Termination w.r.t. Q of the given QTRS could not be shown:

(0) Obligation:

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.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(2) Obligation:

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)

The set Q consists of the following terms:

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fac(x0)
help(x0, x1)
if(true, x0, x1)
if(false, x0, x1)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

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

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

The set Q consists of the following terms:

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fac(x0)
help(x0, x1)
if(true, x0, x1)
if(false, x0, x1)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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)

The set Q consists of the following terms:

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fac(x0)
help(x0, x1)
if(true, x0, x1)
if(false, x0, x1)

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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.
Used ordering: Combined order from the following AFS and order.
LT(x1, x2)  =  x2
s(x1)  =  s(x1)
lt(x1, x2)  =  lt(x1, x2)
0  =  0
true  =  true
false  =  false
fac(x1)  =  fac(x1)
help(x1, x2)  =  help(x1, x2)
if(x1, x2, x3)  =  if(x2)
times(x1, x2)  =  times

Recursive path order with status [RPO].
Quasi-Precedence:

fac₁ > [s₁, lt₂, 0, true, false, help₂, if₁, times]

Status:

s₁: [1]
lt₂: [1,2]
0: multiset
true: multiset
false: multiset
fac₁: multiset
help₂: [1,2]
if₁: [1]
times: multiset

The following usable rules [FROCOS05] were oriented:

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)

(9) Obligation:

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)

The set Q consists of the following terms:

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fac(x0)
help(x0, x1)
if(true, x0, x1)
if(false, x0, x1)

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

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

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)

The set Q consists of the following terms:

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fac(x0)
help(x0, x1)
if(true, x0, x1)
if(false, x0, x1)

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