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

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(+(x, 0)) -> F(x)
+'(x, +(y, z)) -> +'(+(x, y), z)
+'(x, +(y, z)) -> +'(x, y)

Furthermore, R contains two SCCs.

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

Dependency Pair:

F(+(x, 0)) -> F(x)

Rules:

f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)

The following dependency pair can be strictly oriented:

F(+(x, 0)) -> F(x)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(F(x1)) =  x1 POL(+(x1, x2)) =  1 + x1 + x2

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

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

Dependency Pair:

Rules:

f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pairs:

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

Rules:

f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)

The following dependency pair can be strictly oriented:

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

The following usable rule using the Ce-refinement can be oriented:

+(x, +(y, z)) -> +(+(x, y), z)

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

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

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 4
Argument Filtering and Ordering

Dependency Pair:

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

Rules:

f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)

The following dependency pair can be strictly oriented:

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

The following usable rule using the Ce-refinement can be oriented:

+(x, +(y, z)) -> +(+(x, y), z)

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

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

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 4
AFS
...
→DP Problem 5
Dependency Graph

Dependency Pair:

Rules:

f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)

Using the Dependency Graph resulted in no new DP problems.

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