KILLED
Runtime Complexity (full) proof of /tmp/tmpjcTl2y/ExAppendixB_AEL03_C.xml
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
0 CpxTRS
↳1 DecreasingLoopProof (⇔, 8214 ms)
↳2 BOUNDS(n^1, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 typed CpxTrs
↳7 OrderProof (LOWER BOUND(ID), 0 ms)
↳8 typed CpxTrs
↳9 RewriteLemmaProof (LOWER BOUND(ID), 548 ms)
↳10 BEST
↳11 typed CpxTrs
↳12 RewriteLemmaProof (LOWER BOUND(ID), 52 ms)
↳13 BEST
↳14 typed CpxTrs
↳15 RewriteLemmaProof (LOWER BOUND(ID), 21 ms)
↳16 BEST
↳17 typed CpxTrs
↳18 RewriteLemmaProof (LOWER BOUND(ID), 107 ms)
↳19 BEST
↳20 typed CpxTrs
↳21 RewriteLemmaProof (LOWER BOUND(ID), 170 ms)
↳22 BEST
↳23 typed CpxTrs
↳24 RewriteLemmaProof (LOWER BOUND(ID), 72 ms)
↳25 BEST
↳26 typed CpxTrs
↳27 RewriteLemmaProof (LOWER BOUND(ID), 54 ms)
↳28 BEST
↳29 typed CpxTrs
↳30 RewriteLemmaProof (LOWER BOUND(ID), 80 ms)
↳31 BEST
↳32 typed CpxTrs
↳33 RewriteLemmaProof (LOWER BOUND(ID), 55 ms)
↳34 BEST
↳35 typed CpxTrs
↳36 RewriteLemmaProof (LOWER BOUND(ID), 79 ms)
↳37 BEST
↳38 typed CpxTrs
↳39 RewriteLemmaProof (LOWER BOUND(ID), 51 ms)
↳40 BEST
↳41 typed CpxTrs
↳42 RewriteLemmaProof (LOWER BOUND(ID), 14 ms)
↳43 BEST
↳44 typed CpxTrs
↳45 RewriteLemmaProof (LOWER BOUND(ID), 31 ms)
↳46 BEST
↳47 typed CpxTrs
↳48 typed CpxTrs
↳49 typed CpxTrs
↳50 typed CpxTrs
↳51 typed CpxTrs
↳52 typed CpxTrs
↳53 typed CpxTrs
↳54 typed CpxTrs
↳55 typed CpxTrs
↳56 typed CpxTrs
↳57 typed CpxTrs
↳58 typed CpxTrs
↳59 typed CpxTrs
↳60 typed CpxTrs
(0) Obligation:
Runtime Complexity TRS:
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))
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
s(mark(X)) →+ mark(s(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / mark(X)].
The result substitution is [ ].
(2) BOUNDS(n^1, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(4) Obligation:
Runtime Complexity Relative TRS:
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: FULL
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active, cons, from, s, 2ndspos, cons2, rcons, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, topThey will be analysed ascendingly in the following order:
cons < active
from < active
s < active
2ndspos < active
cons2 < active
rcons < active
posrecip < active
2ndsneg < active
negrecip < active
plus < active
times < active
pi < active
square < active
active < top
cons < proper
from < proper
s < proper
2ndspos < proper
cons2 < proper
rcons < proper
posrecip < proper
2ndsneg < proper
negrecip < proper
plus < proper
times < proper
pi < proper
square < proper
proper < top
(8) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
cons, active, from, s, 2ndspos, cons2, rcons, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top
They will be analysed ascendingly in the following order:
cons < active
from < active
s < active
2ndspos < active
cons2 < active
rcons < active
posrecip < active
2ndsneg < active
negrecip < active
plus < active
times < active
pi < active
square < active
active < top
cons < proper
from < proper
s < proper
2ndspos < proper
cons2 < proper
rcons < proper
posrecip < proper
2ndsneg < proper
negrecip < proper
plus < proper
times < proper
pi < proper
square < proper
proper < top
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)Induction Base:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b))
Induction Step:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, +(n5_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) →RΩ(1)
mark(cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
from, active, s, 2ndspos, cons2, rcons, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top
They will be analysed ascendingly in the following order:
from < active
s < active
2ndspos < active
cons2 < active
rcons < active
posrecip < active
2ndsneg < active
negrecip < active
plus < active
times < active
pi < active
square < active
active < top
from < proper
s < proper
2ndspos < proper
cons2 < proper
rcons < proper
posrecip < proper
2ndsneg < proper
negrecip < proper
plus < proper
times < proper
pi < proper
square < proper
proper < top
(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)Induction Base:
from(gen_mark:0':rnil:ok:nil3_0(+(1, 0)))
Induction Step:
from(gen_mark:0':rnil:ok:nil3_0(+(1, +(n1502_0, 1)))) →RΩ(1)
mark(from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(13) Complex Obligation (BEST)
(14) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
s, active, 2ndspos, cons2, rcons, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top
They will be analysed ascendingly in the following order:
s < active
2ndspos < active
cons2 < active
rcons < active
posrecip < active
2ndsneg < active
negrecip < active
plus < active
times < active
pi < active
square < active
active < top
s < proper
2ndspos < proper
cons2 < proper
rcons < proper
posrecip < proper
2ndsneg < proper
negrecip < proper
plus < proper
times < proper
pi < proper
square < proper
proper < top
(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)Induction Base:
s(gen_mark:0':rnil:ok:nil3_0(+(1, 0)))
Induction Step:
s(gen_mark:0':rnil:ok:nil3_0(+(1, +(n2159_0, 1)))) →RΩ(1)
mark(s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(16) Complex Obligation (BEST)
(17) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
2ndspos, active, cons2, rcons, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top
They will be analysed ascendingly in the following order:
2ndspos < active
cons2 < active
rcons < active
posrecip < active
2ndsneg < active
negrecip < active
plus < active
times < active
pi < active
square < active
active < top
2ndspos < proper
cons2 < proper
rcons < proper
posrecip < proper
2ndsneg < proper
negrecip < proper
plus < proper
times < proper
pi < proper
square < proper
proper < top
(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)Induction Base:
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b))
Induction Step:
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, +(n2917_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) →RΩ(1)
mark(2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(19) Complex Obligation (BEST)
(20) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
cons2, active, rcons, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top
They will be analysed ascendingly in the following order:
cons2 < active
rcons < active
posrecip < active
2ndsneg < active
negrecip < active
plus < active
times < active
pi < active
square < active
active < top
cons2 < proper
rcons < proper
posrecip < proper
2ndsneg < proper
negrecip < proper
plus < proper
times < proper
pi < proper
square < proper
proper < top
(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)Induction Base:
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, 0)))
Induction Step:
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, +(n5329_0, 1)))) →RΩ(1)
mark(cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(22) Complex Obligation (BEST)
(23) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
rcons, active, posrecip, 2ndsneg, negrecip, plus, times, pi, square, proper, top
They will be analysed ascendingly in the following order:
rcons < active
posrecip < active
2ndsneg < active
negrecip < active
plus < active
times < active
pi < active
square < active
active < top
rcons < proper
posrecip < proper
2ndsneg < proper
negrecip < proper
plus < proper
times < proper
pi < proper
square < proper
proper < top
(24) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)Induction Base:
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b))
Induction Step:
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, +(n7846_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) →RΩ(1)
mark(rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(25) Complex Obligation (BEST)
(26) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
posrecip, active, 2ndsneg, negrecip, plus, times, pi, square, proper, top
They will be analysed ascendingly in the following order:
posrecip < active
2ndsneg < active
negrecip < active
plus < active
times < active
pi < active
square < active
active < top
posrecip < proper
2ndsneg < proper
negrecip < proper
plus < proper
times < proper
pi < proper
square < proper
proper < top
(27) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)Induction Base:
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, 0)))
Induction Step:
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, +(n10866_0, 1)))) →RΩ(1)
mark(posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(28) Complex Obligation (BEST)
(29) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
2ndsneg, active, negrecip, plus, times, pi, square, proper, top
They will be analysed ascendingly in the following order:
2ndsneg < active
negrecip < active
plus < active
times < active
pi < active
square < active
active < top
2ndsneg < proper
negrecip < proper
plus < proper
times < proper
pi < proper
square < proper
proper < top
(30) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n121750)Induction Base:
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b))
Induction Step:
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, +(n12175_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) →RΩ(1)
mark(2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(31) Complex Obligation (BEST)
(32) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n121750)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
negrecip, active, plus, times, pi, square, proper, top
They will be analysed ascendingly in the following order:
negrecip < active
plus < active
times < active
pi < active
square < active
active < top
negrecip < proper
plus < proper
times < proper
pi < proper
square < proper
proper < top
(33) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n15705_0))) → *4_0, rt ∈ Ω(n157050)Induction Base:
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, 0)))
Induction Step:
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, +(n15705_0, 1)))) →RΩ(1)
mark(negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n15705_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(34) Complex Obligation (BEST)
(35) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n121750)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n15705_0))) → *4_0, rt ∈ Ω(n157050)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
plus, active, times, pi, square, proper, top
They will be analysed ascendingly in the following order:
plus < active
times < active
pi < active
square < active
active < top
plus < proper
times < proper
pi < proper
square < proper
proper < top
(36) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17265_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n172650)Induction Base:
plus(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b))
Induction Step:
plus(gen_mark:0':rnil:ok:nil3_0(+(1, +(n17265_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) →RΩ(1)
mark(plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17265_0)), gen_mark:0':rnil:ok:nil3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(37) Complex Obligation (BEST)
(38) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n121750)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n15705_0))) → *4_0, rt ∈ Ω(n157050)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17265_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n172650)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
times, active, pi, square, proper, top
They will be analysed ascendingly in the following order:
times < active
pi < active
square < active
active < top
times < proper
pi < proper
square < proper
proper < top
(39) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
times(gen_mark:0':rnil:ok:nil3_0(+(1, n21305_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n213050)Induction Base:
times(gen_mark:0':rnil:ok:nil3_0(+(1, 0)), gen_mark:0':rnil:ok:nil3_0(b))
Induction Step:
times(gen_mark:0':rnil:ok:nil3_0(+(1, +(n21305_0, 1))), gen_mark:0':rnil:ok:nil3_0(b)) →RΩ(1)
mark(times(gen_mark:0':rnil:ok:nil3_0(+(1, n21305_0)), gen_mark:0':rnil:ok:nil3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(40) Complex Obligation (BEST)
(41) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n121750)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n15705_0))) → *4_0, rt ∈ Ω(n157050)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17265_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n172650)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n21305_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n213050)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
pi, active, square, proper, top
They will be analysed ascendingly in the following order:
pi < active
square < active
active < top
pi < proper
square < proper
proper < top
(42) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
pi(gen_mark:0':rnil:ok:nil3_0(+(1, n25649_0))) → *4_0, rt ∈ Ω(n256490)Induction Base:
pi(gen_mark:0':rnil:ok:nil3_0(+(1, 0)))
Induction Step:
pi(gen_mark:0':rnil:ok:nil3_0(+(1, +(n25649_0, 1)))) →RΩ(1)
mark(pi(gen_mark:0':rnil:ok:nil3_0(+(1, n25649_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(43) Complex Obligation (BEST)
(44) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n121750)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n15705_0))) → *4_0, rt ∈ Ω(n157050)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17265_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n172650)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n21305_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n213050)
pi(gen_mark:0':rnil:ok:nil3_0(+(1, n25649_0))) → *4_0, rt ∈ Ω(n256490)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
square, active, proper, top
They will be analysed ascendingly in the following order:
square < active
active < top
square < proper
proper < top
(45) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
square(gen_mark:0':rnil:ok:nil3_0(+(1, n27610_0))) → *4_0, rt ∈ Ω(n276100)Induction Base:
square(gen_mark:0':rnil:ok:nil3_0(+(1, 0)))
Induction Step:
square(gen_mark:0':rnil:ok:nil3_0(+(1, +(n27610_0, 1)))) →RΩ(1)
mark(square(gen_mark:0':rnil:ok:nil3_0(+(1, n27610_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(46) Complex Obligation (BEST)
(47) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n121750)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n15705_0))) → *4_0, rt ∈ Ω(n157050)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17265_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n172650)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n21305_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n213050)
pi(gen_mark:0':rnil:ok:nil3_0(+(1, n25649_0))) → *4_0, rt ∈ Ω(n256490)
square(gen_mark:0':rnil:ok:nil3_0(+(1, n27610_0))) → *4_0, rt ∈ Ω(n276100)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
The following defined symbols remain to be analysed:
active, proper, top
They will be analysed ascendingly in the following order:
active < top
proper < top
(48) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n121750)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n15705_0))) → *4_0, rt ∈ Ω(n157050)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17265_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n172650)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n21305_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n213050)
pi(gen_mark:0':rnil:ok:nil3_0(+(1, n25649_0))) → *4_0, rt ∈ Ω(n256490)
square(gen_mark:0':rnil:ok:nil3_0(+(1, n27610_0))) → *4_0, rt ∈ Ω(n276100)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
No more defined symbols left to analyse.
(49) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n121750)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n15705_0))) → *4_0, rt ∈ Ω(n157050)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17265_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n172650)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n21305_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n213050)
pi(gen_mark:0':rnil:ok:nil3_0(+(1, n25649_0))) → *4_0, rt ∈ Ω(n256490)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
No more defined symbols left to analyse.
(50) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n121750)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n15705_0))) → *4_0, rt ∈ Ω(n157050)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17265_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n172650)
times(gen_mark:0':rnil:ok:nil3_0(+(1, n21305_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n213050)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
No more defined symbols left to analyse.
(51) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n121750)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n15705_0))) → *4_0, rt ∈ Ω(n157050)
plus(gen_mark:0':rnil:ok:nil3_0(+(1, n17265_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n172650)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
No more defined symbols left to analyse.
(52) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n121750)
negrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n15705_0))) → *4_0, rt ∈ Ω(n157050)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
No more defined symbols left to analyse.
(53) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
2ndsneg(gen_mark:0':rnil:ok:nil3_0(+(1, n12175_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n121750)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
No more defined symbols left to analyse.
(54) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
posrecip(gen_mark:0':rnil:ok:nil3_0(+(1, n10866_0))) → *4_0, rt ∈ Ω(n108660)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
No more defined symbols left to analyse.
(55) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
rcons(gen_mark:0':rnil:ok:nil3_0(+(1, n7846_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n78460)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
No more defined symbols left to analyse.
(56) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
cons2(gen_mark:0':rnil:ok:nil3_0(a), gen_mark:0':rnil:ok:nil3_0(+(1, n5329_0))) → *4_0, rt ∈ Ω(n53290)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
No more defined symbols left to analyse.
(57) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
2ndspos(gen_mark:0':rnil:ok:nil3_0(+(1, n2917_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n29170)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
No more defined symbols left to analyse.
(58) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
s(gen_mark:0':rnil:ok:nil3_0(+(1, n2159_0))) → *4_0, rt ∈ Ω(n21590)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
No more defined symbols left to analyse.
(59) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
from(gen_mark:0':rnil:ok:nil3_0(+(1, n1502_0))) → *4_0, rt ∈ Ω(n15020)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
No more defined symbols left to analyse.
(60) Obligation:
TRS:
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))
Types:
active :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
from :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
mark :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
cons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
s :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndspos :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
0' :: mark:0':rnil:ok:nil
rnil :: mark:0':rnil:ok:nil
cons2 :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
rcons :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
posrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
2ndsneg :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
negrecip :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
pi :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
plus :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
times :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
square :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
proper :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
ok :: mark:0':rnil:ok:nil → mark:0':rnil:ok:nil
nil :: mark:0':rnil:ok:nil
top :: mark:0':rnil:ok:nil → top
hole_mark:0':rnil:ok:nil1_0 :: mark:0':rnil:ok:nil
hole_top2_0 :: top
gen_mark:0':rnil:ok:nil3_0 :: Nat → mark:0':rnil:ok:nil
Lemmas:
cons(gen_mark:0':rnil:ok:nil3_0(+(1, n5_0)), gen_mark:0':rnil:ok:nil3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_mark:0':rnil:ok:nil3_0(0) ⇔ 0'
gen_mark:0':rnil:ok:nil3_0(+(x, 1)) ⇔ mark(gen_mark:0':rnil:ok:nil3_0(x))
No more defined symbols left to analyse.