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))

Innermost 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

         ↳Forward Instantiation Transformation

       →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))

Strategy:

innermost

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

CONCAT(cons(u, v), y) -> CONCAT(v, y)

one new Dependency Pair is created:

CONCAT(cons(u, cons(u'', v'')), y'') -> CONCAT(cons(u'', v''), y'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 3

             ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

Dependency Pair:

CONCAT(cons(u, cons(u'', v'')), y'') -> CONCAT(cons(u'', 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))

Strategy:

innermost

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

CONCAT(cons(u, cons(u'', v'')), y'') -> CONCAT(cons(u'', v''), y'')

one new Dependency Pair is created:

CONCAT(cons(u, cons(u''0, cons(u'''', v''''))), y'''') -> CONCAT(cons(u''0, cons(u'''', v'''')), y'''')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 3

             ↳FwdInst

...

               →DP Problem 4

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳Nar

Dependency Pair:

CONCAT(cons(u, cons(u''0, cons(u'''', v''''))), y'''') -> CONCAT(cons(u''0, cons(u'''', 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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

CONCAT(cons(u, cons(u''0, cons(u'''', v''''))), y'''') -> CONCAT(cons(u''0, cons(u'''', v'''')), y'''')

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(cons(x₁, x₂)) = 1 + x₂

POL(CONCAT(x₁, x₂)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 3

             ↳FwdInst

...

               →DP Problem 5

                 ↳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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →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))

Strategy:

innermost

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

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 6

             ↳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))

Strategy:

innermost

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

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 7

                 ↳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))

Strategy:

innermost

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

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 8

                 ↳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))

Strategy:

innermost

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

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 9

                 ↳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))

Strategy:

innermost

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

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 10

                 ↳Polynomial Ordering

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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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(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')))

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS 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)) = x₁

POL(leaf) = 1

POL(concat(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 11

                 ↳Polynomial Ordering

Dependency Pairs:

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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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(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))

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS 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)) = x₁

POL(leaf) = 0

POL(concat(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 12

                 ↳Dependency Graph

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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:04 minutes

POL(cons(x₁, x₂))	= 1 + x₂
POL(CONCAT(x₁, x₂))	= 1 + x₁

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

POL(cons(x₁, x₂))	= 1 + x₁ + x₂
POL(LESS_LEAVES(x₁, x₂))	= x₁
POL(leaf)	= 0
POL(concat(x₁, x₂))	= x₁ + x₂