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

         ↳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))

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 w.r.t. to the AFS 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))
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

         ↳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))

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

         ↳AFS

       →DP Problem 2

         ↳Nar

           →DP Problem 4

             ↳Argument Filtering and Ordering

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

The following dependency pairs can be strictly oriented:

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))

The following usable rules for innermost w.r.t. to the AFS 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:

LESS_LEAVES(x₁, x₂) -> LESS_LEAVES(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))
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:
0:01 minutes