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

Termination of R to be shown. 



   R
     ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳OC
       →TRS2
         ↳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
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Usable Rules (Innermost)


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




As we are in the innermost case, we can delete all 2 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 2
                 ↳Rewriting Transformation


Dependency Pair:

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


Rule:


p(s(x)) -> 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
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 3
                 ↳Usable Rules (Innermost)


Dependency Pair:

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


Rule:


p(s(x)) -> x


Strategy:

innermost




As we are in the innermost case, we can delete all 1 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 4
                 ↳Size-Change Principle


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. FAC(s(x)) -> FAC(x)
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems.

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