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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(f, x), app(c, x)), app(c, y)) -> APP(app(app(f, y), y), app(app(app(f, y), x), y))
APP(app(app(f, x), app(c, x)), app(c, y)) -> APP(app(f, y), y)
APP(app(app(f, x), app(c, x)), app(c, y)) -> APP(f, y)
APP(app(app(f, x), app(c, x)), app(c, y)) -> APP(app(app(f, y), x), y)
APP(app(app(f, x), app(c, x)), app(c, y)) -> APP(app(f, y), x)
APP(app(app(f, app(s, x)), y), z) -> APP(app(app(f, x), app(s, app(c, y))), app(c, z))
APP(app(app(f, app(s, x)), y), z) -> APP(app(f, x), app(s, app(c, y)))
APP(app(app(f, app(s, x)), y), z) -> APP(f, x)
APP(app(app(f, app(s, x)), y), z) -> APP(s, app(c, y))
APP(app(app(f, app(s, x)), y), z) -> APP(c, y)
APP(app(app(f, app(s, x)), y), z) -> APP(c, z)
APP(app(app(f, app(c, x)), x), y) -> APP(c, y)

Furthermore, R contains two SCCs. 


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


Dependency Pair:

APP(app(app(f, app(s, x)), y), z) -> APP(app(app(f, x), app(s, app(c, y))), app(c, z))


Rules:


app(app(app(f, x), app(c, x)), app(c, y)) -> app(app(app(f, y), y), app(app(app(f, y), x), y))
app(app(app(f, app(s, x)), y), z) -> app(app(app(f, x), app(s, app(c, y))), app(c, z))
app(app(app(f, app(c, x)), x), y) -> app(c, y)
app(app(g, x), y) -> x
app(app(g, x), y) -> y


Strategy:

innermost




As we are in the innermost case, we can delete all 5 non-usable-rules.


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


Dependency Pair:

APP(app(app(f, app(s, x)), y), z) -> APP(app(app(f, x), app(s, app(c, y))), app(c, z))


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:

F(s(x), y, z) -> F(x, s(c(y)), c(z))


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. F(s(x), y, z) -> F(x, s(c(y)), c(z))
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:
c(x1) -> c(x1)
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, x), app(c, x)), app(c, y)) -> APP(app(app(f, y), x), y)


Rules:


app(app(app(f, x), app(c, x)), app(c, y)) -> app(app(app(f, y), y), app(app(app(f, y), x), y))
app(app(app(f, app(s, x)), y), z) -> app(app(app(f, x), app(s, app(c, y))), app(c, z))
app(app(app(f, app(c, x)), x), y) -> app(c, y)
app(app(g, x), y) -> x
app(app(g, x), y) -> y


Strategy:

innermost




As we are in the innermost case, we can delete all 5 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 5
             ↳A-Transformation


Dependency Pair:

APP(app(app(f, x), app(c, x)), app(c, y)) -> APP(app(app(f, y), x), y)


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
         ↳UsableRules
           →DP Problem 5
             ↳ATrans
             ...
               →DP Problem 6
                 ↳Size-Change Principle


Dependency Pair:

F(x, c(x), c(y)) -> F(y, x, y)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. F(x, c(x), c(y)) -> F(y, x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1=2
2>2
3>1
3>3

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>2
2>2
3>1
3>2
3>3

DP: empty set
Oriented Rules: none

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

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

We obtain no new DP problems.

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