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
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo


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 implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(cons(x1, x2))=  1 + x2  
  POL(CONCAT(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 3
             ↳Dependency Graph
       →DP Problem 2
         ↳Polo


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
         ↳Polo
       →DP Problem 2
         ↳Polynomial Ordering


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




The following dependency pair can be strictly oriented:

LESSLEAVES(cons(u, v), cons(w, z)) -> LESSLEAVES(concat(u, v), 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))=  x2  
  POL(leaf)=  1  
  POL(concat(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 4
             ↳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:
0:00 minutes