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:

half1(0) -> 0
half1(s1(s1(x))) -> s1(half1(x))
log1(s1(0)) -> 0
log1(s1(s1(x))) -> s1(log1(s1(half1(x))))

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

Q restricted rewrite system:
The TRS R consists of the following rules:

half1(0) -> 0
half1(s1(s1(x))) -> s1(half1(x))
log1(s1(0)) -> 0
log1(s1(s1(x))) -> s1(log1(s1(half1(x))))

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:

half1(0) -> 0
half1(s1(s1(x))) -> s1(half1(x))
log1(s1(0)) -> 0
log1(s1(s1(x))) -> s1(log1(s1(half1(x))))

The set Q consists of the following terms:

half1(0)
half1(s1(s1(x0)))
log1(s1(0))
log1(s1(s1(x0)))


Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

LOG1(s1(s1(x))) -> HALF1(x)
LOG1(s1(s1(x))) -> LOG1(s1(half1(x)))
HALF1(s1(s1(x))) -> HALF1(x)

The TRS R consists of the following rules:

half1(0) -> 0
half1(s1(s1(x))) -> s1(half1(x))
log1(s1(0)) -> 0
log1(s1(s1(x))) -> s1(log1(s1(half1(x))))

The set Q consists of the following terms:

half1(0)
half1(s1(s1(x0)))
log1(s1(0))
log1(s1(s1(x0)))

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:

LOG1(s1(s1(x))) -> HALF1(x)
LOG1(s1(s1(x))) -> LOG1(s1(half1(x)))
HALF1(s1(s1(x))) -> HALF1(x)

The TRS R consists of the following rules:

half1(0) -> 0
half1(s1(s1(x))) -> s1(half1(x))
log1(s1(0)) -> 0
log1(s1(s1(x))) -> s1(log1(s1(half1(x))))

The set Q consists of the following terms:

half1(0)
half1(s1(s1(x0)))
log1(s1(0))
log1(s1(s1(x0)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 1 less node.

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPOrderProof
              ↳ 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:

half1(0) -> 0
half1(s1(s1(x))) -> s1(half1(x))
log1(s1(0)) -> 0
log1(s1(s1(x))) -> s1(log1(s1(half1(x))))

The set Q consists of the following terms:

half1(0)
half1(s1(s1(x0)))
log1(s1(0))
log1(s1(s1(x0)))

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.


HALF1(s1(s1(x))) -> HALF1(x)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
HALF1(x1)  =  HALF1(x1)
s1(x1)  =  s1(x1)

Lexicographic Path Order [19].
Precedence:
[HALF1, s1]


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
              ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

half1(0) -> 0
half1(s1(s1(x))) -> s1(half1(x))
log1(s1(0)) -> 0
log1(s1(s1(x))) -> s1(log1(s1(half1(x))))

The set Q consists of the following terms:

half1(0)
half1(s1(s1(x0)))
log1(s1(0))
log1(s1(s1(x0)))

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
                ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

LOG1(s1(s1(x))) -> LOG1(s1(half1(x)))

The TRS R consists of the following rules:

half1(0) -> 0
half1(s1(s1(x))) -> s1(half1(x))
log1(s1(0)) -> 0
log1(s1(s1(x))) -> s1(log1(s1(half1(x))))

The set Q consists of the following terms:

half1(0)
half1(s1(s1(x0)))
log1(s1(0))
log1(s1(s1(x0)))

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.


LOG1(s1(s1(x))) -> LOG1(s1(half1(x)))
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
LOG1(x1)  =  LOG1(x1)
s1(x1)  =  s1(x1)
half1(x1)  =  half1(x1)
0  =  0

Lexicographic Path Order [19].
Precedence:
s1 > LOG1
s1 > half1 > 0


The following usable rules [14] were oriented:

half1(0) -> 0
half1(s1(s1(x))) -> s1(half1(x))



↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

half1(0) -> 0
half1(s1(s1(x))) -> s1(half1(x))
log1(s1(0)) -> 0
log1(s1(s1(x))) -> s1(log1(s1(half1(x))))

The set Q consists of the following terms:

half1(0)
half1(s1(s1(x0)))
log1(s1(0))
log1(s1(s1(x0)))

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.