Term Rewriting System R:
[x, y, z]
f(f(f(a, x), y), z) -> f(f(x, z), f(y, z))
f(f(b, x), y) -> x
f(c, y) -> y

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

F(f(f(a, f(f(a, x''), y'')), y), z'') -> F(f(f(a, x''), y''), z'')

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
FwdInst
→DP Problem 2
Forward Instantiation Transformation

Dependency Pairs:

F(f(f(a, f(f(a, x''), y'')), y), z'') -> F(f(f(a, x''), y''), z'')
F(f(f(a, x), y), z) -> F(f(x, z), f(y, z))
F(f(f(a, x), y), z) -> F(y, z)

Rules:

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

Strategy:

innermost

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

F(f(f(a, x), y), z) -> F(y, z)
two new Dependency Pairs are created:

F(f(f(a, x), f(f(a, x''), y'')), z'') -> F(f(f(a, x''), y''), z'')
F(f(f(a, x), f(f(a, f(f(a, x''''), y'''')), y'')), z') -> F(f(f(a, f(f(a, x''''), y'''')), y''), z')

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
FwdInst
→DP Problem 2
FwdInst
...
→DP Problem 3
Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

F(f(f(a, x), f(f(a, f(f(a, x''''), y'''')), y'')), z') -> F(f(f(a, f(f(a, x''''), y'''')), y''), z')
F(f(f(a, x), f(f(a, x''), y'')), z'') -> F(f(f(a, x''), y''), z'')
F(f(f(a, x), y), z) -> F(f(x, z), f(y, z))
F(f(f(a, f(f(a, x''), y'')), y), z'') -> F(f(f(a, x''), y''), z'')

Rules:

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

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes