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

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

SUM(s(x)) -> +'(sum(x), s(x))
SUM(s(x)) -> SUM(x)
+'(x, s(y)) -> +'(x, y)

Furthermore, R contains two SCCs.

R
DPs
→DP Problem 1
Argument Filtering and Ordering
→DP Problem 2
AFS

Dependency Pair:

+'(x, s(y)) -> +'(x, y)

Rules:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

The following dependency pair can be strictly oriented:

+'(x, s(y)) -> +'(x, y)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(s(x1)) =  1 + x1 POL(+'(x1, x2)) =  x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
+'(x1, x2) -> +'(x1, x2)
s(x1) -> s(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 3
Dependency Graph
→DP Problem 2
AFS

Dependency Pair:

Rules:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

Using the Dependency Graph resulted in no new DP problems.

R
DPs
→DP Problem 1
AFS
→DP Problem 2
Argument Filtering and Ordering

Dependency Pair:

SUM(s(x)) -> SUM(x)

Rules:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

The following dependency pair can be strictly oriented:

SUM(s(x)) -> SUM(x)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(SUM(x1)) =  x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.
Used Argument Filtering System:
SUM(x1) -> SUM(x1)
s(x1) -> s(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 4
Dependency Graph

Dependency Pair:

Rules:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

Using the Dependency Graph resulted in no new DP problems.

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