Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

ACTIVE(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(2ndspos(s(N), cons(X, Z))) -> 2NDSPOS(s(N), cons2(X, Z))
ACTIVE(2ndspos(s(N), cons(X, Z))) -> CONS2(X, Z)
ACTIVE(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> RCONS(posrecip(Y), 2ndsneg(N, Z))
ACTIVE(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> POSRECIP(Y)
ACTIVE(2ndspos(s(N), cons2(X, cons(Y, Z)))) -> 2NDSNEG(N, Z)
ACTIVE(2ndsneg(s(N), cons(X, Z))) -> 2NDSNEG(s(N), cons2(X, Z))
ACTIVE(2ndsneg(s(N), cons(X, Z))) -> CONS2(X, Z)
ACTIVE(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> RCONS(negrecip(Y), 2ndspos(N, Z))
ACTIVE(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> NEGRECIP(Y)
ACTIVE(2ndsneg(s(N), cons2(X, cons(Y, Z)))) -> 2NDSPOS(N, Z)
ACTIVE(pi(X)) -> 2NDSPOS(X, from(0))
ACTIVE(pi(X)) -> FROM(0)
ACTIVE(plus(s(X), Y)) -> S(plus(X, Y))
ACTIVE(plus(s(X), Y)) -> PLUS(X, Y)
ACTIVE(times(s(X), Y)) -> PLUS(Y, times(X, Y))
ACTIVE(times(s(X), Y)) -> TIMES(X, Y)
ACTIVE(square(X)) -> TIMES(X, X)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(posrecip(X)) -> POSRECIP(active(X))
ACTIVE(posrecip(X)) -> ACTIVE(X)
ACTIVE(negrecip(X)) -> NEGRECIP(active(X))
ACTIVE(negrecip(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons2(X1, X2)) -> CONS2(X1, active(X2))
ACTIVE(cons2(X1, X2)) -> ACTIVE(X2)
ACTIVE(rcons(X1, X2)) -> RCONS(active(X1), X2)
ACTIVE(rcons(X1, X2)) -> ACTIVE(X1)
ACTIVE(rcons(X1, X2)) -> RCONS(X1, active(X2))
ACTIVE(rcons(X1, X2)) -> ACTIVE(X2)
ACTIVE(from(X)) -> FROM(active(X))
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(2ndspos(X1, X2)) -> 2NDSPOS(active(X1), X2)
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) -> 2NDSPOS(X1, active(X2))
ACTIVE(2ndspos(X1, X2)) -> ACTIVE(X2)
ACTIVE(2ndsneg(X1, X2)) -> 2NDSNEG(active(X1), X2)
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X1)
ACTIVE(2ndsneg(X1, X2)) -> 2NDSNEG(X1, active(X2))
ACTIVE(2ndsneg(X1, X2)) -> ACTIVE(X2)
ACTIVE(pi(X)) -> PI(active(X))
ACTIVE(pi(X)) -> ACTIVE(X)
ACTIVE(plus(X1, X2)) -> PLUS(active(X1), X2)
ACTIVE(plus(X1, X2)) -> ACTIVE(X1)
ACTIVE(plus(X1, X2)) -> PLUS(X1, active(X2))
ACTIVE(plus(X1, X2)) -> ACTIVE(X2)
ACTIVE(times(X1, X2)) -> TIMES(active(X1), X2)
ACTIVE(times(X1, X2)) -> ACTIVE(X1)
ACTIVE(times(X1, X2)) -> TIMES(X1, active(X2))
ACTIVE(times(X1, X2)) -> ACTIVE(X2)
ACTIVE(square(X)) -> SQUARE(active(X))
ACTIVE(square(X)) -> ACTIVE(X)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
POSRECIP(mark(X)) -> POSRECIP(X)
POSRECIP(ok(X)) -> POSRECIP(X)
NEGRECIP(mark(X)) -> NEGRECIP(X)
NEGRECIP(ok(X)) -> NEGRECIP(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS2(X1, mark(X2)) -> CONS2(X1, X2)
CONS2(ok(X1), ok(X2)) -> CONS2(X1, X2)
RCONS(mark(X1), X2) -> RCONS(X1, X2)
RCONS(X1, mark(X2)) -> RCONS(X1, X2)
RCONS(ok(X1), ok(X2)) -> RCONS(X1, X2)
FROM(mark(X)) -> FROM(X)
FROM(ok(X)) -> FROM(X)
2NDSPOS(mark(X1), X2) -> 2NDSPOS(X1, X2)
2NDSPOS(X1, mark(X2)) -> 2NDSPOS(X1, X2)
2NDSPOS(ok(X1), ok(X2)) -> 2NDSPOS(X1, X2)
2NDSNEG(mark(X1), X2) -> 2NDSNEG(X1, X2)
2NDSNEG(X1, mark(X2)) -> 2NDSNEG(X1, X2)
2NDSNEG(ok(X1), ok(X2)) -> 2NDSNEG(X1, X2)
PI(mark(X)) -> PI(X)
PI(ok(X)) -> PI(X)
PLUS(mark(X1), X2) -> PLUS(X1, X2)
PLUS(X1, mark(X2)) -> PLUS(X1, X2)
PLUS(ok(X1), ok(X2)) -> PLUS(X1, X2)
TIMES(mark(X1), X2) -> TIMES(X1, X2)
TIMES(X1, mark(X2)) -> TIMES(X1, X2)
TIMES(ok(X1), ok(X2)) -> TIMES(X1, X2)
SQUARE(mark(X)) -> SQUARE(X)
SQUARE(ok(X)) -> SQUARE(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(posrecip(X)) -> POSRECIP(proper(X))
PROPER(posrecip(X)) -> PROPER(X)
PROPER(negrecip(X)) -> NEGRECIP(proper(X))
PROPER(negrecip(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons2(X1, X2)) -> CONS2(proper(X1), proper(X2))
PROPER(cons2(X1, X2)) -> PROPER(X1)
PROPER(cons2(X1, X2)) -> PROPER(X2)
PROPER(rcons(X1, X2)) -> RCONS(proper(X1), proper(X2))
PROPER(rcons(X1, X2)) -> PROPER(X1)
PROPER(rcons(X1, X2)) -> PROPER(X2)
PROPER(from(X)) -> FROM(proper(X))
PROPER(from(X)) -> PROPER(X)
PROPER(2ndspos(X1, X2)) -> 2NDSPOS(proper(X1), proper(X2))
PROPER(2ndspos(X1, X2)) -> PROPER(X1)
PROPER(2ndspos(X1, X2)) -> PROPER(X2)
PROPER(2ndsneg(X1, X2)) -> 2NDSNEG(proper(X1), proper(X2))
PROPER(2ndsneg(X1, X2)) -> PROPER(X1)
PROPER(2ndsneg(X1, X2)) -> PROPER(X2)
PROPER(pi(X)) -> PI(proper(X))
PROPER(pi(X)) -> PROPER(X)
PROPER(plus(X1, X2)) -> PLUS(proper(X1), proper(X2))
PROPER(plus(X1, X2)) -> PROPER(X1)
PROPER(plus(X1, X2)) -> PROPER(X2)
PROPER(times(X1, X2)) -> TIMES(proper(X1), proper(X2))
PROPER(times(X1, X2)) -> PROPER(X1)
PROPER(times(X1, X2)) -> PROPER(X2)
PROPER(square(X)) -> SQUARE(proper(X))
PROPER(square(X)) -> PROPER(X)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains 16 SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FROM(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 17

             ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FROM(x₁)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 17

             ↳Polo

...

               →DP Problem 18

                 ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polynomial Ordering

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(2NDSPOS(x₁, x₂)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 19

             ↳Polynomial Ordering

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(2NDSPOS(x₁, x₂)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 19

             ↳Polo

...

               →DP Problem 20

                 ↳Polynomial Ordering

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(2NDSPOS(x₁, x₂)) = x₂

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 19

             ↳Polo

...

               →DP Problem 21

                 ↳Dependency Graph

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polynomial Ordering

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(CONS2(x₁, x₂)) = x₁

POL(mark(x₁)) = 0

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 22

             ↳Polynomial Ordering

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(CONS2(x₁, x₂)) = x₂

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 22

             ↳Polo

...

               →DP Problem 23

                 ↳Dependency Graph

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polynomial Ordering

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Rules:

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

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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = x₁

POL(RCONS(x₁, x₂)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

           →DP Problem 24

             ↳Polynomial Ordering

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 1 + x₁

POL(RCONS(x₁, x₂)) = x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

           →DP Problem 24

             ↳Polo

...

               →DP Problem 25

                 ↳Polynomial Ordering

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 1 + x₁

POL(RCONS(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

           →DP Problem 24

             ↳Polo

...

               →DP Problem 26

                 ↳Dependency Graph

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polynomial Ordering

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 1 + x₁

POL(ok(x₁)) = x₁

POL(POSRECIP(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

           →DP Problem 27

             ↳Polynomial Ordering

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

           →DP Problem 27

             ↳Polo

...

               →DP Problem 28

                 ↳Dependency Graph

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polynomial Ordering

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(2NDSNEG(x₁, x₂)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳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 29

             ↳Polynomial Ordering

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(2NDSNEG(x₁, x₂)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳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 29

             ↳Polo

...

               →DP Problem 30

                 ↳Polynomial Ordering

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(2NDSNEG(x₁, x₂)) = x₂

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳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 29

             ↳Polo

...

               →DP Problem 31

                 ↳Dependency Graph

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polynomial Ordering

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(NEGRECIP(x₁)) = x₁

POL(mark(x₁)) = 1 + x₁

POL(ok(x₁)) = x₁

resulting in one new DP problem.

     ↳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 32

             ↳Polynomial Ordering

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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.

     ↳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 32

             ↳Polo

...

               →DP Problem 33

                 ↳Dependency Graph

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polynomial Ordering

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(S(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳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 34

             ↳Polynomial Ordering

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(S(x₁)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳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 34

             ↳Polo

...

               →DP Problem 35

                 ↳Dependency Graph

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polynomial Ordering

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(PLUS(x₁, x₂)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

           →DP Problem 36

             ↳Polynomial Ordering

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(PLUS(x₁, x₂)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳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

                 ↳Polynomial Ordering

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(PLUS(x₁, x₂)) = x₂

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳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 38

                 ↳Dependency Graph

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polynomial Ordering

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Rules:

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

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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 0

POL(ok(x₁)) = 1 + x₁

POL(CONS(x₁, x₂)) = x₂

resulting in one new DP problem.

     ↳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 39

             ↳Polynomial Ordering

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 1 + x₁

POL(CONS(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳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 39

             ↳Polo

...

               →DP Problem 40

                 ↳Dependency Graph

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polynomial Ordering

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

           →DP Problem 41

             ↳Polynomial Ordering

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

           →DP Problem 41

             ↳Polo

...

               →DP Problem 42

                 ↳Dependency Graph

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

     ↳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

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Rules:

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

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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(TIMES(x₁, x₂)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳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

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(TIMES(x₁, x₂)) = x₂

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳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

                 ↳Polynomial Ordering

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(TIMES(x₁, x₂)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳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 45

                 ↳Dependency Graph

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳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

         ↳Polo

       →DP Problem 16

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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.

     ↳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 46

             ↳Polynomial Ordering

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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.

     ↳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 46

             ↳Polo

...

               →DP Problem 47

                 ↳Dependency Graph

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳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

         ↳Polo

       →DP Problem 16

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₂

POL(s(x₁)) = x₁

POL(pi(x₁)) = x₁

resulting in one new DP problem.

     ↳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 48

             ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

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

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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = x₁

POL(plus(x₁, x₂)) = x₁ + x₂

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)) = 1 + x₁ + x₂

POL(pi(x₁)) = x₁

POL(cons2(x₁, x₂)) = x₂

POL(s(x₁)) = x₁

resulting in one new DP problem.

     ↳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 48

             ↳Polo

...

               →DP Problem 49

                 ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

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

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 w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₂

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.

     ↳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 48

             ↳Polo

...

               →DP Problem 50

                 ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₂

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.

     ↳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 48

             ↳Polo

...

               →DP Problem 51

                 ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

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

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 w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₂

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.

     ↳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 48

             ↳Polo

...

               →DP Problem 52

                 ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

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

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 w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₂

POL(s(x₁)) = x₁

POL(rcons(x₁, x₂)) = x₁ + x₂

POL(2ndspos(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳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 48

             ↳Polo

...

               →DP Problem 53

                 ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₂

POL(s(x₁)) = x₁

POL(rcons(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳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 48

             ↳Polo

...

               →DP Problem 54

                 ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

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 w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₂

POL(s(x₁)) = x₁

POL(rcons(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳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 48

             ↳Polo

...

               →DP Problem 55

                 ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(cons2(X1, X2)) -> ACTIVE(X2)

There are no usable rules for innermost w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = 1 + x₂

POL(s(x₁)) = x₁

resulting in one new DP problem.

     ↳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 48

             ↳Polo

...

               →DP Problem 56

                 ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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.

     ↳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 48

             ↳Polo

...

               →DP Problem 57

                 ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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.

     ↳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 48

             ↳Polo

...

               →DP Problem 58

                 ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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.

     ↳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 48

             ↳Polo

...

               →DP Problem 59

                 ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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.

     ↳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 48

             ↳Polo

...

               →DP Problem 60

                 ↳Dependency Graph

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳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

         ↳Polynomial Ordering

       →DP Problem 16

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = x₁

POL(pi(x₁)) = x₁

resulting in one new DP problem.

     ↳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

         ↳Polo

           →DP Problem 61

             ↳Polynomial Ordering

       →DP Problem 16

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

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 w.r.t. to the implicit AFS 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(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)) = 1 + x₁ + x₂

POL(pi(x₁)) = x₁

POL(cons2(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = x₁

resulting in one new DP problem.

     ↳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

         ↳Polo

           →DP Problem 61

             ↳Polo

...

               →DP Problem 62

                 ↳Polynomial Ordering

       →DP Problem 16

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

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 w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₁ + x₂

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.

     ↳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

         ↳Polo

           →DP Problem 61

             ↳Polo

...

               →DP Problem 63

                 ↳Polynomial Ordering

       →DP Problem 16

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₁ + x₂

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.

     ↳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

         ↳Polo

           →DP Problem 61

             ↳Polo

...

               →DP Problem 64

                 ↳Polynomial Ordering

       →DP Problem 16

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

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 w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₁ + x₂

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.

     ↳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

         ↳Polo

           →DP Problem 61

             ↳Polo

...

               →DP Problem 65

                 ↳Polynomial Ordering

       →DP Problem 16

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

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 w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = x₁

POL(rcons(x₁, x₂)) = x₁ + x₂

POL(2ndspos(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳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

         ↳Polo

           →DP Problem 61

             ↳Polo

...

               →DP Problem 66

                 ↳Polynomial Ordering

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = x₁

POL(rcons(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳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

         ↳Polo

           →DP Problem 61

             ↳Polo

...

               →DP Problem 67

                 ↳Polynomial Ordering

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

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 w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = x₁

POL(rcons(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳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

         ↳Polo

           →DP Problem 61

             ↳Polo

...

               →DP Problem 68

                 ↳Polynomial Ordering

       →DP Problem 16

         ↳Remaining

Dependency Pairs:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PROPER(cons2(X1, X2)) -> PROPER(X2)
PROPER(cons2(X1, X2)) -> PROPER(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS 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(cons2(x₁, x₂)) = 1 + x₁ + x₂

POL(s(x₁)) = x₁

resulting in one new DP problem.

     ↳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

         ↳Polo

           →DP Problem 61

             ↳Polo

...

               →DP Problem 69

                 ↳Polynomial Ordering

       →DP Problem 16

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

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 w.r.t. to the implicit AFS 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.

     ↳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

         ↳Polo

           →DP Problem 61

             ↳Polo

...

               →DP Problem 70

                 ↳Polynomial Ordering

       →DP Problem 16

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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.

     ↳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

         ↳Polo

           →DP Problem 61

             ↳Polo

...

               →DP Problem 71

                 ↳Polynomial Ordering

       →DP Problem 16

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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.

     ↳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

         ↳Polo

           →DP Problem 61

             ↳Polo

...

               →DP Problem 72

                 ↳Polynomial Ordering

       →DP Problem 16

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS 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.

     ↳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

         ↳Polo

           →DP Problem 61

             ↳Polo

...

               →DP Problem 73

                 ↳Dependency Graph

       →DP Problem 16

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳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

         ↳Polo

       →DP Problem 16

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

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:15 minutes

POL(FROM(x₁))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(2NDSPOS(x₁, x₂))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(2NDSPOS(x₁, x₂))	= x₂
POL(mark(x₁))	= 1 + x₁

POL(CONS2(x₁, x₂))	= x₁
POL(mark(x₁))	= 0
POL(ok(x₁))	= 1 + x₁

POL(mark(x₁))	= x₁
POL(RCONS(x₁, x₂))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(2NDSNEG(x₁, x₂))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(NEGRECIP(x₁))	= x₁
POL(mark(x₁))	= 1 + x₁
POL(ok(x₁))	= x₁

POL(S(x₁))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(PLUS(x₁, x₂))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(mark(x₁))	= 1 + x₁
POL(CONS(x₁, x₂))	= x₁

POL(PI(x₁))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(TIMES(x₁, x₂))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(SQUARE(x₁))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

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(cons2(x₁, x₂))	= x₂
POL(s(x₁))	= x₁
POL(rcons(x₁, x₂))	= x₁ + x₂
POL(2ndspos(x₁, x₂))	= x₁ + x₂
POL(2ndsneg(x₁, x₂))	= x₁ + x₂

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(cons2(x₁, x₂))	= x₁ + x₂
POL(s(x₁))	= x₁
POL(rcons(x₁, x₂))	= x₁ + x₂
POL(2ndspos(x₁, x₂))	= x₁ + x₂
POL(2ndsneg(x₁, x₂))	= x₁ + x₂