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`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`

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

Rules:

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

The following dependency pairs can be strictly oriented:

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

The following usable rules using the Ce-refinement can be oriented:

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

Used ordering: Polynomial ordering with 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

resulting in one new DP problem.
Used Argument Filtering System:
A(x1, x2) -> A(x1, x2)
a(x1, x2) -> a(x1, x2)
lambda(x1) -> lambda(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳Remaining Obligation(s)`

The following remains to be proven:
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, 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 could not be shown.
Duration:
0:00 minutes