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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))

Innermost 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, cons(Y, Z)))) -> RCONS(posrecip(Y), 2ndsneg(N, Z))
ACTIVE(2ndspos(s(N), cons(X, cons(Y, Z)))) -> POSRECIP(Y)
ACTIVE(2ndspos(s(N), cons(X, cons(Y, Z)))) -> 2NDSNEG(N, Z)
ACTIVE(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> RCONS(negrecip(Y), 2ndspos(N, Z))
ACTIVE(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> NEGRECIP(Y)
ACTIVE(2ndsneg(s(N), cons(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(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)
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(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 15 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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 16
             ↳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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 16
             ↳Polo
             ...
               →DP Problem 17
                 ↳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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 18
             ↳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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 18
             ↳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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 18
             ↳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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




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
         ↳Remaining


Dependency Pairs:

POSRECIP(mark(X)) -> POSRECIP(X)
POSRECIP(ok(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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

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


Dependency Pair:

POSRECIP(ok(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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  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 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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




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
         ↳Remaining


Dependency Pairs:

2NDSNEG(ok(X1), ok(X2)) -> 2NDSNEG(X1, X2)
2NDSNEG(mark(X1), X2) -> 2NDSNEG(X1, X2)
2NDSNEG(X1, mark(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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 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
         ↳Remaining


Dependency Pairs:

2NDSNEG(mark(X1), X2) -> 2NDSNEG(X1, X2)
2NDSNEG(X1, mark(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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 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
         ↳Remaining


Dependency Pair:

2NDSNEG(X1, mark(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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




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
         ↳Remaining


Dependency Pairs:

NEGRECIP(mark(X)) -> NEGRECIP(X)
NEGRECIP(ok(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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(NEGRECIP(x1))=  x1  
  POL(mark(x1))=  1 + x1  
  POL(ok(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 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
         ↳Remaining


Dependency Pair:

NEGRECIP(ok(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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(NEGRECIP(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 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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




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
         ↳Remaining


Dependency Pairs:

2NDSPOS(ok(X1), ok(X2)) -> 2NDSPOS(X1, X2)
2NDSPOS(mark(X1), X2) -> 2NDSPOS(X1, X2)
2NDSPOS(X1, mark(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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 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
         ↳Remaining


Dependency Pairs:

2NDSPOS(mark(X1), X2) -> 2NDSPOS(X1, X2)
2NDSPOS(X1, mark(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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 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
         ↳Remaining


Dependency Pair:

2NDSPOS(X1, mark(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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




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
         ↳Remaining


Dependency Pairs:

PLUS(ok(X1), ok(X2)) -> PLUS(X1, X2)
PLUS(mark(X1), X2) -> PLUS(X1, X2)
PLUS(X1, mark(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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 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
         ↳Remaining


Dependency Pairs:

PLUS(mark(X1), X2) -> PLUS(X1, X2)
PLUS(X1, mark(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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 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
         ↳Remaining


Dependency Pair:

PLUS(X1, mark(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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




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
         ↳Remaining


Dependency Pairs:

PI(ok(X)) -> PI(X)
PI(mark(X)) -> PI(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PI(ok(X)) -> PI(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PI(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 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
         ↳Remaining


Dependency Pair:

PI(mark(X)) -> PI(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PI(mark(X)) -> PI(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PI(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 10
         ↳Polo
           →DP Problem 38
             ↳Polo
             ...
               →DP Problem 39
                 ↳Dependency Graph
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




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
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 40
             ↳Polynomial Ordering
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 40
             ↳Polo
             ...
               →DP Problem 41
                 ↳Polynomial Ordering
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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 40
             ↳Polo
             ...
               →DP Problem 42
                 ↳Dependency Graph
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polynomial Ordering
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

SQUARE(ok(X)) -> SQUARE(X)
SQUARE(mark(X)) -> SQUARE(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

SQUARE(ok(X)) -> SQUARE(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(SQUARE(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 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
           →DP Problem 43
             ↳Polynomial Ordering
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pair:

SQUARE(mark(X)) -> SQUARE(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

SQUARE(mark(X)) -> SQUARE(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(SQUARE(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 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
           →DP Problem 43
             ↳Polo
             ...
               →DP Problem 44
                 ↳Dependency Graph
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polynomial Ordering
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

ACTIVE(square(X)) -> ACTIVE(X)
ACTIVE(times(X1, X2)) -> ACTIVE(X2)
ACTIVE(times(X1, X2)) -> ACTIVE(X1)
ACTIVE(plus(X1, X2)) -> ACTIVE(X2)
ACTIVE(plus(X1, X2)) -> ACTIVE(X1)
ACTIVE(pi(X)) -> ACTIVE(X)
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X2)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(negrecip(X)) -> ACTIVE(X)
ACTIVE(posrecip(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(square(X)) -> ACTIVE(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(plus(x1, x2))=  x1 + x2  
  POL(square(x1))=  1 + x1  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  
  POL(2ndspos(x1, x2))=  x1 + x2  
  POL(2ndsneg(x1, x2))=  x1 + x2  
  POL(ACTIVE(x1))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(times(x1, x2))=  x1 + x2  
  POL(pi(x1))=  x1  
  POL(s(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
           →DP Problem 45
             ↳Polynomial Ordering
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

ACTIVE(times(X1, X2)) -> ACTIVE(X2)
ACTIVE(times(X1, X2)) -> ACTIVE(X1)
ACTIVE(plus(X1, X2)) -> ACTIVE(X2)
ACTIVE(plus(X1, X2)) -> ACTIVE(X1)
ACTIVE(pi(X)) -> ACTIVE(X)
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X2)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(negrecip(X)) -> ACTIVE(X)
ACTIVE(posrecip(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(ACTIVE(x1))=  x1  
  POL(plus(x1, x2))=  x1 + x2  
  POL(cons(x1, x2))=  x1  
  POL(times(x1, x2))=  1 + x1 + x2  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(pi(x1))=  x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  
  POL(2ndspos(x1, x2))=  x1 + x2  
  POL(2ndsneg(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
           →DP Problem 45
             ↳Polo
             ...
               →DP Problem 46
                 ↳Polynomial Ordering
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

ACTIVE(plus(X1, X2)) -> ACTIVE(X2)
ACTIVE(plus(X1, X2)) -> ACTIVE(X1)
ACTIVE(pi(X)) -> ACTIVE(X)
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X2)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(negrecip(X)) -> ACTIVE(X)
ACTIVE(posrecip(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(ACTIVE(x1))=  x1  
  POL(plus(x1, x2))=  1 + x1 + x2  
  POL(cons(x1, x2))=  x1  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(pi(x1))=  x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  
  POL(2ndspos(x1, x2))=  x1 + x2  
  POL(2ndsneg(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
           →DP Problem 45
             ↳Polo
             ...
               →DP Problem 47
                 ↳Polynomial Ordering
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

ACTIVE(pi(X)) -> ACTIVE(X)
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X2)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(negrecip(X)) -> ACTIVE(X)
ACTIVE(posrecip(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(pi(X)) -> ACTIVE(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(ACTIVE(x1))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(pi(x1))=  1 + x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  
  POL(2ndspos(x1, x2))=  x1 + x2  
  POL(2ndsneg(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
           →DP Problem 45
             ↳Polo
             ...
               →DP Problem 48
                 ↳Polynomial Ordering
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X2)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(negrecip(X)) -> ACTIVE(X)
ACTIVE(posrecip(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X1)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(ACTIVE(x1))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  
  POL(2ndspos(x1, x2))=  x1 + x2  
  POL(2ndsneg(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
           →DP Problem 45
             ↳Polo
             ...
               →DP Problem 49
                 ↳Polynomial Ordering
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X2)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(negrecip(X)) -> ACTIVE(X)
ACTIVE(posrecip(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X1)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(ACTIVE(x1))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  
  POL(2ndspos(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
           →DP Problem 45
             ↳Polo
             ...
               →DP Problem 50
                 ↳Polynomial Ordering
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X2)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(negrecip(X)) -> ACTIVE(X)
ACTIVE(posrecip(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(from(X)) -> ACTIVE(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  1 + x1  
  POL(ACTIVE(x1))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
           →DP Problem 45
             ↳Polo
             ...
               →DP Problem 51
                 ↳Polynomial Ordering
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

ACTIVE(rcons(X1, X2)) -> ACTIVE(X2)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(negrecip(X)) -> ACTIVE(X)
ACTIVE(posrecip(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
           →DP Problem 45
             ↳Polo
             ...
               →DP Problem 52
                 ↳Polynomial Ordering
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(negrecip(X)) -> ACTIVE(X)
ACTIVE(posrecip(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(cons(x1, x2))=  1 + x1  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(s(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
           →DP Problem 45
             ↳Polo
             ...
               →DP Problem 53
                 ↳Polynomial Ordering
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

ACTIVE(negrecip(X)) -> ACTIVE(X)
ACTIVE(posrecip(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(negrecip(X)) -> ACTIVE(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(negrecip(x1))=  1 + x1  
  POL(posrecip(x1))=  x1  
  POL(s(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
           →DP Problem 45
             ↳Polo
             ...
               →DP Problem 54
                 ↳Polynomial Ordering
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(posrecip(X)) -> ACTIVE(X)


There are no usable rules for innermost that need to be oriented.

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
           →DP Problem 45
             ↳Polo
             ...
               →DP Problem 55
                 ↳Polynomial Ordering
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining


Dependency Pair:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(s(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 12
         ↳Polo
       →DP Problem 13
         ↳Polo
           →DP Problem 45
             ↳Polo
             ...
               →DP Problem 56
                 ↳Dependency Graph
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polynomial Ordering
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

PROPER(square(X)) -> PROPER(X)
PROPER(times(X1, X2)) -> PROPER(X2)
PROPER(times(X1, X2)) -> PROPER(X1)
PROPER(plus(X1, X2)) -> PROPER(X2)
PROPER(plus(X1, X2)) -> PROPER(X1)
PROPER(pi(X)) -> PROPER(X)
PROPER(2ndsneg(X1, X2)) -> PROPER(X2)
PROPER(2ndsneg(X1, X2)) -> PROPER(X1)
PROPER(2ndspos(X1, X2)) -> PROPER(X2)
PROPER(2ndspos(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)
PROPER(rcons(X1, X2)) -> PROPER(X2)
PROPER(rcons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(negrecip(X)) -> PROPER(X)
PROPER(posrecip(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(square(X)) -> PROPER(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(plus(x1, x2))=  x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(square(x1))=  1 + x1  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  
  POL(2ndspos(x1, x2))=  x1 + x2  
  POL(2ndsneg(x1, x2))=  x1 + x2  
  POL(cons(x1, x2))=  x1 + x2  
  POL(times(x1, x2))=  x1 + x2  
  POL(pi(x1))=  x1  
  POL(s(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
           →DP Problem 57
             ↳Polynomial Ordering
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

PROPER(times(X1, X2)) -> PROPER(X2)
PROPER(times(X1, X2)) -> PROPER(X1)
PROPER(plus(X1, X2)) -> PROPER(X2)
PROPER(plus(X1, X2)) -> PROPER(X1)
PROPER(pi(X)) -> PROPER(X)
PROPER(2ndsneg(X1, X2)) -> PROPER(X2)
PROPER(2ndsneg(X1, X2)) -> PROPER(X1)
PROPER(2ndspos(X1, X2)) -> PROPER(X2)
PROPER(2ndspos(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)
PROPER(rcons(X1, X2)) -> PROPER(X2)
PROPER(rcons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(negrecip(X)) -> PROPER(X)
PROPER(posrecip(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(plus(x1, x2))=  x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(times(x1, x2))=  1 + x1 + x2  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(pi(x1))=  x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  
  POL(2ndspos(x1, x2))=  x1 + x2  
  POL(2ndsneg(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
           →DP Problem 57
             ↳Polo
             ...
               →DP Problem 58
                 ↳Polynomial Ordering
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

PROPER(plus(X1, X2)) -> PROPER(X2)
PROPER(plus(X1, X2)) -> PROPER(X1)
PROPER(pi(X)) -> PROPER(X)
PROPER(2ndsneg(X1, X2)) -> PROPER(X2)
PROPER(2ndsneg(X1, X2)) -> PROPER(X1)
PROPER(2ndspos(X1, X2)) -> PROPER(X2)
PROPER(2ndspos(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)
PROPER(rcons(X1, X2)) -> PROPER(X2)
PROPER(rcons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(negrecip(X)) -> PROPER(X)
PROPER(posrecip(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(plus(x1, x2))=  1 + x1 + x2  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(pi(x1))=  x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  
  POL(2ndspos(x1, x2))=  x1 + x2  
  POL(2ndsneg(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
           →DP Problem 57
             ↳Polo
             ...
               →DP Problem 59
                 ↳Polynomial Ordering
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

PROPER(pi(X)) -> PROPER(X)
PROPER(2ndsneg(X1, X2)) -> PROPER(X2)
PROPER(2ndsneg(X1, X2)) -> PROPER(X1)
PROPER(2ndspos(X1, X2)) -> PROPER(X2)
PROPER(2ndspos(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)
PROPER(rcons(X1, X2)) -> PROPER(X2)
PROPER(rcons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(negrecip(X)) -> PROPER(X)
PROPER(posrecip(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(pi(X)) -> PROPER(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(pi(x1))=  1 + x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  
  POL(2ndspos(x1, x2))=  x1 + x2  
  POL(2ndsneg(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
           →DP Problem 57
             ↳Polo
             ...
               →DP Problem 60
                 ↳Polynomial Ordering
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

PROPER(2ndsneg(X1, X2)) -> PROPER(X2)
PROPER(2ndsneg(X1, X2)) -> PROPER(X1)
PROPER(2ndspos(X1, X2)) -> PROPER(X2)
PROPER(2ndspos(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)
PROPER(rcons(X1, X2)) -> PROPER(X2)
PROPER(rcons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(negrecip(X)) -> PROPER(X)
PROPER(posrecip(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

PROPER(2ndsneg(X1, X2)) -> PROPER(X2)
PROPER(2ndsneg(X1, X2)) -> PROPER(X1)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  
  POL(2ndspos(x1, x2))=  x1 + x2  
  POL(2ndsneg(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
           →DP Problem 57
             ↳Polo
             ...
               →DP Problem 61
                 ↳Polynomial Ordering
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

PROPER(2ndspos(X1, X2)) -> PROPER(X2)
PROPER(2ndspos(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)
PROPER(rcons(X1, X2)) -> PROPER(X2)
PROPER(rcons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(negrecip(X)) -> PROPER(X)
PROPER(posrecip(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

PROPER(2ndspos(X1, X2)) -> PROPER(X2)
PROPER(2ndspos(X1, X2)) -> PROPER(X1)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  
  POL(2ndspos(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
           →DP Problem 57
             ↳Polo
             ...
               →DP Problem 62
                 ↳Polynomial Ordering
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

PROPER(from(X)) -> PROPER(X)
PROPER(rcons(X1, X2)) -> PROPER(X2)
PROPER(rcons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(negrecip(X)) -> PROPER(X)
PROPER(posrecip(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(from(X)) -> PROPER(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  1 + x1  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
           →DP Problem 57
             ↳Polo
             ...
               →DP Problem 63
                 ↳Polynomial Ordering
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

PROPER(rcons(X1, X2)) -> PROPER(X2)
PROPER(rcons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(negrecip(X)) -> PROPER(X)
PROPER(posrecip(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(s(x1))=  x1  
  POL(rcons(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
           →DP Problem 57
             ↳Polo
             ...
               →DP Problem 64
                 ↳Polynomial Ordering
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(negrecip(X)) -> PROPER(X)
PROPER(posrecip(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  1 + x1 + x2  
  POL(negrecip(x1))=  x1  
  POL(posrecip(x1))=  x1  
  POL(s(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
           →DP Problem 57
             ↳Polo
             ...
               →DP Problem 65
                 ↳Polynomial Ordering
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

PROPER(negrecip(X)) -> PROPER(X)
PROPER(posrecip(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(negrecip(X)) -> PROPER(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(negrecip(x1))=  1 + x1  
  POL(posrecip(x1))=  x1  
  POL(s(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
           →DP Problem 57
             ↳Polo
             ...
               →DP Problem 66
                 ↳Polynomial Ordering
       →DP Problem 15
         ↳Remaining


Dependency Pairs:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(posrecip(X)) -> PROPER(X)


There are no usable rules for innermost that need to be oriented.

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
           →DP Problem 57
             ↳Polo
             ...
               →DP Problem 67
                 ↳Polynomial Ordering
       →DP Problem 15
         ↳Remaining


Dependency Pair:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(s(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 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
           →DP Problem 57
             ↳Polo
             ...
               →DP Problem 68
                 ↳Dependency Graph
       →DP Problem 15
         ↳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, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Polo
       →DP Problem 10
         ↳Polo
       →DP Problem 11
         ↳Polo
       →DP Problem 12
         ↳Polo
       →DP Problem 13
         ↳Polo
       →DP Problem 14
         ↳Polo
       →DP Problem 15
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(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(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))
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(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))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:12 minutes