Term Rewriting System R:
[x]
+(1, x) -> +(+(0, 1), x)
+(0, x) -> x

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

+'(1, x) -> +'(+(0, 1), x)
+'(1, x) -> +'(0, 1)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Usable Rules (Innermost)


Dependency Pair:

+'(1, x) -> +'(+(0, 1), x)


Rules:


+(1, x) -> +(+(0, 1), x)
+(0, x) -> x


Strategy:

innermost




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


   R
DPs
       →DP Problem 1
UsableRules
           →DP Problem 2
Rewriting Transformation


Dependency Pair:

+'(1, x) -> +'(+(0, 1), x)


Rule:


+(0, x) -> x


Strategy:

innermost




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

+'(1, x) -> +'(+(0, 1), x)
one new Dependency Pair is created:

+'(1, x) -> +'(1, x)

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
UsableRules
           →DP Problem 2
Rw
             ...
               →DP Problem 3
Usable Rules (Innermost)


Dependency Pair:

+'(1, x) -> +'(1, x)


Rule:


+(0, x) -> x


Strategy:

innermost




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


   R
DPs
       →DP Problem 1
UsableRules
           →DP Problem 2
Rw
             ...
               →DP Problem 4
A-Transformation


Dependency Pair:

+'(1, x) -> +'(1, x)


Rule:

none


Strategy:

innermost




We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.


   R
DPs
       →DP Problem 1
UsableRules
           →DP Problem 2
Rw
             ...
               →DP Problem 5
Non Termination


Dependency Pair:

1'(x) -> 1'(x)


Rule:

none


Strategy:

innermost




Found an infinite P-chain over R:
P =

1'(x) -> 1'(x)

R = none

s = 1'(x')
evaluates to t =1'(x')

Thus, s starts an infinite chain.

Innermost Non-Termination of R could be shown.
Duration:
0:03 minutes