Term Rewriting System R:
[x, y]
app(intlist, nil) -> nil
app(intlist, app(app(cons, x), y)) -> app(app(cons, app(s, x)), app(intlist, y))
app(app(int, 0), 0) -> app(app(cons, 0), nil)
app(app(int, 0), app(s, y)) -> app(app(cons, 0), app(app(int, app(s, 0)), app(s, y)))
app(app(int, app(s, x)), 0) -> nil
app(app(int, app(s, x)), app(s, y)) -> app(intlist, app(app(int, x), y))

Innermost Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

app(app(int, 0), 0) -> app(app(cons, 0), nil)
app(app(int, app(s, x)), 0) -> nil

where the Polynomial interpretation:
  POL(intlist)=  0  
  POL(0)=  0  
  POL(cons)=  0  
  POL(int)=  1  
  POL(nil)=  0  
  POL(s)=  0  
  POL(app(x1, x2))=  x1 + x2  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(int, app(s, x)), app(s, y)) -> APP(intlist, app(app(int, x), y))
APP(app(int, app(s, x)), app(s, y)) -> APP(app(int, x), y)
APP(app(int, app(s, x)), app(s, y)) -> APP(int, x)
APP(intlist, app(app(cons, x), y)) -> APP(app(cons, app(s, x)), app(intlist, y))
APP(intlist, app(app(cons, x), y)) -> APP(cons, app(s, x))
APP(intlist, app(app(cons, x), y)) -> APP(s, x)
APP(intlist, app(app(cons, x), y)) -> APP(intlist, y)
APP(app(int, 0), app(s, y)) -> APP(app(cons, 0), app(app(int, app(s, 0)), app(s, y)))
APP(app(int, 0), app(s, y)) -> APP(cons, 0)
APP(app(int, 0), app(s, y)) -> APP(app(int, app(s, 0)), app(s, y))
APP(app(int, 0), app(s, y)) -> APP(int, app(s, 0))
APP(app(int, 0), app(s, y)) -> APP(s, 0)

Furthermore, R contains two SCCs. 


   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Usable Rules (Innermost)
           →DP Problem 2
             ↳UsableRules


Dependency Pair:

APP(intlist, app(app(cons, x), y)) -> APP(intlist, y)


Rules:


app(intlist, nil) -> nil
app(app(int, app(s, x)), app(s, y)) -> app(intlist, app(app(int, x), y))
app(intlist, app(app(cons, x), y)) -> app(app(cons, app(s, x)), app(intlist, y))
app(app(int, 0), app(s, y)) -> app(app(cons, 0), app(app(int, app(s, 0)), app(s, y)))


Strategy:

innermost




As we are in the innermost case, we can delete all 4 non-usable-rules.


   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 3
                 ↳A-Transformation
           →DP Problem 2
             ↳UsableRules


Dependency Pair:

APP(intlist, app(app(cons, x), y)) -> APP(intlist, 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
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 4
                 ↳Size-Change Principle
           →DP Problem 2
             ↳UsableRules


Dependency Pair:

INTLIST(cons(x, y)) -> INTLIST(y)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. INTLIST(cons(x, y)) -> INTLIST(y)
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:
cons(x1, x2) -> cons(x1, x2)

We obtain no new DP problems. 


   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
           →DP Problem 2
             ↳Usable Rules (Innermost)


Dependency Pairs:

APP(app(int, 0), app(s, y)) -> APP(app(int, app(s, 0)), app(s, y))
APP(app(int, app(s, x)), app(s, y)) -> APP(app(int, x), y)


Rules:


app(intlist, nil) -> nil
app(app(int, app(s, x)), app(s, y)) -> app(intlist, app(app(int, x), y))
app(intlist, app(app(cons, x), y)) -> app(app(cons, app(s, x)), app(intlist, y))
app(app(int, 0), app(s, y)) -> app(app(cons, 0), app(app(int, app(s, 0)), app(s, y)))


Strategy:

innermost




As we are in the innermost case, we can delete all 4 non-usable-rules.


   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
           →DP Problem 2
             ↳UsableRules
             ...
               →DP Problem 5
                 ↳A-Transformation


Dependency Pairs:

APP(app(int, 0), app(s, y)) -> APP(app(int, app(s, 0)), app(s, y))
APP(app(int, app(s, x)), app(s, y)) -> APP(app(int, 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
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
           →DP Problem 2
             ↳UsableRules
             ...
               →DP Problem 6
                 ↳Size-Change Principle


Dependency Pairs:

INT(0, s(y)) -> INT(s(0), s(y))
INT(s(x), s(y)) -> INT(x, y)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. INT(0, s(y)) -> INT(s(0), s(y))
  2. INT(s(x), s(y)) -> INT(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
2=2
{2} , {2}
1>1
2>2

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

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.

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