Term Rewriting System R:
[x, y]
g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
f(x) -> x

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

G(c(x, s(y))) -> G(c(s(x), y))
F(c(s(x), y)) -> F(c(x, s(y)))
F(f(x)) -> F(d(f(x)))

Furthermore, R contains two SCCs.

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

Dependency Pair:

G(c(x, s(y))) -> G(c(s(x), y))

Rules:

g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
f(x) -> x

The following dependency pair can be strictly oriented:

G(c(x, s(y))) -> G(c(s(x), y))

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

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

g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
f(x) -> 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(c(s(x), y)) -> F(c(x, s(y)))

Rules:

g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
f(x) -> x

The following dependency pair can be strictly oriented:

F(c(s(x), y)) -> F(c(x, s(y)))

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

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

resulting in one new DP problem.

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

Dependency Pair:

Rules:

g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
f(x) -> x

Using the Dependency Graph resulted in no new DP problems.

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