Termination w.r.t. Q proof of /tmp/aprove_benchmark_example

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 16 SCCs with 53 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SQUARE(ok(X)) → SQUARE(X)
SQUARE(mark(X)) → SQUARE(X)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SQUARE(ok(X)) → SQUARE(X)
SQUARE(mark(X)) → SQUARE(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

SQUARE(ok(X)) → SQUARE(X)
The graph contains the following edges 1 > 1
SQUARE(mark(X)) → SQUARE(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

TIMES(ok(X1), ok(X2)) → TIMES(X1, X2)
TIMES(X1, mark(X2)) → TIMES(X1, X2)
TIMES(mark(X1), X2) → TIMES(X1, X2)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

TIMES(X1, mark(X2)) → TIMES(X1, X2)
TIMES(ok(X1), ok(X2)) → TIMES(X1, X2)
TIMES(mark(X1), X2) → TIMES(X1, X2)

From the DPs we obtained the following set of size-change graphs:

TIMES(ok(X1), ok(X2)) → TIMES(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
TIMES(X1, mark(X2)) → TIMES(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
TIMES(mark(X1), X2) → TIMES(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)

From the DPs we obtained the following set of size-change graphs:

PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
PLUS(mark(X1), X2) → PLUS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
PLUS(X1, mark(X2)) → PLUS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PI(mark(X)) → PI(X)
PI(ok(X)) → PI(X)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PI(mark(X)) → PI(X)
PI(ok(X)) → PI(X)

From the DPs we obtained the following set of size-change graphs:

PI(mark(X)) → PI(X)
The graph contains the following edges 1 > 1
PI(ok(X)) → PI(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

2NDSNEG(ok(X1), ok(X2)) → 2NDSNEG(X1, X2)
2NDSNEG(mark(X1), X2) → 2NDSNEG(X1, X2)
2NDSNEG(X1, mark(X2)) → 2NDSNEG(X1, X2)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

2NDSNEG(ok(X1), ok(X2)) → 2NDSNEG(X1, X2)
2NDSNEG(mark(X1), X2) → 2NDSNEG(X1, X2)
2NDSNEG(X1, mark(X2)) → 2NDSNEG(X1, X2)

From the DPs we obtained the following set of size-change graphs:

2NDSNEG(ok(X1), ok(X2)) → 2NDSNEG(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
2NDSNEG(mark(X1), X2) → 2NDSNEG(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
2NDSNEG(X1, mark(X2)) → 2NDSNEG(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

2NDSPOS(X1, mark(X2)) → 2NDSPOS(X1, X2)
2NDSPOS(mark(X1), X2) → 2NDSPOS(X1, X2)
2NDSPOS(ok(X1), ok(X2)) → 2NDSPOS(X1, X2)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

2NDSPOS(X1, mark(X2)) → 2NDSPOS(X1, X2)
2NDSPOS(mark(X1), X2) → 2NDSPOS(X1, X2)
2NDSPOS(ok(X1), ok(X2)) → 2NDSPOS(X1, X2)

From the DPs we obtained the following set of size-change graphs:

2NDSPOS(X1, mark(X2)) → 2NDSPOS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
2NDSPOS(mark(X1), X2) → 2NDSPOS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
2NDSPOS(ok(X1), ok(X2)) → 2NDSPOS(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FROM(mark(X)) → FROM(X)
FROM(ok(X)) → FROM(X)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FROM(mark(X)) → FROM(X)
FROM(ok(X)) → FROM(X)

From the DPs we obtained the following set of size-change graphs:

FROM(mark(X)) → FROM(X)
The graph contains the following edges 1 > 1
FROM(ok(X)) → FROM(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

RCONS(X1, mark(X2)) → RCONS(X1, X2)
RCONS(mark(X1), X2) → RCONS(X1, X2)
RCONS(ok(X1), ok(X2)) → RCONS(X1, X2)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

RCONS(X1, mark(X2)) → RCONS(X1, X2)
RCONS(mark(X1), X2) → RCONS(X1, X2)
RCONS(ok(X1), ok(X2)) → RCONS(X1, X2)

From the DPs we obtained the following set of size-change graphs:

RCONS(X1, mark(X2)) → RCONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
RCONS(mark(X1), X2) → RCONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
RCONS(ok(X1), ok(X2)) → RCONS(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONS2(ok(X1), ok(X2)) → CONS2(X1, X2)
CONS2(X1, mark(X2)) → CONS2(X1, X2)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONS2(ok(X1), ok(X2)) → CONS2(X1, X2)
CONS2(X1, mark(X2)) → CONS2(X1, X2)

From the DPs we obtained the following set of size-change graphs:

CONS2(ok(X1), ok(X2)) → CONS2(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
CONS2(X1, mark(X2)) → CONS2(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)

From the DPs we obtained the following set of size-change graphs:

CONS(mark(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

NEGRECIP(ok(X)) → NEGRECIP(X)
NEGRECIP(mark(X)) → NEGRECIP(X)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

NEGRECIP(ok(X)) → NEGRECIP(X)
NEGRECIP(mark(X)) → NEGRECIP(X)

From the DPs we obtained the following set of size-change graphs:

NEGRECIP(ok(X)) → NEGRECIP(X)
The graph contains the following edges 1 > 1
NEGRECIP(mark(X)) → NEGRECIP(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

POSRECIP(mark(X)) → POSRECIP(X)
POSRECIP(ok(X)) → POSRECIP(X)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

POSRECIP(mark(X)) → POSRECIP(X)
POSRECIP(ok(X)) → POSRECIP(X)

From the DPs we obtained the following set of size-change graphs:

POSRECIP(mark(X)) → POSRECIP(X)
The graph contains the following edges 1 > 1
POSRECIP(ok(X)) → POSRECIP(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

S(ok(X)) → S(X)
S(mark(X)) → S(X)

From the DPs we obtained the following set of size-change graphs:

S(ok(X)) → S(X)
The graph contains the following edges 1 > 1
S(mark(X)) → S(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER(times(X1, X2)) → PROPER(X2)
PROPER(2ndsneg(X1, X2)) → PROPER(X1)
PROPER(2ndspos(X1, X2)) → PROPER(X2)
PROPER(2ndsneg(X1, X2)) → PROPER(X2)
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(rcons(X1, X2)) → PROPER(X1)
PROPER(square(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons2(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(2ndspos(X1, X2)) → PROPER(X1)
PROPER(posrecip(X)) → PROPER(X)
PROPER(from(X)) → PROPER(X)
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(cons2(X1, X2)) → PROPER(X2)
PROPER(pi(X)) → PROPER(X)
PROPER(rcons(X1, X2)) → PROPER(X2)
PROPER(times(X1, X2)) → PROPER(X1)
PROPER(negrecip(X)) → PROPER(X)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER(times(X1, X2)) → PROPER(X2)
PROPER(2ndsneg(X1, X2)) → PROPER(X1)
PROPER(2ndspos(X1, X2)) → PROPER(X2)
PROPER(2ndsneg(X1, X2)) → PROPER(X2)
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(rcons(X1, X2)) → PROPER(X1)
PROPER(square(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons2(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(2ndspos(X1, X2)) → PROPER(X1)
PROPER(from(X)) → PROPER(X)
PROPER(posrecip(X)) → PROPER(X)
PROPER(cons2(X1, X2)) → PROPER(X2)
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(rcons(X1, X2)) → PROPER(X2)
PROPER(pi(X)) → PROPER(X)
PROPER(negrecip(X)) → PROPER(X)
PROPER(times(X1, X2)) → PROPER(X1)

From the DPs we obtained the following set of size-change graphs:

PROPER(times(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(2ndsneg(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(2ndspos(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(2ndsneg(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(plus(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(rcons(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(square(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(s(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(cons(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(cons2(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(cons(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(2ndspos(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(posrecip(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(from(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(plus(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(cons2(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(pi(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(rcons(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(times(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(negrecip(X)) → PROPER(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) → ACTIVE(X2)
ACTIVE(posrecip(X)) → ACTIVE(X)
ACTIVE(times(X1, X2)) → ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) → ACTIVE(X1)
ACTIVE(2ndsneg(X1, X2)) → ACTIVE(X1)
ACTIVE(negrecip(X)) → ACTIVE(X)
ACTIVE(2ndsneg(X1, X2)) → ACTIVE(X2)
ACTIVE(pi(X)) → ACTIVE(X)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(rcons(X1, X2)) → ACTIVE(X2)
ACTIVE(square(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(rcons(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(times(X1, X2)) → ACTIVE(X2)
ACTIVE(cons2(X1, X2)) → ACTIVE(X2)

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) → ACTIVE(X2)
ACTIVE(posrecip(X)) → ACTIVE(X)
ACTIVE(times(X1, X2)) → ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) → ACTIVE(X1)
ACTIVE(2ndsneg(X1, X2)) → ACTIVE(X1)
ACTIVE(negrecip(X)) → ACTIVE(X)
ACTIVE(2ndsneg(X1, X2)) → ACTIVE(X2)
ACTIVE(pi(X)) → ACTIVE(X)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(rcons(X1, X2)) → ACTIVE(X2)
ACTIVE(square(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(rcons(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(cons2(X1, X2)) → ACTIVE(X2)
ACTIVE(times(X1, X2)) → ACTIVE(X2)

From the DPs we obtained the following set of size-change graphs:

ACTIVE(plus(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(2ndspos(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(posrecip(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(times(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(2ndspos(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(2ndsneg(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(negrecip(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(2ndsneg(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(pi(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(s(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(rcons(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(square(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(rcons(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(from(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(times(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(cons2(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following 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))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(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))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(2ndsneg(x₁, x₂)) = x₁ + x₂
POL(2ndspos(x₁, x₂)) = 2·x₁ + 2·x₂
POL(TOP(x₁)) = x₁
POL(active(x₁)) = 2·x₁
POL(cons(x₁, x₂)) = x₁ + x₂
POL(cons2(x₁, x₂)) = 2·x₁ + 2·x₂
POL(from(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(negrecip(x₁)) = 2·x₁
POL(nil) = 0
POL(ok(x₁)) = 2·x₁
POL(pi(x₁)) = 2·x₁
POL(plus(x₁, x₂)) = 2·x₁ + x₂
POL(posrecip(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(rcons(x₁, x₂)) = 2·x₁ + x₂
POL(rnil) = 0
POL(s(x₁)) = x₁
POL(square(x₁)) = 2·x₁
POL(times(x₁, x₂)) = x₁ + 2·x₂

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following 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))
square(mark(X)) → mark(square(X))
square(ok(X)) → ok(square(X))
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))
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))
pi(mark(X)) → mark(pi(X))
pi(ok(X)) → ok(pi(X))
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))
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))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
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))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
negrecip(mark(X)) → mark(negrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
posrecip(mark(X)) → mark(posrecip(X))
posrecip(ok(X)) → ok(posrecip(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(mark(X)) → TOP(proper(X)) at position [0] we obtained the following new rules:

TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(mark(rnil)) → TOP(ok(rnil))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(mark(rnil)) → TOP(ok(rnil))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following 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))
square(mark(X)) → mark(square(X))
square(ok(X)) → ok(square(X))
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))
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))
pi(mark(X)) → mark(pi(X))
pi(ok(X)) → ok(pi(X))
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))
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))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
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))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
negrecip(mark(X)) → mark(negrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
posrecip(mark(X)) → mark(posrecip(X))
posrecip(ok(X)) → ok(posrecip(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(ok(X)) → TOP(active(X)) at position [0] we obtained the following new rules:

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(pi(x0))) → TOP(mark(2ndspos(x0, from(0))))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(ok(2ndsneg(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(negrecip(x2), 2ndspos(x0, x3))))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(ok(2ndsneg(0, x0))) → TOP(mark(rnil))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(square(x0))) → TOP(mark(times(x0, x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(times(0, x0))) → TOP(mark(0))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(ok(2ndspos(s(x0), cons(x1, x2)))) → TOP(mark(2ndspos(s(x0), cons2(x1, x2))))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))
TOP(ok(2ndspos(0, x0))) → TOP(mark(rnil))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(pi(x0))) → TOP(mark(2ndspos(x0, from(0))))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(ok(2ndsneg(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(negrecip(x2), 2ndspos(x0, x3))))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(mark(rnil)) → TOP(ok(rnil))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(2ndsneg(0, x0))) → TOP(mark(rnil))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(square(x0))) → TOP(mark(times(x0, x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(times(0, x0))) → TOP(mark(0))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndspos(s(x0), cons(x1, x2)))) → TOP(mark(2ndspos(s(x0), cons2(x1, x2))))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(2ndspos(0, x0))) → TOP(mark(rnil))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

The TRS R consists of the following 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))
square(mark(X)) → mark(square(X))
square(ok(X)) → ok(square(X))
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))
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))
pi(mark(X)) → mark(pi(X))
pi(ok(X)) → ok(pi(X))
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))
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))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
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))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
negrecip(mark(X)) → mark(negrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
posrecip(mark(X)) → mark(posrecip(X))
posrecip(ok(X)) → ok(posrecip(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 6 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(pi(x0))) → TOP(mark(2ndspos(x0, from(0))))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(ok(2ndsneg(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(negrecip(x2), 2ndspos(x0, x3))))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(square(x0))) → TOP(mark(times(x0, x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndspos(s(x0), cons(x1, x2)))) → TOP(mark(2ndspos(s(x0), cons2(x1, x2))))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

The TRS R consists of the following 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))
square(mark(X)) → mark(square(X))
square(ok(X)) → ok(square(X))
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))
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))
pi(mark(X)) → mark(pi(X))
pi(ok(X)) → ok(pi(X))
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))
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))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
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))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
negrecip(mark(X)) → mark(negrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
posrecip(mark(X)) → mark(posrecip(X))
posrecip(ok(X)) → ok(posrecip(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))

The remaining pairs can at least be oriented weakly.

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(pi(x0))) → TOP(mark(2ndspos(x0, from(0))))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(ok(2ndsneg(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(negrecip(x2), 2ndspos(x0, x3))))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(square(x0))) → TOP(mark(times(x0, x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndspos(s(x0), cons(x1, x2)))) → TOP(mark(2ndspos(s(x0), cons2(x1, x2))))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(2ndsneg(x₁, x₂)) = 0
POL(2ndspos(x₁, x₂)) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(cons2(x₁, x₂)) = 0
POL(from(x₁)) = 1
POL(mark(x₁)) = x₁
POL(negrecip(x₁)) = 0
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(pi(x₁)) = 0
POL(plus(x₁, x₂)) = x₂
POL(posrecip(x₁)) = 0
POL(proper(x₁)) = x₁
POL(rcons(x₁, x₂)) = 0
POL(rnil) = 0
POL(s(x₁)) = 0
POL(square(x₁)) = 0
POL(times(x₁, x₂)) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(pi(x0))) → TOP(mark(2ndspos(x0, from(0))))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(2ndsneg(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(negrecip(x2), 2ndspos(x0, x3))))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(square(x0))) → TOP(mark(times(x0, x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndspos(s(x0), cons(x1, x2)))) → TOP(mark(2ndspos(s(x0), cons2(x1, x2))))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

The TRS R consists of the following 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))
square(mark(X)) → mark(square(X))
square(ok(X)) → ok(square(X))
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))
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))
pi(mark(X)) → mark(pi(X))
pi(ok(X)) → ok(pi(X))
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))
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))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
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))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
negrecip(mark(X)) → mark(negrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
posrecip(mark(X)) → mark(posrecip(X))
posrecip(ok(X)) → ok(posrecip(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(pi(x0))) → TOP(mark(2ndspos(x0, from(0))))

The remaining pairs can at least be oriented weakly.

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(2ndsneg(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(negrecip(x2), 2ndspos(x0, x3))))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(square(x0))) → TOP(mark(times(x0, x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndspos(s(x0), cons(x1, x2)))) → TOP(mark(2ndspos(s(x0), cons2(x1, x2))))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(2ndsneg(x₁, x₂)) = 0
POL(2ndspos(x₁, x₂)) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(cons2(x₁, x₂)) = 0
POL(from(x₁)) = 0
POL(mark(x₁)) = x₁
POL(negrecip(x₁)) = 0
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(pi(x₁)) = 1
POL(plus(x₁, x₂)) = x₂
POL(posrecip(x₁)) = 0
POL(proper(x₁)) = x₁
POL(rcons(x₁, x₂)) = 0
POL(rnil) = 0
POL(s(x₁)) = 0
POL(square(x₁)) = 0
POL(times(x₁, x₂)) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(ok(2ndsneg(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(negrecip(x2), 2ndspos(x0, x3))))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(square(x0))) → TOP(mark(times(x0, x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndspos(s(x0), cons(x1, x2)))) → TOP(mark(2ndspos(s(x0), cons2(x1, x2))))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

The TRS R consists of the following 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))
square(mark(X)) → mark(square(X))
square(ok(X)) → ok(square(X))
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))
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))
pi(mark(X)) → mark(pi(X))
pi(ok(X)) → ok(pi(X))
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))
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))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
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))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
negrecip(mark(X)) → mark(negrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
posrecip(mark(X)) → mark(posrecip(X))
posrecip(ok(X)) → ok(posrecip(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(2ndspos(s(x0), cons(x1, x2)))) → TOP(mark(2ndspos(s(x0), cons2(x1, x2))))

The remaining pairs can at least be oriented weakly.

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(ok(2ndsneg(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(negrecip(x2), 2ndspos(x0, x3))))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(square(x0))) → TOP(mark(times(x0, x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(2ndsneg(x₁, x₂)) = 0
POL(2ndspos(x₁, x₂)) = x₂
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = 1
POL(cons2(x₁, x₂)) = 0
POL(from(x₁)) = 1
POL(mark(x₁)) = x₁
POL(negrecip(x₁)) = 0
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(pi(x₁)) = 1
POL(plus(x₁, x₂)) = x₂
POL(posrecip(x₁)) = 0
POL(proper(x₁)) = x₁
POL(rcons(x₁, x₂)) = 0
POL(rnil) = 0
POL(s(x₁)) = 0
POL(square(x₁)) = x₁
POL(times(x₁, x₂)) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(2ndsneg(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(negrecip(x2), 2ndspos(x0, x3))))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(square(x0))) → TOP(mark(times(x0, x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

The TRS R consists of the following 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))
square(mark(X)) → mark(square(X))
square(ok(X)) → ok(square(X))
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))
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))
pi(mark(X)) → mark(pi(X))
pi(ok(X)) → ok(pi(X))
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))
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))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
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))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
negrecip(mark(X)) → mark(negrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
posrecip(mark(X)) → mark(posrecip(X))
posrecip(ok(X)) → ok(posrecip(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(2ndsneg(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(negrecip(x2), 2ndspos(x0, x3))))

The remaining pairs can at least be oriented weakly.

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(square(x0))) → TOP(mark(times(x0, x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(2ndsneg(x₁, x₂)) = 1
POL(2ndspos(x₁, x₂)) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(cons2(x₁, x₂)) = 0
POL(from(x₁)) = 0
POL(mark(x₁)) = x₁
POL(negrecip(x₁)) = 0
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(pi(x₁)) = 0
POL(plus(x₁, x₂)) = x₂
POL(posrecip(x₁)) = 0
POL(proper(x₁)) = x₁
POL(rcons(x₁, x₂)) = 0
POL(rnil) = 0
POL(s(x₁)) = 0
POL(square(x₁)) = 0
POL(times(x₁, x₂)) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(square(x0))) → TOP(mark(times(x0, x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

The TRS R consists of the following 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))
square(mark(X)) → mark(square(X))
square(ok(X)) → ok(square(X))
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))
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))
pi(mark(X)) → mark(pi(X))
pi(ok(X)) → ok(pi(X))
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))
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))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
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))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
negrecip(mark(X)) → mark(negrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
posrecip(mark(X)) → mark(posrecip(X))
posrecip(ok(X)) → ok(posrecip(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(square(x0))) → TOP(mark(times(x0, x0)))

The remaining pairs can at least be oriented weakly.

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(2ndsneg(x₁, x₂)) = 0
POL(2ndspos(x₁, x₂)) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(cons2(x₁, x₂)) = 0
POL(from(x₁)) = 0
POL(mark(x₁)) = x₁
POL(negrecip(x₁)) = 0
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(pi(x₁)) = 0
POL(plus(x₁, x₂)) = x₂
POL(posrecip(x₁)) = 0
POL(proper(x₁)) = x₁
POL(rcons(x₁, x₂)) = 0
POL(rnil) = 0
POL(s(x₁)) = 0
POL(square(x₁)) = 1
POL(times(x₁, x₂)) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

The TRS R consists of the following 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))
square(mark(X)) → mark(square(X))
square(ok(X)) → ok(square(X))
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))
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))
pi(mark(X)) → mark(pi(X))
pi(ok(X)) → ok(pi(X))
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))
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))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
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))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
negrecip(mark(X)) → mark(negrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
posrecip(mark(X)) → mark(posrecip(X))
posrecip(ok(X)) → ok(posrecip(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(2ndspos(s(x0), cons2(x1, cons(x2, x3))))) → TOP(mark(rcons(posrecip(x2), 2ndsneg(x0, x3))))

The remaining pairs can at least be oriented weakly.

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(2ndsneg(x₁, x₂)) = 0
POL(2ndspos(x₁, x₂)) = 1
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(cons2(x₁, x₂)) = 0
POL(from(x₁)) = 0
POL(mark(x₁)) = x₁
POL(negrecip(x₁)) = 0
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(pi(x₁)) = 1
POL(plus(x₁, x₂)) = x₂
POL(posrecip(x₁)) = 0
POL(proper(x₁)) = x₁
POL(rcons(x₁, x₂)) = 0
POL(rnil) = 0
POL(s(x₁)) = 0
POL(square(x₁)) = 0
POL(times(x₁, x₂)) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

The TRS R consists of the following 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))
square(mark(X)) → mark(square(X))
square(ok(X)) → ok(square(X))
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))
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))
pi(mark(X)) → mark(pi(X))
pi(ok(X)) → ok(pi(X))
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))
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))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
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))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
negrecip(mark(X)) → mark(negrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
posrecip(mark(X)) → mark(posrecip(X))
posrecip(ok(X)) → ok(posrecip(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(2ndsneg(s(x0), cons(x1, x2)))) → TOP(mark(2ndsneg(s(x0), cons2(x1, x2))))

The remaining pairs can at least be oriented weakly.

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(2ndsneg(x₁, x₂)) = x₂
POL(2ndspos(x₁, x₂)) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = 1
POL(cons2(x₁, x₂)) = 0
POL(from(x₁)) = 1
POL(mark(x₁)) = x₁
POL(negrecip(x₁)) = 0
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(pi(x₁)) = 0
POL(plus(x₁, x₂)) = x₂
POL(posrecip(x₁)) = 0
POL(proper(x₁)) = x₁
POL(rcons(x₁, x₂)) = 0
POL(rnil) = 0
POL(s(x₁)) = 0
POL(square(x₁)) = 0
POL(times(x₁, x₂)) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

The TRS R consists of the following 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))
square(mark(X)) → mark(square(X))
square(ok(X)) → ok(square(X))
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))
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))
pi(mark(X)) → mark(pi(X))
pi(ok(X)) → ok(pi(X))
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))
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))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
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))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
negrecip(mark(X)) → mark(negrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
posrecip(mark(X)) → mark(posrecip(X))
posrecip(ok(X)) → ok(posrecip(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(plus(s(x0), x1))) → TOP(mark(s(plus(x0, x1))))
TOP(ok(plus(0, x0))) → TOP(mark(x0))
TOP(ok(times(s(x0), x1))) → TOP(mark(plus(x1, times(x0, x1))))

The remaining pairs can at least be oriented weakly.

TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))

Used ordering: Combined order from the following AFS and order.
TOP(x1) = TOP(x1)
ok(x1) = x1
plus(x1, x2) = plus(x1, x2)
s(x1) = s(x1)
mark(x1) = x1
square(x1) = square(x1)
proper(x1) = x1
times(x1, x2) = times(x1, x2)
active(x1) = x1
from(x1) = from
rcons(x1, x2) = x2
cons2(x1, x2) = x2
0 = 0
negrecip(x1) = negrecip
cons(x1, x2) = x2
2ndsneg(x1, x2) = 2ndsneg(x1, x2)
2ndspos(x1, x2) = 2ndspos(x1, x2)
posrecip(x1) = x1
pi(x1) = pi(x1)
rnil = rnil
nil = nil

Recursive path order with status [2].
Quasi-Precedence:

square₁ > times₂ > TOP₁ > plus₂ > s₁ > [2ndsneg₂, 2ndspos₂] > negrecip
square₁ > times₂ > TOP₁ > from > s₁ > [2ndsneg₂, 2ndspos₂] > negrecip
[0, pi₁] > TOP₁ > plus₂ > s₁ > [2ndsneg₂, 2ndspos₂] > negrecip
[0, pi₁] > TOP₁ > from > s₁ > [2ndsneg₂, 2ndspos₂] > negrecip
[0, pi₁] > rnil

Status:

plus₂: [2,1]
rnil: multiset
pi₁: multiset
square₁: multiset
0: multiset
2ndsneg₂: [1,2]
negrecip: multiset
from: []
2ndspos₂: [1,2]
times₂: multiset
s₁: multiset
TOP₁: multiset
nil: multiset

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(square(x0))) → TOP(square(proper(x0)))
TOP(ok(times(x0, x1))) → TOP(times(x0, active(x1)))
TOP(ok(times(x0, x1))) → TOP(times(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(times(x0, x1))) → TOP(times(proper(x0), proper(x1)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(x0, active(x1)))
TOP(ok(cons2(x0, x1))) → TOP(cons2(x0, active(x1)))
TOP(ok(square(x0))) → TOP(square(active(x0)))
TOP(mark(negrecip(x0))) → TOP(negrecip(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(cons2(x0, x1))) → TOP(cons2(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(2ndsneg(x0, x1))) → TOP(2ndsneg(proper(x0), proper(x1)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(active(x0), x1))
TOP(mark(2ndspos(x0, x1))) → TOP(2ndspos(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(rcons(x0, x1))) → TOP(rcons(active(x0), x1))
TOP(mark(posrecip(x0))) → TOP(posrecip(proper(x0)))
TOP(mark(rcons(x0, x1))) → TOP(rcons(proper(x0), proper(x1)))
TOP(ok(posrecip(x0))) → TOP(posrecip(active(x0)))
TOP(mark(pi(x0))) → TOP(pi(proper(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(2ndsneg(x0, x1))) → TOP(2ndsneg(active(x0), x1))
TOP(ok(pi(x0))) → TOP(pi(active(x0)))
TOP(ok(2ndspos(x0, x1))) → TOP(2ndspos(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(negrecip(x0))) → TOP(negrecip(active(x0)))

The TRS R consists of the following 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))
square(mark(X)) → mark(square(X))
square(ok(X)) → ok(square(X))
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))
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))
pi(mark(X)) → mark(pi(X))
pi(ok(X)) → ok(pi(X))
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))
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))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
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))
cons2(X1, mark(X2)) → mark(cons2(X1, X2))
cons2(ok(X1), ok(X2)) → ok(cons2(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
negrecip(mark(X)) → mark(negrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
posrecip(mark(X)) → mark(posrecip(X))
posrecip(ok(X)) → ok(posrecip(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.