Termination Proof using AProVE

Term Rewriting System R:
[y, u, v, w, z]
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

CONCAT(cons(u, v), y) -> CONCAT(v, y)
LESS_LEAVES(cons(u, v), cons(w, z)) -> LESS_LEAVES(concat(u, v), concat(w, z))
LESS_LEAVES(cons(u, v), cons(w, z)) -> CONCAT(u, v)
LESS_LEAVES(cons(u, v), cons(w, z)) -> CONCAT(w, z)

Furthermore, R contains two SCCs.

     ↳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))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(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: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

CONCAT(x₁, x₂) -> CONCAT(x₁, x₂)
cons(x₁, x₂) -> cons(x₁, x₂)

     ↳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))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Narrowing Transformation

Dependency Pair:

LESS_LEAVES(cons(u, v), cons(w, z)) -> LESS_LEAVES(concat(u, v), concat(w, z))

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

LESS_LEAVES(cons(u, v), cons(w, z)) -> LESS_LEAVES(concat(u, v), concat(w, z))

four new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

LESS_LEAVES(cons(leaf, v'), cons(w, z)) -> LESS_LEAVES(v', concat(w, z))

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

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

four new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 6

                 ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 7

                 ↳Narrowing Transformation

Dependency Pairs:

LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), concat(w, z))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(w, z)) -> LESS_LEAVES(cons(u'', v0'), concat(w, z))
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESS_LEAVES(concat(u, v), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(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))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

LESS_LEAVES(cons(u, v), cons(cons(u'', v''), z')) -> LESS_LEAVES(concat(u, v), cons(u'', concat(v'', z')))

four new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 8

                 ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 9

                 ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

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

four new Dependency Pairs are created:

LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', v0'')), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), z')
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(cons(u0, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), cons(u0, concat(v'', z')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 10

                 ↳Narrowing Transformation

Dependency Pairs:

LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(cons(u0, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), cons(u0, concat(v'', z')))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', v0'')), concat(w, z))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', v0'), z')
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(u, v), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(concat(u, v), cons(u'', z''))
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', v0'), cons(u', concat(v', z')))

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 11

                 ↳Narrowing Transformation

Dependency Pairs:

LESS_LEAVES(cons(cons(u''', v''), v0), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(cons(u''', concat(v'', v0)), cons(u'', z''))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(v', cons(u'', z''))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', v0'')), concat(w, z))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', v0'), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', v0'), z')
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(cons(u0, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), cons(u0, concat(v'', z')))

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', concat(v''', z''))))

four new Dependency Pairs are created:

LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(v', cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

Dependency Pairs:

LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(v', cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(v', cons(u'', z''))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(cons(u0, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), cons(u0, concat(v'', z')))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', v0'')), concat(w, z))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', v0'), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', v0'), z')
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u''', v''), v0), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(cons(u''', concat(v'', v0)), cons(u'', z''))

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

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

two new Dependency Pairs are created:

LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), z')
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(cons(u0, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), cons(u0, concat(v', z')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 13

                 ↳Narrowing Transformation

Dependency Pairs:

LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(cons(u0, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), cons(u0, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), z')
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(v', cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(cons(u''', v''), v0), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(cons(u''', concat(v'', v0)), cons(u'', z''))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(v', cons(u'', z''))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(cons(u0, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), cons(u0, concat(v'', z')))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), concat(w, z))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', v0'), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', v0'), z')
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), concat(w, z))

four new Dependency Pairs are created:

LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(cons(u1, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), cons(u1, concat(v', z')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 14

                 ↳Narrowing Transformation

Dependency Pairs:

LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(cons(u1, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), cons(u1, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), z')
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(v', cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(cons(u''', v''), v0), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(cons(u''', concat(v'', v0)), cons(u'', z''))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(v', cons(u'', z''))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(cons(u0, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), cons(u0, concat(v'', z')))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), z')
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', v0'), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', v0'), z')
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(cons(u0, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), cons(u0, concat(v', z')))

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

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

two new Dependency Pairs are created:

LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(v', cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', z''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 15

                 ↳Narrowing Transformation

Dependency Pairs:

LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(v', cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(cons(u0, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), cons(u0, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), z')
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(v', cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(cons(u''', v''), v0), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(cons(u''', concat(v'', v0)), cons(u'', z''))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(v', cons(u'', z''))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(cons(u0, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), cons(u0, concat(v'', z')))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), z')
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', v0'), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', v0'), z')
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(cons(u1, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), cons(u1, concat(v', z')))

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))

four new Dependency Pairs are created:

LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(v', cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(cons(u'1, v'''), v0), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(cons(u'1, concat(v''', v0)), cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, leaf))), z'''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, z''''))))
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, v''')))), z'''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, cons(u'1, concat(v''', z''''))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 16

                 ↳Narrowing Transformation

Dependency Pairs:

LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, v''')))), z'''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, cons(u'1, concat(v''', z''''))))))
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, leaf))), z'''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, z''''))))
LESS_LEAVES(cons(cons(u'1, v'''), v0), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(cons(u'1, concat(v''', v0)), cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(v', cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(v', cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(cons(u1, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), cons(u1, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(cons(u0, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), cons(u0, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), z')
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(v', cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(cons(u''', v''), v0), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(cons(u''', concat(v'', v0)), cons(u'', z''))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(v', cons(u'', z''))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(cons(u0, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), cons(u0, concat(v'', z')))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), z')
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', v0'), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', v0'), z')
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', z''')))

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), concat(w, z))

two new Dependency Pairs are created:

LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(cons(u1, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), cons(u1, concat(v', z')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 17

                 ↳Narrowing Transformation

Dependency Pairs:

LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(cons(u1, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), cons(u1, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), z')
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, leaf))), z'''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, z''''))))
LESS_LEAVES(cons(cons(u'1, v'''), v0), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(cons(u'1, concat(v''', v0)), cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(v', cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(v', cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(cons(u1, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), cons(u1, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(cons(u0, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), cons(u0, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), z')
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(v', cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(cons(u''', v''), v0), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(cons(u''', concat(v'', v0)), cons(u'', z''))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(v', cons(u'', z''))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(cons(u0, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), cons(u0, concat(v'', z')))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), z')
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', v0'), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', v0'), z')
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, v''')))), z'''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, cons(u'1, concat(v''', z''''))))))

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), concat(w, z))

four new Dependency Pairs are created:

LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, leaf)))), v0''''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, v0'''')))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, cons(u2, v''))))), v0''''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, cons(u2, concat(v'', v0'''')))))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(cons(u2, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), cons(u2, concat(v'', z')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 18

                 ↳Narrowing Transformation

Dependency Pairs:

LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(cons(u2, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), cons(u2, concat(v'', z')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, cons(u2, v''))))), v0''''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, cons(u2, concat(v'', v0'''')))))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, leaf)))), v0''''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, v0'''')))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), z')
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, v''')))), z'''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, cons(u'1, concat(v''', z''''))))))
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, leaf))), z'''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, z''''))))
LESS_LEAVES(cons(cons(u'1, v'''), v0), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(cons(u'1, concat(v''', v0)), cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(v', cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(v', cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(cons(u1, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), cons(u1, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), z')
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(cons(u0, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), cons(u0, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), z')
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(v', cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(cons(u''', v''), v0), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(cons(u''', concat(v'', v0)), cons(u'', z''))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(v', cons(u'', z''))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(cons(u0, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), cons(u0, concat(v'', z')))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), z')
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', v0'), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', v0'), z')
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(cons(u1, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), cons(u1, concat(v', z')))

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, leaf))), z'''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, z''''))))

two new Dependency Pairs are created:

LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', cons(u'0, leaf))), z'''')) -> LESS_LEAVES(v', cons(u'', cons(u''', cons(u'0, z''''))))
LESS_LEAVES(cons(cons(u'1, v''), v0), cons(cons(u'', cons(u''', cons(u'0, leaf))), z'''')) -> LESS_LEAVES(cons(u'1, concat(v'', v0)), cons(u'', cons(u''', cons(u'0, z''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 19

                 ↳Narrowing Transformation

Dependency Pairs:

LESS_LEAVES(cons(cons(u'1, v''), v0), cons(cons(u'', cons(u''', cons(u'0, leaf))), z'''')) -> LESS_LEAVES(cons(u'1, concat(v'', v0)), cons(u'', cons(u''', cons(u'0, z''''))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', cons(u'0, leaf))), z'''')) -> LESS_LEAVES(v', cons(u'', cons(u''', cons(u'0, z''''))))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, cons(u2, v''))))), v0''''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, cons(u2, concat(v'', v0'''')))))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, leaf)))), v0''''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, v0'''')))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(cons(u1, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), cons(u1, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), z')
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, v''')))), z'''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, cons(u'1, concat(v''', z''''))))))
LESS_LEAVES(cons(cons(u'1, v'''), v0), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(cons(u'1, concat(v''', v0)), cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(v', cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(v', cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(cons(u1, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), cons(u1, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), z')
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(cons(u0, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), cons(u0, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), z')
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(v', cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(cons(u''', v''), v0), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(cons(u''', concat(v'', v0)), cons(u'', z''))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(v', cons(u'', z''))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(cons(u0, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), cons(u0, concat(v'', z')))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), z')
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', v0'), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', v0'), z')
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(cons(u2, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), cons(u2, concat(v'', z')))

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, v''')))), z'''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, cons(u'1, concat(v''', z''''))))))

four new Dependency Pairs are created:

LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, v''')))), z'''')) -> LESS_LEAVES(v', cons(u'', cons(u''', cons(u'0, cons(u'1, concat(v''', z''''))))))
LESS_LEAVES(cons(cons(u'2, v''), v0), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, v''')))), z'''')) -> LESS_LEAVES(cons(u'2, concat(v'', v0)), cons(u'', cons(u''', cons(u'0, cons(u'1, concat(v''', z''''))))))
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, leaf)))), z''''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, cons(u'1, z''''')))))
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, cons(u'2, v''))))), z''''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, cons(u'1, cons(u'2, concat(v'', z''''')))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 20

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, cons(u'2, v''))))), z''''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, cons(u'1, cons(u'2, concat(v'', z''''')))))))
LESS_LEAVES(cons(u, v), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, leaf)))), z''''')) -> LESS_LEAVES(concat(u, v), cons(u'', cons(u''', cons(u'0, cons(u'1, z''''')))))
LESS_LEAVES(cons(cons(u'2, v''), v0), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, v''')))), z'''')) -> LESS_LEAVES(cons(u'2, concat(v'', v0)), cons(u'', cons(u''', cons(u'0, cons(u'1, concat(v''', z''''))))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', cons(u'0, cons(u'1, v''')))), z'''')) -> LESS_LEAVES(v', cons(u'', cons(u''', cons(u'0, cons(u'1, concat(v''', z''''))))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', cons(u'0, leaf))), z'''')) -> LESS_LEAVES(v', cons(u'', cons(u''', cons(u'0, z''''))))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(cons(u2, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), cons(u2, concat(v'', z')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, v')))), v0'''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, concat(v', v0'''))))), z')
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, cons(u2, v''))))), v0''''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, cons(u2, concat(v'', v0'''')))))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, cons(u1, leaf)))), v0''''), cons(w, z)) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, cons(u1, v0'''')))), concat(w, z))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(cons(u1, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), cons(u1, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, leaf))), v0'''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, v0'''))), z')
LESS_LEAVES(cons(cons(u'1, v'''), v0), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(cons(u'1, concat(v''', v0)), cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', cons(u'0, v''))), z''')) -> LESS_LEAVES(v', cons(u'', cons(u''', cons(u'0, concat(v'', z''')))))
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', leaf)), z''')) -> LESS_LEAVES(v', cons(u'', cons(u''', z''')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(cons(u1, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), cons(u1, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', cons(u0, v''))), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', cons(u0, concat(v'', v0'')))), z')
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(cons(u0, v'), z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), cons(u0, concat(v', z')))
LESS_LEAVES(cons(cons(u'', cons(u', leaf)), v0''), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', v0'')), z')
LESS_LEAVES(cons(cons(u'0, v''), v0), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(cons(u'0, concat(v'', v0)), cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', cons(u''', v''')), z'')) -> LESS_LEAVES(v', cons(u'', cons(u''', concat(v''', z''))))
LESS_LEAVES(cons(cons(u''', v''), v0), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(cons(u''', concat(v'', v0)), cons(u'', z''))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', leaf), z'')) -> LESS_LEAVES(v', cons(u'', z''))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(cons(u0, v''), z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), cons(u0, concat(v'', z')))
LESS_LEAVES(cons(cons(u'', cons(u', v')), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', cons(u', concat(v', v0'))), z')
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', v0'), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', leaf), v0'), cons(leaf, z')) -> LESS_LEAVES(cons(u'', v0'), z')
LESS_LEAVES(cons(cons(u''', v'''), v0), cons(cons(u'', v''), z')) -> LESS_LEAVES(cons(u''', concat(v''', v0)), cons(u'', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(cons(u'', v''), z')) -> LESS_LEAVES(v', cons(u'', concat(v'', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'', v''), v0), cons(cons(u', v'), z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), cons(u', concat(v', z')))
LESS_LEAVES(cons(cons(u'', v''), v0), cons(leaf, z')) -> LESS_LEAVES(cons(u'', concat(v'', v0)), z')
LESS_LEAVES(cons(leaf, v'), cons(cons(u', v''), z')) -> LESS_LEAVES(v', cons(u', concat(v'', z')))
LESS_LEAVES(cons(leaf, v'), cons(leaf, z')) -> LESS_LEAVES(v', z')
LESS_LEAVES(cons(cons(u'1, v''), v0), cons(cons(u'', cons(u''', cons(u'0, leaf))), z'''')) -> LESS_LEAVES(cons(u'1, concat(v'', v0)), cons(u'', cons(u''', cons(u'0, z''''))))

Rules:

concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Termination of R could not be shown.
Duration:
0:09 minutes