Term Rewriting System R:
[x, y, z]
a(lambda(x), y) -> lambda(a(x, 1))
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(a(x, y), z) -> a(x, a(y, z))
a(x, y) -> x
a(x, y) -> y
lambda(x) -> x

Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

lambda(x) -> x

where the Polynomial interpretation:
 POL(t) =  0 POL(1) =  0 POL(lambda(x1)) =  1 + x1 POL(a(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:

A(lambda(x), y) -> A(x, a(y, t))
A(lambda(x), y) -> A(y, t)
A(lambda(x), y) -> A(x, 1)
A(a(x, y), z) -> A(x, a(y, z))
A(a(x, y), z) -> A(y, z)

Furthermore, R contains one SCC.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳Modular Removal of Rules`

Dependency Pairs:

A(a(x, y), z) -> A(y, z)
A(a(x, y), z) -> A(x, a(y, z))
A(lambda(x), y) -> A(x, 1)
A(lambda(x), y) -> A(y, t)
A(lambda(x), y) -> A(x, a(y, t))

Rules:

a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(x, y) -> y
a(lambda(x), y) -> lambda(a(x, 1))
a(x, y) -> x
a(a(x, y), z) -> a(x, a(y, z))

We have the following set of usable rules:

a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(x, y) -> y
a(lambda(x), y) -> lambda(a(x, 1))
a(x, y) -> x
a(a(x, y), z) -> a(x, a(y, z))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(t) =  0 POL(1) =  0 POL(lambda(x1)) =  1 + x1 POL(a(x1, x2)) =  x1 + x2 POL(A(x1, x2)) =  1 + x1 + x2

We have the following set D of usable symbols: {t, 1, lambda, a, A}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:

A(lambda(x), y) -> A(x, 1)
A(lambda(x), y) -> A(y, t)
A(lambda(x), y) -> A(x, a(y, t))

No Rules can be deleted.

The result of this processor delivers one new DP problem.

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

Dependency Pairs:

A(a(x, y), z) -> A(y, z)
A(a(x, y), z) -> A(x, a(y, z))

Rules:

a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(x, y) -> y
a(lambda(x), y) -> lambda(a(x, 1))
a(x, y) -> x
a(a(x, y), z) -> a(x, a(y, z))

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

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

DP: empty set
Oriented Rules: none

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

We obtain no new DP problems.

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