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

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳Nar

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

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))
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

         ↳AFS

       →DP Problem 2

         ↳Narrowing Transformation

Dependency Pair:

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

Rules:

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

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

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Argument Filtering and Ordering

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

The following dependency pairs can be strictly oriented:

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))

The following usable rules for innermost can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{concat, cons}

resulting in one new DP problem.
Used Argument Filtering System:

LESSLEAVES(x₁, x₂) -> LESSLEAVES(x₁, x₂)
cons(x₁, x₂) -> cons(x₁, x₂)
concat(x₁, x₂) -> concat(x₁, x₂)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳AFS

...

               →DP Problem 5

                 ↳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:
0:01 minutes