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:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fac(x) → help(x, 0)
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fac(x) → help(x, 0)
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0)

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

HELP(x, c) → IF(lt(c, x), x, c)
LT(s(x), s(y)) → LT(x, y)
FAC(x) → HELP(x, 0)
HELP(x, c) → LT(c, x)
IF(true, x, c) → HELP(x, s(c))

The TRS R consists of the following rules:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fac(x) → help(x, 0)
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0)

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:

HELP(x, c) → IF(lt(c, x), x, c)
LT(s(x), s(y)) → LT(x, y)
FAC(x) → HELP(x, 0)
HELP(x, c) → LT(c, x)
IF(true, x, c) → HELP(x, s(c))

The TRS R consists of the following rules:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fac(x) → help(x, 0)
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 2 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP

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

LT(s(x), s(y)) → LT(x, y)

The TRS R consists of the following rules:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fac(x) → help(x, 0)
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


LT(s(x), s(y)) → LT(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(s(x1)) = 1 + (4)x_1   
POL(LT(x1, x2)) = (3)x_2   
The value of delta used in the strict ordering is 3.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP

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

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fac(x) → help(x, 0)
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0)

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

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

HELP(x, c) → IF(lt(c, x), x, c)
IF(true, x, c) → HELP(x, s(c))

The TRS R consists of the following rules:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fac(x) → help(x, 0)
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.