Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳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.

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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))

Strategy:

innermost

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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 6

             ↳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))

Strategy:

innermost

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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 7

                 ↳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))

Strategy:

innermost

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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 8

                 ↳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))

Strategy:

innermost

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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 9

                 ↳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))

Strategy:

innermost

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:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 10

                 ↳Polynomial 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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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'', 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')))

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(leaf) = 1

POL(concat(x₁, x₂)) = x₁ + x₂

POL(LESSLEAVES(x₁, x₂)) = x₁

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

Strategy:

innermost

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(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(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))

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(leaf) = 0

POL(concat(x₁, x₂)) = x₁ + x₂

POL(LESSLEAVES(x₁, x₂)) = x₁

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))
lessleaves(X, leaf) -> false
lessleaves(leaf, cons(W, Z)) -> true
lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(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(leaf)	= 1
POL(concat(x₁, x₂))	= x₁ + x₂
POL(LESSLEAVES(x₁, x₂))	= x₁

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