f(

f(a(

f(b(

f(c(

R

↳Dependency Pair Analysis

F(x, a(b(c(y)))) -> F(b(c(a(b(x)))),y)

F(a(x),y) -> F(x, a(y))

F(b(x),y) -> F(x, b(y))

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Usable Rules (Innermost)

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

f(x, a(b(c(y)))) -> f(b(c(a(b(x)))),y)

f(a(x),y) -> f(x, a(y))

f(b(x),y) -> f(x, b(y))

f(c(x),y) -> f(x, c(y))

innermost

As we are in the innermost case, we can delete all 4 non-usable-rules.

R

↳DPs

→DP Problem 1

↳UsableRules

→DP Problem 2

↳Scc To SRS

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

none

innermost

It has been determined that showing finiteness of this DP problem is equivalent to showing termination of a string rewrite system.

(Re)applying the dependency pair method (incl. the dependency graph) for the following SRS:

The graph does not contain any SCC and, thus, we obtain no new DP problems.

a(b(c(x))) -> b(a(c(b(x))))

Duration:

0:00 minutes