Term Rewriting System R:
[Y, U, V, X, W, Z]
concat(leaf, Y) -> Y
concat(cons(U, V), Y) -> cons(U, concat(V, Y))
lessleaves(X, leaf) -> false
lessleaves(leaf, cons(W, Z)) -> true
lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

CONCAT(cons(U, V), Y) -> CONCAT(V, Y)
LESSLEAVES(cons(U, V), cons(W, Z)) -> LESSLEAVES(concat(U, V), concat(W, Z))
LESSLEAVES(cons(U, V), cons(W, Z)) -> CONCAT(U, V)
LESSLEAVES(cons(U, V), cons(W, Z)) -> CONCAT(W, Z)

Furthermore, R contains two SCCs.

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

Dependency Pair:

CONCAT(cons(U, V), Y) -> CONCAT(V, Y)

Rules:

concat(leaf, Y) -> Y
concat(cons(U, V), Y) -> cons(U, concat(V, Y))
lessleaves(X, leaf) -> false
lessleaves(leaf, cons(W, Z)) -> true
lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))

The following dependency pair can be strictly oriented:

CONCAT(cons(U, V), Y) -> CONCAT(V, Y)

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(cons(x1, x2)) =  1 + x2 POL(CONCAT(x1, x2)) =  x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

concat(leaf, Y) -> Y
concat(cons(U, V), Y) -> cons(U, concat(V, Y))
lessleaves(X, leaf) -> false
lessleaves(leaf, cons(W, Z)) -> true
lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

LESSLEAVES(cons(U, V), cons(W, Z)) -> LESSLEAVES(concat(U, V), concat(W, Z))

Rules:

concat(leaf, Y) -> Y
concat(cons(U, V), Y) -> cons(U, concat(V, Y))
lessleaves(X, leaf) -> false
lessleaves(leaf, cons(W, Z)) -> true
lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))

The following dependency pair can be strictly oriented:

LESSLEAVES(cons(U, V), cons(W, Z)) -> LESSLEAVES(concat(U, V), concat(W, Z))

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

concat(leaf, Y) -> Y
concat(cons(U, V), Y) -> cons(U, concat(V, Y))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(cons(x1, x2)) =  1 + x1 + x2 POL(leaf) =  0 POL(concat(x1, x2)) =  x1 + x2 POL(LESSLEAVES(x1, x2)) =  x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`           →DP Problem 4`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

concat(leaf, Y) -> Y
concat(cons(U, V), Y) -> cons(U, concat(V, Y))
lessleaves(X, leaf) -> false
lessleaves(leaf, cons(W, Z)) -> true
lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))

Using the Dependency Graph resulted in no new DP problems.

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