Term Rewriting System R:
[x, y]
times(x, 0) -> 0
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0) -> x
plus(0, x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs.


   R
DPs
       →DP Problem 1
Polynomial Ordering
       →DP Problem 2
Polo


Dependency Pairs:

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


Rules:


times(x, 0) -> 0
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0) -> x
plus(0, x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))





The following dependency pair can be strictly oriented:

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


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PLUS(x1, x2))=  x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 3
Polynomial Ordering
       →DP Problem 2
Polo


Dependency Pair:

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


Rules:


times(x, 0) -> 0
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0) -> x
plus(0, x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))





The following dependency pair can be strictly oriented:

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


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PLUS(x1, x2))=  x2  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 3
Polo
             ...
               →DP Problem 4
Dependency Graph
       →DP Problem 2
Polo


Dependency Pair:


Rules:


times(x, 0) -> 0
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0) -> x
plus(0, x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polynomial Ordering


Dependency Pair:

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


Rules:


times(x, 0) -> 0
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0) -> x
plus(0, x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))





The following dependency pair can be strictly oriented:

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


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(TIMES(x1, x2))=  x2  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
           →DP Problem 5
Dependency Graph


Dependency Pair:


Rules:


times(x, 0) -> 0
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0) -> x
plus(0, x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))





Using the Dependency Graph resulted in no new DP problems.

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