge2(0, s1(x0))
ge2(s1(x0), s1(x1))
-2(x0, 0)
-2(s1(x0), s1(x1))
fact1(x0)
iffact2(x0, true)
iffact2(x0, false)
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: Precedence:
trivial
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
The TRS R consists of the following rules:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
The set Q consists of the following terms:
+2(x0, 0)
+2(x0, s1(x1))
*2(x0, 0)
*2(x0, s1(x1))
ge2(x0, 0)
ge2(0, s1(x0))
ge2(s1(x0), s1(x1))
-2(x0, 0)
-2(s1(x0), s1(x1))
fact1(x0)
iffact2(x0, true)
iffact2(x0, false)
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GE2(s1(x), s1(y)) -> GE2(x, y)
The TRS R consists of the following rules:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
The set Q consists of the following terms:
+2(x0, 0)
+2(x0, s1(x1))
*2(x0, 0)
*2(x0, s1(x1))
ge2(x0, 0)
ge2(0, s1(x0))
ge2(s1(x0), s1(x1))
-2(x0, 0)
-2(s1(x0), s1(x1))
fact1(x0)
iffact2(x0, true)
iffact2(x0, false)
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.
GE2(s1(x), s1(y)) -> GE2(x, y)
Used argument filtering: GE2(x1, x2) = x2
s1(x1) = s1(x1)
Used ordering: Precedence:
trivial
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
The TRS R consists of the following rules:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
The set Q consists of the following terms:
+2(x0, 0)
+2(x0, s1(x1))
*2(x0, 0)
*2(x0, s1(x1))
ge2(x0, 0)
ge2(0, s1(x0))
ge2(s1(x0), s1(x1))
-2(x0, 0)
-2(s1(x0), s1(x1))
fact1(x0)
iffact2(x0, true)
iffact2(x0, false)
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
+12(x, s1(y)) -> +12(x, y)
The TRS R consists of the following rules:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
The set Q consists of the following terms:
+2(x0, 0)
+2(x0, s1(x1))
*2(x0, 0)
*2(x0, s1(x1))
ge2(x0, 0)
ge2(0, s1(x0))
ge2(s1(x0), s1(x1))
-2(x0, 0)
-2(s1(x0), s1(x1))
fact1(x0)
iffact2(x0, true)
iffact2(x0, false)
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(x, s1(y)) -> +12(x, y)
Used argument filtering: +12(x1, x2) = x2
s1(x1) = s1(x1)
Used ordering: Precedence:
trivial
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
Q DP problem:
P is empty.
The TRS R consists of the following rules:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
The set Q consists of the following terms:
+2(x0, 0)
+2(x0, s1(x1))
*2(x0, 0)
*2(x0, s1(x1))
ge2(x0, 0)
ge2(0, s1(x0))
ge2(s1(x0), s1(x1))
-2(x0, 0)
-2(s1(x0), s1(x1))
fact1(x0)
iffact2(x0, true)
iffact2(x0, false)
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
*12(x, s1(y)) -> *12(x, y)
The TRS R consists of the following rules:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
The set Q consists of the following terms:
+2(x0, 0)
+2(x0, s1(x1))
*2(x0, 0)
*2(x0, s1(x1))
ge2(x0, 0)
ge2(0, s1(x0))
ge2(s1(x0), s1(x1))
-2(x0, 0)
-2(s1(x0), s1(x1))
fact1(x0)
iffact2(x0, true)
iffact2(x0, false)
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(x, s1(y)) -> *12(x, y)
Used argument filtering: *12(x1, x2) = x2
s1(x1) = s1(x1)
Used ordering: Precedence:
trivial
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
↳ QDP
Q DP problem:
P is empty.
The TRS R consists of the following rules:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
The set Q consists of the following terms:
+2(x0, 0)
+2(x0, s1(x1))
*2(x0, 0)
*2(x0, s1(x1))
ge2(x0, 0)
ge2(0, s1(x0))
ge2(s1(x0), s1(x1))
-2(x0, 0)
-2(s1(x0), s1(x1))
fact1(x0)
iffact2(x0, true)
iffact2(x0, false)
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
FACT1(x) -> IFFACT2(x, ge2(x, s1(s1(0))))
IFFACT2(x, true) -> FACT1(-2(x, s1(0)))
The TRS R consists of the following rules:
+2(x, 0) -> x
+2(x, s1(y)) -> s1(+2(x, y))
*2(x, 0) -> 0
*2(x, s1(y)) -> +2(*2(x, y), x)
ge2(x, 0) -> true
ge2(0, s1(y)) -> false
ge2(s1(x), s1(y)) -> ge2(x, y)
-2(x, 0) -> x
-2(s1(x), s1(y)) -> -2(x, y)
fact1(x) -> iffact2(x, ge2(x, s1(s1(0))))
iffact2(x, true) -> *2(x, fact1(-2(x, s1(0))))
iffact2(x, false) -> s1(0)
The set Q consists of the following terms:
+2(x0, 0)
+2(x0, s1(x1))
*2(x0, 0)
*2(x0, s1(x1))
ge2(x0, 0)
ge2(0, s1(x0))
ge2(s1(x0), s1(x1))
-2(x0, 0)
-2(s1(x0), s1(x1))
fact1(x0)
iffact2(x0, true)
iffact2(x0, false)
We have to consider all minimal (P,Q,R)-chains.