Term Rewriting System R:
[x, y]
app(app(app(f, app(g, x)), app(s, 0)), y) -> app(app(app(f, app(g, app(s, 0))), y), app(g, x))
app(g, app(s, x)) -> app(s, app(g, x))
app(g, 0) -> 0

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(f, app(g, x)), app(s, 0)), y) -> APP(app(app(f, app(g, app(s, 0))), y), app(g, x))
APP(app(app(f, app(g, x)), app(s, 0)), y) -> APP(app(f, app(g, app(s, 0))), y)
APP(app(app(f, app(g, x)), app(s, 0)), y) -> APP(f, app(g, app(s, 0)))
APP(app(app(f, app(g, x)), app(s, 0)), y) -> APP(g, app(s, 0))
APP(g, app(s, x)) -> APP(s, app(g, x))
APP(g, app(s, x)) -> APP(g, x)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)
       →DP Problem 2
         ↳UsableRules


Dependency Pair:

APP(g, app(s, x)) -> APP(g, x)


Rules:


app(app(app(f, app(g, x)), app(s, 0)), y) -> app(app(app(f, app(g, app(s, 0))), y), app(g, x))
app(g, app(s, x)) -> app(s, app(g, x))
app(g, 0) -> 0


Strategy:

innermost




As we are in the innermost case, we can delete all 3 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 3
             ↳A-Transformation
       →DP Problem 2
         ↳UsableRules


Dependency Pair:

APP(g, app(s, x)) -> APP(g, 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 3
             ↳ATrans
             ...
               →DP Problem 4
                 ↳Size-Change Principle
       →DP Problem 2
         ↳UsableRules


Dependency Pair:

G(s(x)) -> G(x)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. G(s(x)) -> G(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. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Usable Rules (Innermost)


Dependency Pair:

APP(app(app(f, app(g, x)), app(s, 0)), y) -> APP(app(app(f, app(g, app(s, 0))), y), app(g, x))


Rules:


app(app(app(f, app(g, x)), app(s, 0)), y) -> app(app(app(f, app(g, app(s, 0))), y), app(g, x))
app(g, app(s, x)) -> app(s, app(g, x))
app(g, 0) -> 0


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
         ↳UsableRules
           →DP Problem 5
             ↳A-Transformation


Dependency Pair:

APP(app(app(f, app(g, x)), app(s, 0)), y) -> APP(app(app(f, app(g, app(s, 0))), y), app(g, x))


Rules:


app(g, 0) -> 0
app(g, app(s, x)) -> app(s, app(g, x))


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
         ↳UsableRules
           →DP Problem 5
             ↳ATrans
             ...
               →DP Problem 6
                 ↳Narrowing Transformation


Dependency Pair:

F(g(x), s(0), y) -> F(g(s(0)), y, g(x))


Rules:


g(0) -> 0
g(s(x)) -> s(g(x))


Strategy:

innermost




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

F(g(x), s(0), y) -> F(g(s(0)), y, g(x))
one new Dependency Pair is created:

F(g(x), s(0), y) -> F(s(g(0)), y, g(x))

The transformation is resulting in no new DP problems.


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