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
                 ↳Argument Filtering and 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 that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
FAC(x1) -> FAC(x1)
s(x1) -> s(x1)


   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