Term Rewriting System R:
[X, XS, N, X1, X2]
active(from(X)) -> mark(cons(X, from(s(X))))
active(after(0, XS)) -> mark(XS)
active(after(s(N), cons(X, XS))) -> mark(after(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(after(X1, X2)) -> after(active(X1), X2)
active(after(X1, X2)) -> after(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
after(mark(X1), X2) -> mark(after(X1, X2))
after(X1, mark(X2)) -> mark(after(X1, X2))
after(ok(X1), ok(X2)) -> ok(after(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(after(X1, X2)) -> after(proper(X1), proper(X2))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(after(s(N), cons(X, XS))) -> AFTER(N, XS)
ACTIVE(from(X)) -> FROM(active(X))
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(after(X1, X2)) -> AFTER(active(X1), X2)
ACTIVE(after(X1, X2)) -> ACTIVE(X1)
ACTIVE(after(X1, X2)) -> AFTER(X1, active(X2))
ACTIVE(after(X1, X2)) -> ACTIVE(X2)
FROM(mark(X)) -> FROM(X)
FROM(ok(X)) -> FROM(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
AFTER(mark(X1), X2) -> AFTER(X1, X2)
AFTER(X1, mark(X2)) -> AFTER(X1, X2)
AFTER(ok(X1), ok(X2)) -> AFTER(X1, X2)
PROPER(from(X)) -> FROM(proper(X))
PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(after(X1, X2)) -> AFTER(proper(X1), proper(X2))
PROPER(after(X1, X2)) -> PROPER(X1)
PROPER(after(X1, X2)) -> PROPER(X2)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains seven SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Remaining
       →DP Problem 3
         ↳Remaining
       →DP Problem 4
         ↳Remaining
       →DP Problem 5
         ↳Remaining
       →DP Problem 6
         ↳Remaining
       →DP Problem 7
         ↳Remaining


Dependency Pairs:

CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(after(0, XS)) -> mark(XS)
active(after(s(N), cons(X, XS))) -> mark(after(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(after(X1, X2)) -> after(active(X1), X2)
active(after(X1, X2)) -> after(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
after(mark(X1), X2) -> mark(after(X1, X2))
after(X1, mark(X2)) -> mark(after(X1, X2))
after(ok(X1), ok(X2)) -> ok(after(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(after(X1, X2)) -> after(proper(X1), proper(X2))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

CONS(ok(X1), ok(X2)) -> CONS(X1, X2)


Additionally, the following rules can be oriented:

active(from(X)) -> mark(cons(X, from(s(X))))
active(after(0, XS)) -> mark(XS)
active(after(s(N), cons(X, XS))) -> mark(after(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(after(X1, X2)) -> after(active(X1), X2)
active(after(X1, X2)) -> after(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
after(mark(X1), X2) -> mark(after(X1, X2))
after(X1, mark(X2)) -> mark(after(X1, X2))
after(ok(X1), ok(X2)) -> ok(after(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(after(X1, X2)) -> after(proper(X1), proper(X2))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(top(x1))=  0  
  POL(active(x1))=  x1  
  POL(proper(x1))=  1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x2  
  POL(s(x1))=  x1  
  POL(after(x1, x2))=  x2  
  POL(mark(x1))=  0  
  POL(ok(x1))=  1 + x1  
  POL(CONS(x1, x2))=  x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Remaining Obligation(s)
       →DP Problem 3
         ↳Remaining Obligation(s)
       →DP Problem 4
         ↳Remaining Obligation(s)
       →DP Problem 5
         ↳Remaining Obligation(s)
       →DP Problem 6
         ↳Remaining Obligation(s)
       →DP Problem 7
         ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Remaining Obligation(s)
       →DP Problem 3
         ↳Remaining Obligation(s)
       →DP Problem 4
         ↳Remaining Obligation(s)
       →DP Problem 5
         ↳Remaining Obligation(s)
       →DP Problem 6
         ↳Remaining Obligation(s)
       →DP Problem 7
         ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Remaining Obligation(s)
       →DP Problem 3
         ↳Remaining Obligation(s)
       →DP Problem 4
         ↳Remaining Obligation(s)
       →DP Problem 5
         ↳Remaining Obligation(s)
       →DP Problem 6
         ↳Remaining Obligation(s)
       →DP Problem 7
         ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Remaining Obligation(s)
       →DP Problem 3
         ↳Remaining Obligation(s)
       →DP Problem 4
         ↳Remaining Obligation(s)
       →DP Problem 5
         ↳Remaining Obligation(s)
       →DP Problem 6
         ↳Remaining Obligation(s)
       →DP Problem 7
         ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Remaining Obligation(s)
       →DP Problem 3
         ↳Remaining Obligation(s)
       →DP Problem 4
         ↳Remaining Obligation(s)
       →DP Problem 5
         ↳Remaining Obligation(s)
       →DP Problem 6
         ↳Remaining Obligation(s)
       →DP Problem 7
         ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Remaining Obligation(s)
       →DP Problem 3
         ↳Remaining Obligation(s)
       →DP Problem 4
         ↳Remaining Obligation(s)
       →DP Problem 5
         ↳Remaining Obligation(s)
       →DP Problem 6
         ↳Remaining Obligation(s)
       →DP Problem 7
         ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Remaining Obligation(s)
       →DP Problem 3
         ↳Remaining Obligation(s)
       →DP Problem 4
         ↳Remaining Obligation(s)
       →DP Problem 5
         ↳Remaining Obligation(s)
       →DP Problem 6
         ↳Remaining Obligation(s)
       →DP Problem 7
         ↳Remaining Obligation(s)




The following remains to be proven:

Termination of R could not be shown.
Duration:
0:00 minutes