Term Rewriting System R:
[f, x, xs, y]
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) -> x
app(app(div, 0), app(s, y)) -> 0
app(app(div, app(s, x)), app(s, y)) -> app(s, app(app(div, app(app(minus, x), y)), app(s, y)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(cons, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, app(p, app(s, x))), app(p, app(s, y)))
APP(app(minus, app(s, x)), app(s, y)) -> APP(minus, app(p, app(s, x)))
APP(app(minus, app(s, x)), app(s, y)) -> APP(p, app(s, x))
APP(app(minus, app(s, x)), app(s, y)) -> APP(p, app(s, y))
APP(app(div, app(s, x)), app(s, y)) -> APP(s, app(app(div, app(app(minus, x), y)), app(s, y)))
APP(app(div, app(s, x)), app(s, y)) -> APP(app(div, app(app(minus, x), y)), app(s, y))
APP(app(div, app(s, x)), app(s, y)) -> APP(div, app(app(minus, x), y))
APP(app(div, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(div, app(s, x)), app(s, y)) -> APP(minus, x)

Furthermore, R contains three SCCs. 


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


Dependency Pair:

APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, app(p, app(s, x))), app(p, app(s, y)))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) -> x
app(app(div, 0), app(s, y)) -> 0
app(app(div, app(s, x)), app(s, y)) -> app(s, app(app(div, app(app(minus, x), y)), app(s, y)))


Strategy:

innermost




As we are in the innermost case, we can delete all 6 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 4
             ↳A-Transformation
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules


Dependency Pair:

APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, app(p, app(s, x))), app(p, app(s, y)))


Rule:


app(p, app(s, x)) -> 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 4
             ↳ATrans
             ...
               →DP Problem 5
                 ↳Modular Removal of Rules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules


Dependency Pair:

MINUS(s(x), s(y)) -> MINUS(p(s(x)), p(s(y)))


Rule:


p(s(x)) -> x


Strategy:

innermost




We have the following set of usable rules:

p(s(x)) -> x
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
  POL(MINUS(x1, x2))=  1 + x1 + x2  
  POL(s(x1))=  1 + x1  
  POL(p(x1))=  x1  

We have the following set D of usable symbols: {MINUS, s, p}
No Dependency Pairs can be deleted.
The following rules can be deleted as the lhs is strictly greater than the corresponding rhs:

p(s(x)) -> x


The result of this processor delivers one new DP problem.



   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 4
             ↳ATrans
             ...
               →DP Problem 6
                 ↳Dependency Graph
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules


Dependency Pair:

MINUS(s(x), s(y)) -> MINUS(p(s(x)), p(s(y)))


Rule:

none


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

APP(app(div, app(s, x)), app(s, y)) -> APP(app(div, app(app(minus, x), y)), app(s, y))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) -> x
app(app(div, 0), app(s, y)) -> 0
app(app(div, app(s, x)), app(s, y)) -> app(s, app(app(div, app(app(minus, x), y)), app(s, y)))


Strategy:

innermost




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


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 7
             ↳A-Transformation
       →DP Problem 3
         ↳UsableRules


Dependency Pair:

APP(app(div, app(s, x)), app(s, y)) -> APP(app(div, app(app(minus, x), y)), app(s, y))


Rules:


app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) -> 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 7
             ↳ATrans
             ...
               →DP Problem 8
                 ↳Negative Polynomial Order
       →DP Problem 3
         ↳UsableRules


Dependency Pair:

DIV(s(x), s(y)) -> DIV(minus(x, y), s(y))


Rules:


minus(x, 0) -> x
minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
p(s(x)) -> x


Strategy:

innermost




The following Dependency Pair can be strictly oriented using the given order.

DIV(s(x), s(y)) -> DIV(minus(x, y), s(y))


Moreover, the following usable rules (regarding the implicit AFS) are oriented.

minus(x, 0) -> x
p(s(x)) -> x
minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))


Used ordering:
Polynomial Order with Interpretation:

POL( DIV(x1, x2) ) = x1

POL( s(x1) ) = x1 + 1

POL( minus(x1, x2) ) = x1

POL( p(x1) ) = x1


This results in one new DP problem. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 7
             ↳ATrans
             ...
               →DP Problem 9
                 ↳Dependency Graph
       →DP Problem 3
         ↳UsableRules


Dependency Pair:


Rules:


minus(x, 0) -> x
minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
p(s(x)) -> x


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pairs:

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) -> x
app(app(div, 0), app(s, y)) -> 0
app(app(div, app(s, x)), app(s, y)) -> app(s, app(app(div, app(app(minus, x), y)), app(s, y)))


Strategy:

innermost




As we are in the innermost case, we can delete all 7 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
           →DP Problem 10
             ↳Size-Change Principle


Dependency Pairs:

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
  2. APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1=1
2>2
{1, 2} , {1, 2}
1>1
2>2

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

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
app(x1, x2) -> app(x1, x2)

We obtain no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes