Term Rewriting System R:
[x]
f(f(a, x), a) -> f(f(f(a, f(a, a)), x), a)

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Narrowing Transformation

Dependency Pairs:

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

Rule:

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

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(f(a, x), a) -> F(f(f(a, f(a, a)), x), a)
one new Dependency Pair is created:

F(f(a, a), a) -> F(f(f(f(a, f(a, a)), f(a, a)), a), a)

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Narrowing Transformation

Dependency Pairs:

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

Rule:

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

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(f(a, a), a) -> F(f(f(f(a, f(a, a)), f(a, a)), a), a)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 3
Forward Instantiation Transformation

Dependency Pair:

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

Rule:

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

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

F(f(a, x), a) -> F(f(a, f(a, a)), x)
one new Dependency Pair is created:

F(f(a, a), a) -> F(f(a, f(a, a)), a)

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 4
Narrowing Transformation

Dependency Pair:

F(f(a, a), a) -> F(f(a, f(a, a)), a)

Rule:

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

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(f(a, a), a) -> F(f(a, f(a, a)), a)
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.

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