Term Rewriting System R:
[X, Z, N, Y, X1, X2]
active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(2ndspos(s(N), cons(X, Z))) -> 2NDSPOS(s(N), cons2(X, Z))
ACTIVE(2ndspos(s(N), cons(X, Z))) -> CONS2(X, Z)
ACTIVE(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> RCONS(posrecip(Y), 2ndsneg(N, Z))
ACTIVE(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> POSRECIP(Y)
ACTIVE(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> 2NDSNEG(N, Z)
ACTIVE(2ndsneg(s(N), cons(X, Z))) -> 2NDSNEG(s(N), cons2(X, Z))
ACTIVE(2ndsneg(s(N), cons(X, Z))) -> CONS2(X, Z)
ACTIVE(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> RCONS(negrecip(Y), 2ndspos(N, Z))
ACTIVE(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> NEGRECIP(Y)
ACTIVE(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> 2NDSPOS(N, Z)
ACTIVE(pi(X)) -> 2NDSPOS(X, from(0))
ACTIVE(pi(X)) -> FROM(0)
ACTIVE(plus(s(X), Y)) -> S(plus(X, Y))
ACTIVE(plus(s(X), Y)) -> PLUS(X, Y)
ACTIVE(times(s(X), Y)) -> PLUS(Y, times(X, Y))
ACTIVE(times(s(X), Y)) -> TIMES(X, Y)
ACTIVE(square(X)) -> TIMES(X, X)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(posrecip(X)) -> POSRECIP(active(X))
ACTIVE(posrecip(X)) -> ACTIVE(X)
ACTIVE(negrecip(X)) -> NEGRECIP(active(X))
ACTIVE(negrecip(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons2(X1, X2)) -> CONS2(X1, active(X2))
ACTIVE(cons2(X1, X2)) -> ACTIVE(X2)
ACTIVE(rcons(X1, X2)) -> RCONS(active(X1), X2)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X1)
ACTIVE(rcons(X1, X2)) -> RCONS(X1, active(X2))
ACTIVE(rcons(X1, X2)) -> ACTIVE(X2)
ACTIVE(from(X)) -> FROM(active(X))
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(2ndspos(X1, X2)) -> 2NDSPOS(active(X1), X2)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) -> 2NDSPOS(X1, active(X2))
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndsneg(X1, X2)) -> 2NDSNEG(active(X1), X2)
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X1)
ACTIVE(2ndsneg(X1, X2)) -> 2NDSNEG(X1, active(X2))
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X2)
ACTIVE(pi(X)) -> PI(active(X))
ACTIVE(pi(X)) -> ACTIVE(X)
ACTIVE(plus(X1, X2)) -> PLUS(active(X1), X2)
ACTIVE(plus(X1, X2)) -> ACTIVE(X1)
ACTIVE(plus(X1, X2)) -> PLUS(X1, active(X2))
ACTIVE(plus(X1, X2)) -> ACTIVE(X2)
ACTIVE(times(X1, X2)) -> TIMES(active(X1), X2)
ACTIVE(times(X1, X2)) -> ACTIVE(X1)
ACTIVE(times(X1, X2)) -> TIMES(X1, active(X2))
ACTIVE(times(X1, X2)) -> ACTIVE(X2)
ACTIVE(square(X)) -> SQUARE(active(X))
ACTIVE(square(X)) -> ACTIVE(X)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
POSRECIP(mark(X)) -> POSRECIP(X)
POSRECIP(ok(X)) -> POSRECIP(X)
NEGRECIP(mark(X)) -> NEGRECIP(X)
NEGRECIP(ok(X)) -> NEGRECIP(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS2(X1, mark(X2)) -> CONS2(X1, X2)
CONS2(ok(X1), ok(X2)) -> CONS2(X1, X2)
RCONS(mark(X1), X2) -> RCONS(X1, X2)
RCONS(X1, mark(X2)) -> RCONS(X1, X2)
RCONS(ok(X1), ok(X2)) -> RCONS(X1, X2)
FROM(mark(X)) -> FROM(X)
FROM(ok(X)) -> FROM(X)
2NDSPOS(mark(X1), X2) -> 2NDSPOS(X1, X2)
2NDSPOS(X1, mark(X2)) -> 2NDSPOS(X1, X2)
2NDSPOS(ok(X1), ok(X2)) -> 2NDSPOS(X1, X2)
2NDSNEG(mark(X1), X2) -> 2NDSNEG(X1, X2)
2NDSNEG(X1, mark(X2)) -> 2NDSNEG(X1, X2)
2NDSNEG(ok(X1), ok(X2)) -> 2NDSNEG(X1, X2)
PI(mark(X)) -> PI(X)
PI(ok(X)) -> PI(X)
PLUS(mark(X1), X2) -> PLUS(X1, X2)
PLUS(X1, mark(X2)) -> PLUS(X1, X2)
PLUS(ok(X1), ok(X2)) -> PLUS(X1, X2)
TIMES(mark(X1), X2) -> TIMES(X1, X2)
TIMES(X1, mark(X2)) -> TIMES(X1, X2)
TIMES(ok(X1), ok(X2)) -> TIMES(X1, X2)
SQUARE(mark(X)) -> SQUARE(X)
SQUARE(ok(X)) -> SQUARE(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(posrecip(X)) -> POSRECIP(proper(X))
PROPER(posrecip(X)) -> PROPER(X)
PROPER(negrecip(X)) -> NEGRECIP(proper(X))
PROPER(negrecip(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(cons2(X1, X2)) -> CONS2(proper(X1), proper(X2))
PROPER(cons2(X1, X2)) -> PROPER(X1)
PROPER(cons2(X1, X2)) -> PROPER(X2)
PROPER(rcons(X1, X2)) -> RCONS(proper(X1), proper(X2))
PROPER(rcons(X1, X2)) -> PROPER(X1)
PROPER(rcons(X1, X2)) -> PROPER(X2)
PROPER(from(X)) -> FROM(proper(X))
PROPER(from(X)) -> PROPER(X)
PROPER(2ndspos(X1, X2)) -> 2NDSPOS(proper(X1), proper(X2))
PROPER(2ndspos(X1, X2)) -> PROPER(X1)
PROPER(2ndspos(X1, X2)) -> PROPER(X2)
PROPER(2ndsneg(X1, X2)) -> 2NDSNEG(proper(X1), proper(X2))
PROPER(2ndsneg(X1, X2)) -> PROPER(X1)
PROPER(2ndsneg(X1, X2)) -> PROPER(X2)
PROPER(pi(X)) -> PI(proper(X))
PROPER(pi(X)) -> PROPER(X)
PROPER(plus(X1, X2)) -> PLUS(proper(X1), proper(X2))
PROPER(plus(X1, X2)) -> PROPER(X1)
PROPER(plus(X1, X2)) -> PROPER(X2)
PROPER(times(X1, X2)) -> TIMES(proper(X1), proper(X2))
PROPER(times(X1, X2)) -> PROPER(X1)
PROPER(times(X1, X2)) -> PROPER(X2)
PROPER(square(X)) -> SQUARE(proper(X))
PROPER(square(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 16 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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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 w.r.t. to the implicit AFS 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 17
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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 w.r.t. to the implicit AFS 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 17
Polo
             ...
               →DP Problem 18
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

FROM(ok(X)) -> FROM(X)
FROM(mark(X)) -> FROM(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

FROM(ok(X)) -> FROM(X)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FROM(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 19
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:

FROM(mark(X)) -> FROM(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

FROM(mark(X)) -> FROM(X)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FROM(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 19
Polo
             ...
               →DP Problem 20
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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 w.r.t. to the implicit AFS 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 21
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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 w.r.t. to the implicit AFS 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 21
Polo
             ...
               →DP Problem 22
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

2NDSPOS(ok(X1), ok(X2)) -> 2NDSPOS(X1, X2)
2NDSPOS(X1, mark(X2)) -> 2NDSPOS(X1, X2)
2NDSPOS(mark(X1), X2) -> 2NDSPOS(X1, X2)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

2NDSPOS(ok(X1), ok(X2)) -> 2NDSPOS(X1, X2)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(2NDSPOS(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 23
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

2NDSPOS(X1, mark(X2)) -> 2NDSPOS(X1, X2)
2NDSPOS(mark(X1), X2) -> 2NDSPOS(X1, X2)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

2NDSPOS(X1, mark(X2)) -> 2NDSPOS(X1, X2)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(2NDSPOS(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 23
Polo
             ...
               →DP Problem 24
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:

2NDSPOS(mark(X1), X2) -> 2NDSPOS(X1, X2)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

2NDSPOS(mark(X1), X2) -> 2NDSPOS(X1, X2)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(2NDSPOS(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 23
Polo
             ...
               →DP Problem 25
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

CONS2(ok(X1), ok(X2)) -> CONS2(X1, X2)
CONS2(X1, mark(X2)) -> CONS2(X1, X2)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

CONS2(ok(X1), ok(X2)) -> CONS2(X1, X2)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(CONS2(x1, x2))=  x1  
  POL(mark(x1))=  0  
  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 26
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:

CONS2(X1, mark(X2)) -> CONS2(X1, X2)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

CONS2(X1, mark(X2)) -> CONS2(X1, X2)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(CONS2(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 26
Polo
             ...
               →DP Problem 27
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  x1  
  POL(RCONS(x1, x2))=  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 28
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(RCONS(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 28
Polo
             ...
               →DP Problem 29
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(RCONS(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 28
Polo
             ...
               →DP Problem 30
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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

POSRECIP(ok(X)) -> POSRECIP(X)
POSRECIP(mark(X)) -> POSRECIP(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

POSRECIP(ok(X)) -> POSRECIP(X)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  
  POL(POSRECIP(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 31
Polynomial Ordering
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:

POSRECIP(mark(X)) -> POSRECIP(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

POSRECIP(mark(X)) -> POSRECIP(X)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(POSRECIP(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 31
Polo
             ...
               →DP Problem 32
Dependency Graph
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

2NDSNEG(ok(X1), ok(X2)) -> 2NDSNEG(X1, X2)
2NDSNEG(X1, mark(X2)) -> 2NDSNEG(X1, X2)
2NDSNEG(mark(X1), X2) -> 2NDSNEG(X1, X2)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

2NDSNEG(ok(X1), ok(X2)) -> 2NDSNEG(X1, X2)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(2NDSNEG(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 33
Polynomial Ordering
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

2NDSNEG(X1, mark(X2)) -> 2NDSNEG(X1, X2)
2NDSNEG(mark(X1), X2) -> 2NDSNEG(X1, X2)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

2NDSNEG(X1, mark(X2)) -> 2NDSNEG(X1, X2)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(2NDSNEG(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 33
Polo
             ...
               →DP Problem 34
Polynomial Ordering
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:

2NDSNEG(mark(X1), X2) -> 2NDSNEG(X1, X2)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

2NDSNEG(mark(X1), X2) -> 2NDSNEG(X1, X2)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(2NDSNEG(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 33
Polo
             ...
               →DP Problem 35
Dependency Graph
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

NEGRECIP(ok(X)) -> NEGRECIP(X)
NEGRECIP(mark(X)) -> NEGRECIP(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

NEGRECIP(ok(X)) -> NEGRECIP(X)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(NEGRECIP(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 36
Polynomial Ordering
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:

NEGRECIP(mark(X)) -> NEGRECIP(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

NEGRECIP(mark(X)) -> NEGRECIP(X)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(NEGRECIP(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 36
Polo
             ...
               →DP Problem 37
Dependency Graph
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PLUS(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 9
Polo
       →DP Problem 10
Polo
           →DP Problem 38
Polynomial Ordering
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PLUS(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 9
Polo
       →DP Problem 10
Polo
           →DP Problem 38
Polo
             ...
               →DP Problem 39
Polynomial Ordering
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PLUS(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 9
Polo
       →DP Problem 10
Polo
           →DP Problem 38
Polo
             ...
               →DP Problem 40
Dependency Graph
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(TIMES(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 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
           →DP Problem 41
Polynomial Ordering
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pairs:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(TIMES(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 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
           →DP Problem 41
Polo
             ...
               →DP Problem 42
Polynomial Ordering
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(TIMES(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 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
           →DP Problem 41
Polo
             ...
               →DP Problem 43
Dependency Graph
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Remaining


Dependency Pair:


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, Z))) -> mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) -> mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(cons2(X1, X2)) -> cons2(X1, active(X2))
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons2(X1, mark(X2)) -> mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) -> ok(cons2(X1, X2))
rcons(mark(X1), X2) -> mark(rcons(X1, X2))
rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
times(mark(X1), X2) -> mark(times(X1, X2))
times(X1, mark(X2)) -> mark(times(X1, X2))
times(ok(X1), ok(X2)) -> ok(times(X1, X2))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(cons2(X1, X2)) -> cons2(proper(X1), proper(X2))
proper(rnil) -> ok(rnil)
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper