Term Rewriting System R:
[X]
f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(s(X), X) -> F(X, a(X))
F(X, c(X)) -> F(s(X), X)

Furthermore, R contains two SCCs.

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

Dependency Pair:

F(s(X), X) -> F(X, a(X))

Rules:

f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)

The following dependency pair can be strictly oriented:

F(s(X), X) -> F(X, a(X))

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(s(x1)) =  1 + x1 POL(a(x1)) =  0 POL(F(x1, x2)) =  x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

F(X, c(X)) -> F(s(X), X)

Rules:

f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)

The following dependency pair can be strictly oriented:

F(X, c(X)) -> F(s(X), X)

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(c(x1)) =  1 + x1 POL(s(x1)) =  0 POL(F(x1, x2)) =  x2

resulting in one new DP problem.

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

Dependency Pair:

Rules:

f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)

Using the Dependency Graph resulted in no new DP problems.

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