Term Rewriting System R:
[y, z, x]
f(cons(nil, y)) -> y
f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
copy(0, y, z) -> f(z)
copy(s(x), y, z) -> copy(x, y, cons(f(y), z))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(cons(f(cons(nil, y)), z)) -> COPY(n, y, z)
COPY(0, y, z) -> F(z)
COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))
COPY(s(x), y, z) -> F(y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pair:

COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))


Rules:


f(cons(nil, y)) -> y
f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
copy(0, y, z) -> f(z)
copy(s(x), y, z) -> copy(x, y, cons(f(y), z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

COPY(s(x), y, z) -> COPY(x, y, cons(f(y), z))


The following usable rules for innermost can be oriented:

f(cons(nil, y)) -> y
f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
copy(0, y, z) -> f(z)
copy(s(x), y, z) -> copy(x, y, cons(f(y), z))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(COPY(x1, x2, x3))=  x1 + x2 + x3  
  POL(s(x1))=  1 + x1  
  POL(f(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
COPY(x1, x2, x3) -> COPY(x1, x2, x3)
s(x1) -> s(x1)
cons(x1, x2) -> x2
f(x1) -> f(x1)
copy(x1, x2, x3) -> x3


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:


Rules:


f(cons(nil, y)) -> y
f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
copy(0, y, z) -> f(z)
copy(s(x), y, z) -> copy(x, y, cons(f(y), z))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes