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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

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

Rules:

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

Strategy:

innermost

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

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

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

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in no new DP problems.

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

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

F(f(x'', f(a, x'')), f(x, f(a, x))) -> F(f(f(a, x), f(x, a)), f(f(f(a, x''), f(x'', a)), f(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 5`
`                 ↳Narrowing Transformation`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

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