Term Rewriting System R:
[x]
p(s(x)) -> x
fac(0) -> s(0)
fac(s(x)) -> times(s(x), fac(p(s(x))))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

FAC(s(x)) -> FAC(p(s(x)))
FAC(s(x)) -> P(s(x))

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Rewriting Transformation


Dependency Pair:

FAC(s(x)) -> FAC(p(s(x)))


Rules:


p(s(x)) -> x
fac(0) -> s(0)
fac(s(x)) -> times(s(x), fac(p(s(x))))


Strategy:

innermost




On this DP problem, a Rewriting SCC transformation can be performed.
As a result of transforming the rule

FAC(s(x)) -> FAC(p(s(x)))
one new Dependency Pair is created:

FAC(s(x)) -> FAC(x)

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Rw
           →DP Problem 2
Forward Instantiation Transformation


Dependency Pair:

FAC(s(x)) -> FAC(x)


Rules:


p(s(x)) -> x
fac(0) -> s(0)
fac(s(x)) -> times(s(x), fac(p(s(x))))


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FAC(s(x)) -> FAC(x)
one new Dependency Pair is created:

FAC(s(s(x''))) -> FAC(s(x''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Rw
           →DP Problem 2
FwdInst
             ...
               →DP Problem 3
Polynomial Ordering


Dependency Pair:

FAC(s(s(x''))) -> FAC(s(x''))


Rules:


p(s(x)) -> x
fac(0) -> s(0)
fac(s(x)) -> times(s(x), fac(p(s(x))))


Strategy:

innermost




The following dependency pair can be strictly oriented:

FAC(s(s(x''))) -> FAC(s(x''))


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FAC(x1))=  1 + x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Rw
           →DP Problem 2
FwdInst
             ...
               →DP Problem 4
Dependency Graph


Dependency Pair:


Rules:


p(s(x)) -> x
fac(0) -> s(0)
fac(s(x)) -> times(s(x), fac(p(s(x))))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes