Term Rewriting System R:
[y, u, v, 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.



   R
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.


   R
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 w.r.t. to the AFS 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)


   R
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.


   R
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:



   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
Nar
           →DP Problem 4
Remaining Obligation(s)




The following remains to be proven:
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



Innermost Termination of R could not be shown.
Duration:
0:01 minutes