Term Rewriting System R:
[y, u, v, 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`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳Nar`

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 + x1 + x2 POL(CONCAT(x1, x2)) =  x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
CONCAT(x1, x2) -> CONCAT(x1, x2)
cons(x1, x2) -> cons(x1, x2)

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

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`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Narrowing Transformation`

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))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

LESSLEAVES(cons(u, v), cons(w, z)) -> LESSLEAVES(concat(u, v), concat(w, z))
four new Dependency Pairs are created:

LESSLEAVES(cons(leaf, v'), cons(w, z)) -> LESSLEAVES(v', concat(w, z))
LESSLEAVES(cons(cons(u'', v''), v0), cons(w, z)) -> LESSLEAVES(cons(u'', concat(v'', v0)), concat(w, z))
LESSLEAVES(cons(u, v), cons(leaf, z')) -> LESSLEAVES(concat(u, v), z')
LESSLEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESSLEAVES(concat(u, v), cons(u'', concat(v'', z')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 4`
`             ↳Narrowing Transformation`

Dependency Pairs:

LESSLEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESSLEAVES(concat(u, v), cons(u'', concat(v'', z')))
LESSLEAVES(cons(u, v), cons(leaf, z')) -> LESSLEAVES(concat(u, v), z')
LESSLEAVES(cons(cons(u'', v''), v0), cons(w, z)) -> LESSLEAVES(cons(u'', concat(v'', v0)), concat(w, z))
LESSLEAVES(cons(leaf, v'), cons(w, z)) -> LESSLEAVES(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))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

LESSLEAVES(cons(leaf, v'), cons(w, z)) -> LESSLEAVES(v', concat(w, z))
two new Dependency Pairs are created:

LESSLEAVES(cons(leaf, v'), cons(leaf, z')) -> LESSLEAVES(v', z')
LESSLEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESSLEAVES(v', cons(u', concat(v'', z')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 4`
`             ↳Nar`
`             ...`
`               →DP Problem 5`
`                 ↳Narrowing Transformation`

Dependency Pairs:

LESSLEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESSLEAVES(v', cons(u', concat(v'', z')))
LESSLEAVES(cons(leaf, v'), cons(leaf, z')) -> LESSLEAVES(v', z')
LESSLEAVES(cons(u, v), cons(leaf, z')) -> LESSLEAVES(concat(u, v), z')
LESSLEAVES(cons(cons(u'', v''), v0), cons(w, z)) -> LESSLEAVES(cons(u'', concat(v'', v0)), concat(w, z))
LESSLEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESSLEAVES(concat(u, v), cons(u'', concat(v'', 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))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

LESSLEAVES(cons(cons(u'', v''), v0), cons(w, z)) -> LESSLEAVES(cons(u'', concat(v'', v0)), concat(w, z))
four new Dependency Pairs are created:

LESSLEAVES(cons(cons(u'', leaf), v0'), cons(w, z)) -> LESSLEAVES(cons(u'', v0'), concat(w, z))
LESSLEAVES(cons(cons(u'', cons(u', v')), v0'), cons(w, z)) -> LESSLEAVES(cons(u'', cons(u', concat(v', v0'))), concat(w, z))
LESSLEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), z')
LESSLEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 4`
`             ↳Nar`
`             ...`
`               →DP Problem 6`
`                 ↳Narrowing Transformation`

Dependency Pairs:

LESSLEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESSLEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), z')
LESSLEAVES(cons(cons(u'', cons(u', v')), v0'), cons(w, z)) -> LESSLEAVES(cons(u'', cons(u', concat(v', v0'))), concat(w, z))
LESSLEAVES(cons(cons(u'', leaf), v0'), cons(w, z)) -> LESSLEAVES(cons(u'', v0'), concat(w, z))
LESSLEAVES(cons(leaf, v'), cons(leaf, z')) -> LESSLEAVES(v', z')
LESSLEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESSLEAVES(concat(u, v), cons(u'', concat(v'', z')))
LESSLEAVES(cons(u, v), cons(leaf, z')) -> LESSLEAVES(concat(u, v), z')
LESSLEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESSLEAVES(v', cons(u', concat(v'', 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))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

LESSLEAVES(cons(u, v), cons(leaf, z')) -> LESSLEAVES(concat(u, v), z')
two new Dependency Pairs are created:

LESSLEAVES(cons(leaf, v'), cons(leaf, z')) -> LESSLEAVES(v', z')
LESSLEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), z')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 4`
`             ↳Nar`
`             ...`
`               →DP Problem 7`
`                 ↳Narrowing Transformation`

Dependency Pairs:

LESSLEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), z')
LESSLEAVES(cons(leaf, v'), cons(leaf, z')) -> LESSLEAVES(v', z')
LESSLEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), z')
LESSLEAVES(cons(cons(u'', cons(u', v')), v0'), cons(w, z)) -> LESSLEAVES(cons(u'', cons(u', concat(v', v0'))), concat(w, z))
LESSLEAVES(cons(cons(u'', leaf), v0'), cons(w, z)) -> LESSLEAVES(cons(u'', v0'), concat(w, z))
LESSLEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESSLEAVES(v', cons(u', concat(v'', z')))
LESSLEAVES(cons(leaf, v'), cons(leaf, z')) -> LESSLEAVES(v', z')
LESSLEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESSLEAVES(concat(u, v), cons(u'', concat(v'', z')))
LESSLEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', 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))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

LESSLEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESSLEAVES(concat(u, v), cons(u'', concat(v'', z')))
four new Dependency Pairs are created:

LESSLEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESSLEAVES(v', cons(u'', concat(v'', z')))
LESSLEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESSLEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESSLEAVES(cons(u, v), cons(cons(u'', leaf), z'')) -> LESSLEAVES(concat(u, v), cons(u'', z''))
LESSLEAVES(cons(u, v), cons(cons(u'', cons(u''', v''')), z'')) -> LESSLEAVES(concat(u, v), cons(u'', cons(u''', concat(v''', z''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 4`
`             ↳Nar`
`             ...`
`               →DP Problem 8`
`                 ↳Argument Filtering and Ordering`

Dependency Pairs:

LESSLEAVES(cons(u, v), cons(cons(u'', cons(u''', v''')), z'')) -> LESSLEAVES(concat(u, v), cons(u'', cons(u''', concat(v''', z''))))
LESSLEAVES(cons(u, v), cons(cons(u'', leaf), z'')) -> LESSLEAVES(concat(u, v), cons(u'', z''))
LESSLEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESSLEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESSLEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESSLEAVES(v', cons(u'', concat(v'', z')))
LESSLEAVES(cons(leaf, v'), cons(leaf, z')) -> LESSLEAVES(v', z')
LESSLEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESSLEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), z')
LESSLEAVES(cons(cons(u'', cons(u', v')), v0'), cons(w, z)) -> LESSLEAVES(cons(u'', cons(u', concat(v', v0'))), concat(w, z))
LESSLEAVES(cons(cons(u'', leaf), v0'), cons(w, z)) -> LESSLEAVES(cons(u'', v0'), concat(w, z))
LESSLEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESSLEAVES(v', cons(u', concat(v'', z')))
LESSLEAVES(cons(leaf, v'), cons(leaf, z')) -> LESSLEAVES(v', z')
LESSLEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), 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 pairs can be strictly oriented:

LESSLEAVES(cons(u, v), cons(cons(u'', leaf), z'')) -> LESSLEAVES(concat(u, v), cons(u'', z''))
LESSLEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESSLEAVES(v', cons(u'', concat(v'', z')))
LESSLEAVES(cons(leaf, v'), cons(leaf, z')) -> LESSLEAVES(v', z')
LESSLEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), z')
LESSLEAVES(cons(cons(u'', leaf), v0'), cons(w, z)) -> LESSLEAVES(cons(u'', v0'), concat(w, z))
LESSLEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESSLEAVES(v', cons(u', concat(v'', z')))

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)) =  x1 + x2 POL(LESS_LEAVES(x1, x2)) =  1 + x1 + x2 POL(leaf) =  1 POL(concat(x1, x2)) =  x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
LESSLEAVES(x1, x2) -> LESSLEAVES(x1, x2)
cons(x1, x2) -> cons(x1, x2)
concat(x1, x2) -> concat(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 4`
`             ↳Nar`
`             ...`
`               →DP Problem 9`
`                 ↳Argument Filtering and Ordering`

Dependency Pairs:

LESSLEAVES(cons(u, v), cons(cons(u'', cons(u''', v''')), z'')) -> LESSLEAVES(concat(u, v), cons(u'', cons(u''', concat(v''', z''))))
LESSLEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESSLEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESSLEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESSLEAVES(cons(cons(u'', cons(u', v')), v0'), cons(w, z)) -> LESSLEAVES(cons(u'', cons(u', concat(v', v0'))), 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 pairs can be strictly oriented:

LESSLEAVES(cons(u, v), cons(cons(u'', cons(u''', v''')), z'')) -> LESSLEAVES(concat(u, v), cons(u'', cons(u''', concat(v''', z''))))
LESSLEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESSLEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESSLEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESSLEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESSLEAVES(cons(cons(u'', cons(u', v')), v0'), cons(w, z)) -> LESSLEAVES(cons(u'', cons(u', concat(v', v0'))), concat(w, z))

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(LESS_LEAVES(x1, x2)) =  1 + x1 + x2 POL(leaf) =  1 POL(concat(x1, x2)) =  x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
LESSLEAVES(x1, x2) -> LESSLEAVES(x1, x2)
cons(x1, x2) -> cons(x1, x2)
concat(x1, x2) -> concat(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Nar`
`           →DP Problem 4`
`             ↳Nar`
`             ...`
`               →DP Problem 10`
`                 ↳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:
0:01 minutes