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`
`         ↳Polynomial 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)

Additionally, the following usable rules w.r.t. to the implicit AFS 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.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Polynomial Ordering`

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

The following dependency pairs can be strictly oriented:

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

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(t) =  0 POL(1) =  0 POL(lambda(x1)) =  0 POL(a(x1, x2)) =  1 + x1 + x2 POL(A(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pair:

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

Using the Dependency Graph resulted in no new DP problems.

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