Term Rewriting System R:
[x, y, z]
app(app(., 1), x) -> x
app(app(., x), 1) -> x
app(app(., app(i, x)), x) -> 1
app(app(., x), app(i, x)) -> 1
app(app(., app(i, y)), app(app(., y), z)) -> z
app(app(., y), app(app(., app(i, y)), z)) -> z
app(app(., app(app(., x), y)), z) -> app(app(., x), app(app(., y), z))
app(i, 1) -> 1
app(i, app(i, x)) -> x
app(i, app(app(., x), y)) -> app(app(., app(i, y)), app(i, x))

Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

app(app(., 1), x) -> x
app(app(., x), 1) -> x
app(app(., app(i, x)), x) -> 1
app(app(., x), app(i, x)) -> 1
app(app(., app(i, y)), app(app(., y), z)) -> z
app(app(., y), app(app(., app(i, y)), z)) -> z

where the Polynomial interpretation:
 POL(i) =  0 POL(1) =  0 POL(.) =  1 POL(app(x1, x2)) =  x1 + x2
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

APP(app(., app(app(., x), y)), z) -> APP(app(., x), app(app(., y), z))
APP(app(., app(app(., x), y)), z) -> APP(app(., y), z)
APP(app(., app(app(., x), y)), z) -> APP(., y)
APP(i, app(app(., x), y)) -> APP(app(., app(i, y)), app(i, x))
APP(i, app(app(., x), y)) -> APP(., app(i, y))
APP(i, app(app(., x), y)) -> APP(i, y)
APP(i, app(app(., x), y)) -> APP(i, x)

Furthermore, R contains two SCCs.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳Size-Change Principle`
`           →DP Problem 2`
`             ↳SCP`

Dependency Pairs:

APP(app(., app(app(., x), y)), z) -> APP(app(., y), z)
APP(app(., app(app(., x), y)), z) -> APP(app(., x), app(app(., y), z))

Rules:

app(app(., app(app(., x), y)), z) -> app(app(., x), app(app(., y), z))
app(i, app(app(., x), y)) -> app(app(., app(i, y)), app(i, x))
app(i, app(i, x)) -> x
app(i, 1) -> 1

The original DP problem is in applicative form. Its DPs and usable rules are the following.

APP(app(., app(app(., x), y)), z) -> APP(app(., y), z)
APP(app(., app(app(., x), y)), z) -> APP(app(., x), app(app(., y), z))

app(app(., app(app(., x), y)), z) -> app(app(., x), app(app(., y), z))

It is proper and hence, it can be A-transformed which results in the DP problem

.'(.(x, y), z) -> .'(y, z)
.'(.(x, y), z) -> .'(x, .(y, z))

.(.(x, y), z) -> .(x, .(y, z))

We number the DPs as follows:
1. .'(.(x, y), z) -> .'(y, z)
2. .'(.(x, y), z) -> .'(x, .(y, z))
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1
2=2
{1, 2} , {1, 2}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1
2=2
{1, 2} , {1, 2}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

We obtain no new DP problems.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳SCP`
`           →DP Problem 2`
`             ↳Size-Change Principle`

Dependency Pairs:

APP(i, app(app(., x), y)) -> APP(i, x)
APP(i, app(app(., x), y)) -> APP(i, y)

Rules:

app(app(., app(app(., x), y)), z) -> app(app(., x), app(app(., y), z))
app(i, app(app(., x), y)) -> app(app(., app(i, y)), app(i, x))
app(i, app(i, x)) -> x
app(i, 1) -> 1

The original DP problem is in applicative form. Its DPs and usable rules are the following.

APP(i, app(app(., x), y)) -> APP(i, x)
APP(i, app(app(., x), y)) -> APP(i, y)

none

It is proper and hence, it can be A-transformed which results in the DP problem

I(.(x, y)) -> I(x)
I(.(x, y)) -> I(y)

none

We number the DPs as follows:
1. I(.(x, y)) -> I(x)
2. I(.(x, y)) -> I(y)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
.(x1, x2) -> .(x1, x2)

We obtain no new DP problems.

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