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))

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
         ↳Size-Change Principle
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar


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))





We number the DPs as follows:
  1. CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
  2. CONS(mark(X1), X2) -> CONS(X1, X2)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1
2>2
{2, 1} , {2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1
2>2
{2, 1} , {2, 1}
1>1
2=2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳Size-Change Principle
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar


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))





We number the DPs as follows:
  1. FROM(ok(X)) -> FROM(X)
  2. FROM(mark(X)) -> FROM(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳Size-Change Principle
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar


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))





We number the DPs as follows:
  1. S(ok(X)) -> S(X)
  2. S(mark(X)) -> S(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳Size-Change Principle
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar


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))





We number the DPs as follows:
  1. RCONS(ok(X1), ok(X2)) -> RCONS(X1, X2)
  2. RCONS(X1, mark(X2)) -> RCONS(X1, X2)
  3. RCONS(mark(X1), X2) -> RCONS(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳Size-Change Principle
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar


Dependency Pairs:

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


Rules:


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





We number the DPs as follows:
  1. POSRECIP(ok(X)) -> POSRECIP(X)
  2. POSRECIP(mark(X)) -> POSRECIP(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳Size-Change Principle
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar


Dependency Pairs:

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


Rules:


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





We number the DPs as follows:
  1. 2NDSNEG(ok(X1), ok(X2)) -> 2NDSNEG(X1, X2)
  2. 2NDSNEG(X1, mark(X2)) -> 2NDSNEG(X1, X2)
  3. 2NDSNEG(mark(X1), X2) -> 2NDSNEG(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳Size-Change Principle
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar


Dependency Pairs:

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


Rules:


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





We number the DPs as follows:
  1. NEGRECIP(ok(X)) -> NEGRECIP(X)
  2. NEGRECIP(mark(X)) -> NEGRECIP(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳Size-Change Principle
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar


Dependency Pairs:

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


Rules:


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





We number the DPs as follows:
  1. 2NDSPOS(ok(X1), ok(X2)) -> 2NDSPOS(X1, X2)
  2. 2NDSPOS(X1, mark(X2)) -> 2NDSPOS(X1, X2)
  3. 2NDSPOS(mark(X1), X2) -> 2NDSPOS(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Size-Change Principle
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar


Dependency Pairs:

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


Rules:


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





We number the DPs as follows:
  1. PLUS(ok(X1), ok(X2)) -> PLUS(X1, X2)
  2. PLUS(X1, mark(X2)) -> PLUS(X1, X2)
  3. PLUS(mark(X1), X2) -> PLUS(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Size-Change Principle
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar


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))





We number the DPs as follows:
  1. TIMES(ok(X1), ok(X2)) -> TIMES(X1, X2)
  2. TIMES(X1, mark(X2)) -> TIMES(X1, X2)
  3. TIMES(mark(X1), X2) -> TIMES(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳Size-Change Principle
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar


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))





We number the DPs as follows:
  1. PI(ok(X)) -> PI(X)
  2. PI(mark(X)) -> PI(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳Size-Change Principle
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar


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))





We number the DPs as follows:
  1. SQUARE(ok(X)) -> SQUARE(X)
  2. SQUARE(mark(X)) -> SQUARE(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳Size-Change Principle
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar


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))





We number the DPs as follows:
  1. ACTIVE(square(X)) -> ACTIVE(X)
  2. ACTIVE(times(X1, X2)) -> ACTIVE(X2)
  3. ACTIVE(times(X1, X2)) -> ACTIVE(X1)
  4. ACTIVE(plus(X1, X2)) -> ACTIVE(X2)
  5. ACTIVE(plus(X1, X2)) -> ACTIVE(X1)
  6. ACTIVE(pi(X)) -> ACTIVE(X)
  7. ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X2)
  8. ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X1)
  9. ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X2)
  10. ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X1)
  11. ACTIVE(from(X)) -> ACTIVE(X)
  12. ACTIVE(rcons(X1, X2)) -> ACTIVE(X2)
  13. ACTIVE(rcons(X1, X2)) -> ACTIVE(X1)
  14. ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
  15. ACTIVE(negrecip(X)) -> ACTIVE(X)
  16. ACTIVE(posrecip(X)) -> ACTIVE(X)
  17. ACTIVE(s(X)) -> ACTIVE(X)
and get the following Size-Change Graph(s):
{17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
from(x1) -> from(x1)
plus(x1, x2) -> plus(x1, x2)
square(x1) -> square(x1)
negrecip(x1) -> negrecip(x1)
posrecip(x1) -> posrecip(x1)
rcons(x1, x2) -> rcons(x1, x2)
2ndspos(x1, x2) -> 2ndspos(x1, x2)
2ndsneg(x1, x2) -> 2ndsneg(x1, x2)
cons(x1, x2) -> cons(x1, x2)
times(x1, x2) -> times(x1, x2)
s(x1) -> s(x1)
pi(x1) -> pi(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳Size-Change Principle
       →DP Problem 15
         ↳Nar


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))





We number the DPs as follows:
  1. PROPER(square(X)) -> PROPER(X)
  2. PROPER(times(X1, X2)) -> PROPER(X2)
  3. PROPER(times(X1, X2)) -> PROPER(X1)
  4. PROPER(plus(X1, X2)) -> PROPER(X2)
  5. PROPER(plus(X1, X2)) -> PROPER(X1)
  6. PROPER(pi(X)) -> PROPER(X)
  7. PROPER(2ndsneg(X1, X2)) -> PROPER(X2)
  8. PROPER(2ndsneg(X1, X2)) -> PROPER(X1)
  9. PROPER(2ndspos(X1, X2)) -> PROPER(X2)
  10. PROPER(2ndspos(X1, X2)) -> PROPER(X1)
  11. PROPER(from(X)) -> PROPER(X)
  12. PROPER(rcons(X1, X2)) -> PROPER(X2)
  13. PROPER(rcons(X1, X2)) -> PROPER(X1)
  14. PROPER(cons(X1, X2)) -> PROPER(X2)
  15. PROPER(cons(X1, X2)) -> PROPER(X1)
  16. PROPER(negrecip(X)) -> PROPER(X)
  17. PROPER(posrecip(X)) -> PROPER(X)
  18. PROPER(s(X)) -> PROPER(X)
and get the following Size-Change Graph(s):
{18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
from(x1) -> from(x1)
plus(x1, x2) -> plus(x1, x2)
square(x1) -> square(x1)
negrecip(x1) -> negrecip(x1)
posrecip(x1) -> posrecip(x1)
rcons(x1, x2) -> rcons(x1, x2)
2ndspos(x1, x2) -> 2ndspos(x1, x2)
2ndsneg(x1, x2) -> 2ndsneg(x1, x2)
cons(x1, x2) -> cons(x1, x2)
times(x1, x2) -> times(x1, x2)
pi(x1) -> pi(x1)
s(x1) -> s(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Narrowing Transformation


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))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

TOP(mark(X)) -> TOP(proper(X))
15 new Dependency Pairs are created:

TOP(mark(0)) -> TOP(ok(0))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(posrecip(X''))) -> TOP(posrecip(proper(X'')))
TOP(mark(negrecip(X''))) -> TOP(negrecip(proper(X'')))
TOP(mark(nil)) -> TOP(ok(nil))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(rnil)) -> TOP(ok(rnil))
TOP(mark(rcons(X1', X2'))) -> TOP(rcons(proper(X1'), proper(X2')))
TOP(mark(from(X''))) -> TOP(from(proper(X'')))
TOP(mark(2ndspos(X1', X2'))) -> TOP(2ndspos(proper(X1'), proper(X2')))
TOP(mark(2ndsneg(X1', X2'))) -> TOP(2ndsneg(proper(X1'), proper(X2')))
TOP(mark(pi(X''))) -> TOP(pi(proper(X'')))
TOP(mark(plus(X1', X2'))) -> TOP(plus(proper(X1'), proper(X2')))
TOP(mark(times(X1', X2'))) -> TOP(times(proper(X1'), proper(X2')))
TOP(mark(square(X''))) -> TOP(square(proper(X'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar
           →DP Problem 16
             ↳Narrowing Transformation


Dependency Pairs:

TOP(mark(square(X''))) -> TOP(square(proper(X'')))
TOP(mark(times(X1', X2'))) -> TOP(times(proper(X1'), proper(X2')))
TOP(mark(plus(X1', X2'))) -> TOP(plus(proper(X1'), proper(X2')))
TOP(mark(pi(X''))) -> TOP(pi(proper(X'')))
TOP(mark(2ndsneg(X1', X2'))) -> TOP(2ndsneg(proper(X1'), proper(X2')))
TOP(mark(2ndspos(X1', X2'))) -> TOP(2ndspos(proper(X1'), proper(X2')))
TOP(mark(from(X''))) -> TOP(from(proper(X'')))
TOP(mark(rcons(X1', X2'))) -> TOP(rcons(proper(X1'), proper(X2')))
TOP(mark(rnil)) -> TOP(ok(rnil))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(nil)) -> TOP(ok(nil))
TOP(mark(negrecip(X''))) -> TOP(negrecip(proper(X'')))
TOP(mark(posrecip(X''))) -> TOP(posrecip(proper(X'')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))
TOP(ok(X)) -> TOP(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))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

TOP(ok(X)) -> TOP(active(X))
28 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Nar
           →DP Problem 16
             ↳Nar
             ...
               →DP Problem 17
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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




The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:01 minutes