Term Rewriting System R:
[y, x, f, xs, g]
app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), 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(app(curry, g), x), y) -> app(app(g, x), y)
inc -> app(map, app(app(curry, plus), app(s, 0)))

Termination of R to be shown.



   R
Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.


   R
OC
       →TRS2
Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(plus, app(s, x)), y) -> APP(s, app(app(plus, x), y))
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) -> APP(plus, x)
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(app(curry, g), x), y) -> APP(app(g, x), y)
APP(app(app(curry, g), x), y) -> APP(g, x)
INC -> APP(map, app(app(curry, plus), app(s, 0)))
INC -> APP(app(curry, plus), app(s, 0))
INC -> APP(curry, plus)
INC -> APP(s, 0)

Furthermore, R contains two SCCs.


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


Dependency Pair:

APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)


Rules:


app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), 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(app(curry, g), x), y) -> app(app(g, x), y)
inc -> app(map, app(app(curry, plus), app(s, 0)))


Strategy:

innermost




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


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


Dependency Pair:

APP(app(plus, app(s, x)), y) -> APP(app(plus, 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
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
             ...
               →DP Problem 4
Size-Change Principle
           →DP Problem 2
UsableRules


Dependency Pair:

PLUS(s(x), y) -> PLUS(x, y)


Rule:

none


Strategy:

innermost




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

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
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:
s(x1) -> s(x1)

We obtain no new DP problems.


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


Dependency Pairs:

APP(app(app(curry, g), x), y) -> APP(g, x)
APP(app(app(curry, g), x), y) -> APP(app(g, x), y)
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(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), 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(app(curry, g), x), y) -> app(app(g, x), y)
inc -> app(map, app(app(curry, plus), app(s, 0)))


Strategy:

innermost




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


   R
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
           →DP Problem 2
UsableRules
             ...
               →DP Problem 5
Polynomial Ordering


Dependency Pairs:

APP(app(app(curry, g), x), y) -> APP(g, x)
APP(app(app(curry, g), x), y) -> APP(app(g, x), y)
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), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(plus, 0), y) -> y
app(app(map, f), nil) -> nil
app(app(app(curry, g), x), y) -> app(app(g, x), y)


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


Additionally, the following usable rules w.r.t. the implicit AFS can be oriented:

app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(plus, 0), y) -> y
app(app(map, f), nil) -> nil
app(app(app(curry, g), x), y) -> app(app(g, x), y)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(plus)=  0  
  POL(0)=  0  
  POL(curry)=  1  
  POL(cons)=  0  
  POL(map)=  1  
  POL(nil)=  0  
  POL(s)=  0  
  POL(app(x1, x2))=  1 + x1 + x1·x2  
  POL(APP(x1, x2))=  x1  

resulting in one new DP problem.



   R
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
           →DP Problem 2
UsableRules
             ...
               →DP Problem 6
Usable Rules (Innermost)


Dependency Pair:

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


Rules:


app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(plus, 0), y) -> y
app(app(map, f), nil) -> nil
app(app(app(curry, g), x), y) -> app(app(g, x), y)


Strategy:

innermost




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


   R
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
           →DP Problem 2
UsableRules
             ...
               →DP Problem 7
A-Transformation


Dependency Pair:

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


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
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
           →DP Problem 2
UsableRules
             ...
               →DP Problem 8
Size-Change Principle


Dependency Pair:

MAP(f, cons(x, xs)) -> MAP(f, xs)


Rule:

none


Strategy:

innermost




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

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
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:
cons(x1, x2) -> cons(x1, x2)

We obtain no new DP problems.

Termination of R successfully shown.
Duration:
0:07 minutes