/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 571 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


----------------------------------------

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


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))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(0) -> 0
   encArg(rnil) -> rnil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_posrecip(x_1)) -> posrecip(encArg(x_1))
   encArg(cons_negrecip(x_1)) -> negrecip(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_cons2(x_1, x_2)) -> cons2(encArg(x_1), encArg(x_2))
   encArg(cons_rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2))
   encArg(cons_2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encArg(cons_pi(x_1)) -> pi(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_square(x_1)) -> square(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_rnil -> rnil
   encode_cons2(x_1, x_2) -> cons2(encArg(x_1), encArg(x_2))
   encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2))
   encode_posrecip(x_1) -> posrecip(encArg(x_1))
   encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encode_negrecip(x_1) -> negrecip(encArg(x_1))
   encode_pi(x_1) -> pi(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_square(x_1) -> square(encArg(x_1))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_nil -> nil
   encode_top(x_1) -> top(encArg(x_1))


----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


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))

The (relative) TRS S consists of the following rules:

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(0) -> 0
   encArg(rnil) -> rnil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_posrecip(x_1)) -> posrecip(encArg(x_1))
   encArg(cons_negrecip(x_1)) -> negrecip(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_cons2(x_1, x_2)) -> cons2(encArg(x_1), encArg(x_2))
   encArg(cons_rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2))
   encArg(cons_2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encArg(cons_pi(x_1)) -> pi(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_square(x_1)) -> square(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_rnil -> rnil
   encode_cons2(x_1, x_2) -> cons2(encArg(x_1), encArg(x_2))
   encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2))
   encode_posrecip(x_1) -> posrecip(encArg(x_1))
   encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encode_negrecip(x_1) -> negrecip(encArg(x_1))
   encode_pi(x_1) -> pi(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_square(x_1) -> square(encArg(x_1))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_nil -> nil
   encode_top(x_1) -> top(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


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))

The (relative) TRS S consists of the following rules:

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(0) -> 0
   encArg(rnil) -> rnil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_posrecip(x_1)) -> posrecip(encArg(x_1))
   encArg(cons_negrecip(x_1)) -> negrecip(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_cons2(x_1, x_2)) -> cons2(encArg(x_1), encArg(x_2))
   encArg(cons_rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2))
   encArg(cons_2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encArg(cons_pi(x_1)) -> pi(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_square(x_1)) -> square(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_rnil -> rnil
   encode_cons2(x_1, x_2) -> cons2(encArg(x_1), encArg(x_2))
   encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2))
   encode_posrecip(x_1) -> posrecip(encArg(x_1))
   encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encode_negrecip(x_1) -> negrecip(encArg(x_1))
   encode_pi(x_1) -> pi(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_square(x_1) -> square(encArg(x_1))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_nil -> nil
   encode_top(x_1) -> top(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


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))

The (relative) TRS S consists of the following rules:

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(0) -> 0
   encArg(rnil) -> rnil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_posrecip(x_1)) -> posrecip(encArg(x_1))
   encArg(cons_negrecip(x_1)) -> negrecip(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_cons2(x_1, x_2)) -> cons2(encArg(x_1), encArg(x_2))
   encArg(cons_rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2))
   encArg(cons_2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encArg(cons_pi(x_1)) -> pi(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_square(x_1)) -> square(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_rnil -> rnil
   encode_cons2(x_1, x_2) -> cons2(encArg(x_1), encArg(x_2))
   encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2))
   encode_posrecip(x_1) -> posrecip(encArg(x_1))
   encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encode_negrecip(x_1) -> negrecip(encArg(x_1))
   encode_pi(x_1) -> pi(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_square(x_1) -> square(encArg(x_1))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_nil -> nil
   encode_top(x_1) -> top(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(7) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

times(ok(X1), ok(X2)) ->^+ ok(times(X1, X2))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [X1 / ok(X1), X2 / ok(X2)].

The result substitution is [ ].




----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


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))

The (relative) TRS S consists of the following rules:

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(0) -> 0
   encArg(rnil) -> rnil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_posrecip(x_1)) -> posrecip(encArg(x_1))
   encArg(cons_negrecip(x_1)) -> negrecip(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_cons2(x_1, x_2)) -> cons2(encArg(x_1), encArg(x_2))
   encArg(cons_rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2))
   encArg(cons_2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encArg(cons_pi(x_1)) -> pi(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_square(x_1)) -> square(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_rnil -> rnil
   encode_cons2(x_1, x_2) -> cons2(encArg(x_1), encArg(x_2))
   encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2))
   encode_posrecip(x_1) -> posrecip(encArg(x_1))
   encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encode_negrecip(x_1) -> negrecip(encArg(x_1))
   encode_pi(x_1) -> pi(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_square(x_1) -> square(encArg(x_1))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_nil -> nil
   encode_top(x_1) -> top(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


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))

The (relative) TRS S consists of the following rules:

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(0) -> 0
   encArg(rnil) -> rnil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_posrecip(x_1)) -> posrecip(encArg(x_1))
   encArg(cons_negrecip(x_1)) -> negrecip(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_cons2(x_1, x_2)) -> cons2(encArg(x_1), encArg(x_2))
   encArg(cons_rcons(x_1, x_2)) -> rcons(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_2ndspos(x_1, x_2)) -> 2ndspos(encArg(x_1), encArg(x_2))
   encArg(cons_2ndsneg(x_1, x_2)) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encArg(cons_pi(x_1)) -> pi(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_square(x_1)) -> square(encArg(x_1))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_2ndspos(x_1, x_2) -> 2ndspos(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_rnil -> rnil
   encode_cons2(x_1, x_2) -> cons2(encArg(x_1), encArg(x_2))
   encode_rcons(x_1, x_2) -> rcons(encArg(x_1), encArg(x_2))
   encode_posrecip(x_1) -> posrecip(encArg(x_1))
   encode_2ndsneg(x_1, x_2) -> 2ndsneg(encArg(x_1), encArg(x_2))
   encode_negrecip(x_1) -> negrecip(encArg(x_1))
   encode_pi(x_1) -> pi(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_square(x_1) -> square(encArg(x_1))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_nil -> nil
   encode_top(x_1) -> top(encArg(x_1))

Rewrite Strategy: INNERMOST