Term Rewriting System R:
[N, X, Y, Z, X1, X2]
active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
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(terms(N)) -> CONS(recip(sqr(N)), terms(s(N)))
ACTIVE(terms(N)) -> RECIP(sqr(N))
ACTIVE(terms(N)) -> SQR(N)
ACTIVE(terms(N)) -> TERMS(s(N))
ACTIVE(terms(N)) -> S(N)
ACTIVE(sqr(s(X))) -> S(add(sqr(X), dbl(X)))
ACTIVE(sqr(s(X))) -> ADD(sqr(X), dbl(X))
ACTIVE(sqr(s(X))) -> SQR(X)
ACTIVE(sqr(s(X))) -> DBL(X)
ACTIVE(dbl(s(X))) -> S(s(dbl(X)))
ACTIVE(dbl(s(X))) -> S(dbl(X))
ACTIVE(dbl(s(X))) -> DBL(X)
ACTIVE(add(s(X), Y)) -> S(add(X, Y))
ACTIVE(add(s(X), Y)) -> ADD(X, Y)
ACTIVE(first(s(X), cons(Y, Z))) -> CONS(Y, first(X, Z))
ACTIVE(first(s(X), cons(Y, Z))) -> FIRST(X, Z)
ACTIVE(half(s(s(X)))) -> S(half(X))
ACTIVE(half(s(s(X)))) -> HALF(X)
ACTIVE(terms(X)) -> TERMS(active(X))
ACTIVE(terms(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(recip(X)) -> RECIP(active(X))
ACTIVE(recip(X)) -> ACTIVE(X)
ACTIVE(sqr(X)) -> SQR(active(X))
ACTIVE(sqr(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(add(X1, X2)) -> ADD(active(X1), X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ADD(X1, active(X2))
ACTIVE(add(X1, X2)) -> ACTIVE(X2)
ACTIVE(dbl(X)) -> DBL(active(X))
ACTIVE(dbl(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> FIRST(active(X1), X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(first(X1, X2)) -> FIRST(X1, active(X2))
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(half(X)) -> HALF(active(X))
ACTIVE(half(X)) -> ACTIVE(X)
TERMS(mark(X)) -> TERMS(X)
TERMS(ok(X)) -> TERMS(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
RECIP(mark(X)) -> RECIP(X)
RECIP(ok(X)) -> RECIP(X)
SQR(mark(X)) -> SQR(X)
SQR(ok(X)) -> SQR(X)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
ADD(mark(X1), X2) -> ADD(X1, X2)
ADD(X1, mark(X2)) -> ADD(X1, X2)
ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
DBL(mark(X)) -> DBL(X)
DBL(ok(X)) -> DBL(X)
FIRST(mark(X1), X2) -> FIRST(X1, X2)
FIRST(X1, mark(X2)) -> FIRST(X1, X2)
FIRST(ok(X1), ok(X2)) -> FIRST(X1, X2)
HALF(mark(X)) -> HALF(X)
HALF(ok(X)) -> HALF(X)
PROPER(terms(X)) -> TERMS(proper(X))
PROPER(terms(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(recip(X)) -> RECIP(proper(X))
PROPER(recip(X)) -> PROPER(X)
PROPER(sqr(X)) -> SQR(proper(X))
PROPER(sqr(X)) -> PROPER(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(add(X1, X2)) -> ADD(proper(X1), proper(X2))
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(dbl(X)) -> DBL(proper(X))
PROPER(dbl(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> FIRST(proper(X1), proper(X2))
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(half(X)) -> HALF(proper(X))
PROPER(half(X)) -> PROPER(X)
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 12 SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

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


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
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)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  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 13
             ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:

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


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

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


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(CONS(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 13
             ↳Polo
             ...
               →DP Problem 14
                 ↳Dependency Graph
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polynomial Ordering
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

RECIP(ok(X)) -> RECIP(X)
RECIP(mark(X)) -> RECIP(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

RECIP(ok(X)) -> RECIP(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(RECIP(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 15
             ↳Polynomial Ordering
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:

RECIP(mark(X)) -> RECIP(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

RECIP(mark(X)) -> RECIP(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(RECIP(x1))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 15
             ↳Polo
             ...
               →DP Problem 16
                 ↳Dependency Graph
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polynomial Ordering
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

SQR(ok(X)) -> SQR(X)
SQR(mark(X)) -> SQR(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

SQR(ok(X)) -> SQR(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  x1  
  POL(SQR(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
           →DP Problem 17
             ↳Polynomial Ordering
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:

SQR(mark(X)) -> SQR(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

SQR(mark(X)) -> SQR(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(SQR(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
           →DP Problem 17
             ↳Polo
             ...
               →DP Problem 18
                 ↳Dependency Graph
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polynomial Ordering
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

TERMS(ok(X)) -> TERMS(X)
TERMS(mark(X)) -> TERMS(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

TERMS(ok(X)) -> TERMS(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(TERMS(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
           →DP Problem 19
             ↳Polynomial Ordering
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:

TERMS(mark(X)) -> TERMS(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

TERMS(mark(X)) -> TERMS(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(TERMS(x1))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
           →DP Problem 19
             ↳Polo
             ...
               →DP Problem 20
                 ↳Dependency Graph
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polynomial Ordering
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

S(ok(X)) -> S(X)
S(mark(X)) -> S(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

S(ok(X)) -> S(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(S(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
           →DP Problem 21
             ↳Polynomial Ordering
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:

S(mark(X)) -> S(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

S(mark(X)) -> S(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(S(x1))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
           →DP Problem 21
             ↳Polo
             ...
               →DP Problem 22
                 ↳Dependency Graph
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polynomial Ordering
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

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


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

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


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  
  POL(ADD(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
           →DP Problem 23
             ↳Polynomial Ordering
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

ADD(X1, mark(X2)) -> ADD(X1, X2)
ADD(mark(X1), X2) -> ADD(X1, X2)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ADD(mark(X1), X2) -> ADD(X1, X2)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(ADD(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
           →DP Problem 23
             ↳Polo
             ...
               →DP Problem 24
                 ↳Polynomial Ordering
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:

ADD(X1, mark(X2)) -> ADD(X1, X2)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ADD(X1, mark(X2)) -> ADD(X1, X2)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(ADD(x1, x2))=  x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
           →DP Problem 23
             ↳Polo
             ...
               →DP Problem 25
                 ↳Dependency Graph
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

DBL(ok(X)) -> DBL(X)
DBL(mark(X)) -> DBL(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

DBL(ok(X)) -> DBL(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  
  POL(DBL(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 26
             ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:

DBL(mark(X)) -> DBL(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

DBL(mark(X)) -> DBL(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(DBL(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 26
             ↳Polo
             ...
               →DP Problem 27
                 ↳Dependency Graph
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polynomial Ordering
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

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


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

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


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FIRST(x1, x2))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 28
             ↳Polynomial Ordering
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

FIRST(X1, mark(X2)) -> FIRST(X1, X2)
FIRST(mark(X1), X2) -> FIRST(X1, X2)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

FIRST(mark(X1), X2) -> FIRST(X1, X2)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FIRST(x1, x2))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 28
             ↳Polo
             ...
               →DP Problem 29
                 ↳Polynomial Ordering
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:

FIRST(X1, mark(X2)) -> FIRST(X1, X2)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

FIRST(X1, mark(X2)) -> FIRST(X1, X2)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FIRST(x1, x2))=  x2  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 28
             ↳Polo
             ...
               →DP Problem 30
                 ↳Dependency Graph
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polynomial Ordering
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

HALF(ok(X)) -> HALF(X)
HALF(mark(X)) -> HALF(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

HALF(ok(X)) -> HALF(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(HALF(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
           →DP Problem 31
             ↳Polynomial Ordering
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:

HALF(mark(X)) -> HALF(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

HALF(mark(X)) -> HALF(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(HALF(x1))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
           →DP Problem 31
             ↳Polo
             ...
               →DP Problem 32
                 ↳Dependency Graph
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polynomial Ordering
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

ACTIVE(half(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbl(X)) -> ACTIVE(X)
ACTIVE(add(X1, X2)) -> ACTIVE(X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(sqr(X)) -> ACTIVE(X)
ACTIVE(recip(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(terms(X)) -> ACTIVE(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

ACTIVE(add(X1, X2)) -> ACTIVE(X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(first(x1, x2))=  x1 + x2  
  POL(cons(x1, x2))=  x1  
  POL(sqr(x1))=  x1  
  POL(dbl(x1))=  x1  
  POL(s(x1))=  x1  
  POL(terms(x1))=  x1  
  POL(recip(x1))=  x1  
  POL(half(x1))=  x1  
  POL(add(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
           →DP Problem 33
             ↳Polynomial Ordering
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

ACTIVE(half(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbl(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(sqr(X)) -> ACTIVE(X)
ACTIVE(recip(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(terms(X)) -> ACTIVE(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ACTIVE(s(X)) -> ACTIVE(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(first(x1, x2))=  x1 + x2  
  POL(cons(x1, x2))=  x1  
  POL(sqr(x1))=  x1  
  POL(dbl(x1))=  x1  
  POL(s(x1))=  1 + x1  
  POL(terms(x1))=  x1  
  POL(recip(x1))=  x1  
  POL(half(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
           →DP Problem 33
             ↳Polo
             ...
               →DP Problem 34
                 ↳Polynomial Ordering
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

ACTIVE(half(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbl(X)) -> ACTIVE(X)
ACTIVE(sqr(X)) -> ACTIVE(X)
ACTIVE(recip(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(terms(X)) -> ACTIVE(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ACTIVE(sqr(X)) -> ACTIVE(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(first(x1, x2))=  x1 + x2  
  POL(cons(x1, x2))=  x1  
  POL(sqr(x1))=  1 + x1  
  POL(dbl(x1))=  x1  
  POL(terms(x1))=  x1  
  POL(recip(x1))=  x1  
  POL(half(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
           →DP Problem 33
             ↳Polo
             ...
               →DP Problem 35
                 ↳Polynomial Ordering
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

ACTIVE(half(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbl(X)) -> ACTIVE(X)
ACTIVE(recip(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(terms(X)) -> ACTIVE(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ACTIVE(recip(X)) -> ACTIVE(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(first(x1, x2))=  x1 + x2  
  POL(cons(x1, x2))=  x1  
  POL(dbl(x1))=  x1  
  POL(terms(x1))=  x1  
  POL(recip(x1))=  1 + x1  
  POL(half(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
           →DP Problem 33
             ↳Polo
             ...
               →DP Problem 36
                 ↳Polynomial Ordering
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

ACTIVE(half(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbl(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(terms(X)) -> ACTIVE(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ACTIVE(cons(X1, X2)) -> ACTIVE(X1)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(first(x1, x2))=  x1 + x2  
  POL(cons(x1, x2))=  1 + x1  
  POL(dbl(x1))=  x1  
  POL(terms(x1))=  x1  
  POL(half(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
           →DP Problem 33
             ↳Polo
             ...
               →DP Problem 37
                 ↳Polynomial Ordering
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

ACTIVE(half(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbl(X)) -> ACTIVE(X)
ACTIVE(terms(X)) -> ACTIVE(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ACTIVE(terms(X)) -> ACTIVE(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(first(x1, x2))=  x1 + x2  
  POL(dbl(x1))=  x1  
  POL(terms(x1))=  1 + x1  
  POL(half(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
           →DP Problem 33
             ↳Polo
             ...
               →DP Problem 38
                 ↳Polynomial Ordering
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

ACTIVE(half(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbl(X)) -> ACTIVE(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ACTIVE(half(X)) -> ACTIVE(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(first(x1, x2))=  x1 + x2  
  POL(dbl(x1))=  x1  
  POL(half(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
           →DP Problem 33
             ↳Polo
             ...
               →DP Problem 39
                 ↳Polynomial Ordering
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbl(X)) -> ACTIVE(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(first(x1, x2))=  1 + x1 + x2  
  POL(dbl(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
           →DP Problem 33
             ↳Polo
             ...
               →DP Problem 40
                 ↳Polynomial Ordering
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:

ACTIVE(dbl(X)) -> ACTIVE(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ACTIVE(dbl(X)) -> ACTIVE(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(dbl(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
           →DP Problem 33
             ↳Polo
             ...
               →DP Problem 41
                 ↳Dependency Graph
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining


Dependency Pair:


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polynomial Ordering
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

PROPER(half(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(dbl(X)) -> PROPER(X)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(sqr(X)) -> PROPER(X)
PROPER(recip(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(terms(X)) -> PROPER(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(first(x1, x2))=  x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(sqr(x1))=  x1  
  POL(dbl(x1))=  x1  
  POL(s(x1))=  x1  
  POL(terms(x1))=  x1  
  POL(recip(x1))=  x1  
  POL(half(x1))=  x1  
  POL(add(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
           →DP Problem 42
             ↳Polynomial Ordering
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

PROPER(half(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(dbl(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(sqr(X)) -> PROPER(X)
PROPER(recip(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(terms(X)) -> PROPER(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

PROPER(s(X)) -> PROPER(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(first(x1, x2))=  x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(sqr(x1))=  x1  
  POL(dbl(x1))=  x1  
  POL(s(x1))=  1 + x1  
  POL(terms(x1))=  x1  
  POL(recip(x1))=  x1  
  POL(half(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
           →DP Problem 42
             ↳Polo
             ...
               →DP Problem 43
                 ↳Polynomial Ordering
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

PROPER(half(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(dbl(X)) -> PROPER(X)
PROPER(sqr(X)) -> PROPER(X)
PROPER(recip(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(terms(X)) -> PROPER(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

PROPER(sqr(X)) -> PROPER(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(first(x1, x2))=  x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(sqr(x1))=  1 + x1  
  POL(dbl(x1))=  x1  
  POL(terms(x1))=  x1  
  POL(recip(x1))=  x1  
  POL(half(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
           →DP Problem 42
             ↳Polo
             ...
               →DP Problem 44
                 ↳Polynomial Ordering
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

PROPER(half(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(dbl(X)) -> PROPER(X)
PROPER(recip(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(terms(X)) -> PROPER(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

PROPER(recip(X)) -> PROPER(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(first(x1, x2))=  x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(dbl(x1))=  x1  
  POL(terms(x1))=  x1  
  POL(recip(x1))=  1 + x1  
  POL(half(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
           →DP Problem 42
             ↳Polo
             ...
               →DP Problem 45
                 ↳Polynomial Ordering
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

PROPER(half(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(dbl(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(terms(X)) -> PROPER(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(first(x1, x2))=  x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  1 + x1 + x2  
  POL(dbl(x1))=  x1  
  POL(terms(x1))=  x1  
  POL(half(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
           →DP Problem 42
             ↳Polo
             ...
               →DP Problem 46
                 ↳Polynomial Ordering
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

PROPER(half(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(dbl(X)) -> PROPER(X)
PROPER(terms(X)) -> PROPER(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

PROPER(terms(X)) -> PROPER(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(first(x1, x2))=  x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(dbl(x1))=  x1  
  POL(terms(x1))=  1 + x1  
  POL(half(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
           →DP Problem 42
             ↳Polo
             ...
               →DP Problem 47
                 ↳Polynomial Ordering
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

PROPER(half(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(dbl(X)) -> PROPER(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

PROPER(half(X)) -> PROPER(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(first(x1, x2))=  x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(dbl(x1))=  x1  
  POL(half(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
           →DP Problem 42
             ↳Polo
             ...
               →DP Problem 48
                 ↳Polynomial Ordering
       →DP Problem 12
         ↳Remaining


Dependency Pairs:

PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(dbl(X)) -> PROPER(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(first(x1, x2))=  1 + x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(dbl(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
           →DP Problem 42
             ↳Polo
             ...
               →DP Problem 49
                 ↳Polynomial Ordering
       →DP Problem 12
         ↳Remaining


Dependency Pair:

PROPER(dbl(X)) -> PROPER(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

PROPER(dbl(X)) -> PROPER(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(dbl(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
           →DP Problem 42
             ↳Polo
             ...
               →DP Problem 50
                 ↳Dependency Graph
       →DP Problem 12
         ↳Remaining


Dependency Pair:


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))




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