Term Rewriting System R:
[t, u, v, x, f]
app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(rectuv, t), u), v)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(rectuv, t), u)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(rectuv, t)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(app(rec, t), u), v), n)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)
APP(app(app(app(rectuv, t), u), v), n) -> APP(rec, t)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Narrowing Transformation

Dependency Pairs:

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(app(rec, t), u), v), n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(rectuv, t), u)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(rectuv, t), u), v)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))

Rules:

app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)

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

APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(rectuv, t), u), v)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

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

Dependency Pairs:

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(app(rec, t), u), v), n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(rectuv, t), u)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)

Rules:

app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)

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

APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(rectuv, t), u)
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
Narrowing Transformation

Dependency Pairs:

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(app(rec, t), u), v), n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)

Rules:

app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)

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

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(app(rec, t), u), v), n)
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 4
Narrowing Transformation

Dependency Pairs:

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)

Rules:

app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)

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

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)
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 Pairs:

APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)

Rules:

app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)

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

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)
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 6
Narrowing Transformation

Dependency Pairs:

APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))

Rules:

app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)

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

APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
one new Dependency Pair is created:

APP(app(app(app(rec, t), u), v), app(lim, app(app(app(rectuv, t''), u''), v''))) -> APP(app(app(app(rectuv, t), u), v), app(app(app(app(rec, t''), u''), v''), n))

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 7
Narrowing Transformation

Dependency Pairs:

APP(app(app(app(rec, t), u), v), app(lim, app(app(app(rectuv, t''), u''), v''))) -> APP(app(app(app(rectuv, t), u), v), app(app(app(app(rec, t''), u''), v''), n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)

Rules:

app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)

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

APP(app(app(app(rec, t), u), v), app(lim, app(app(app(rectuv, t''), u''), v''))) -> APP(app(app(app(rectuv, t), u), v), app(app(app(app(rec, t''), u''), v''), n))
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 8
Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))

Rules:

app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)

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