KILLEDRuntime Complexity (full) proof of /tmp/tmp3d3lGy/ExAppendixB_AEL03_C.xml
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).0 CpxTRS↳1 RenamingProof (⇔, 0 ms)↳2 CpxRelTRS↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)↳4 typed CpxTrs↳5 OrderProof (LOWER BOUND(ID), 6 ms)↳6 typed CpxTrs↳7 RewriteLemmaProof (LOWER BOUND(ID), 585 ms)↳8 BEST↳9 typed CpxTrs↳10 RewriteLemmaProof (LOWER BOUND(ID), 164 ms)↳11 BEST↳12 typed CpxTrs↳13 RewriteLemmaProof (LOWER BOUND(ID), 97 ms)↳14 BEST↳15 typed CpxTrs↳16 RewriteLemmaProof (LOWER BOUND(ID), 234 ms)↳17 BEST↳18 typed CpxTrs↳19 RewriteLemmaProof (LOWER BOUND(ID), 184 ms)↳20 BEST↳21 typed CpxTrs↳22 RewriteLemmaProof (LOWER BOUND(ID), 148 ms)↳23 BEST↳24 typed CpxTrs↳25 RewriteLemmaProof (LOWER BOUND(ID), 124 ms)↳26 BEST↳27 typed CpxTrs↳28 RewriteLemmaProof (LOWER BOUND(ID), 76 ms)↳29 BEST↳30 typed CpxTrs↳31 RewriteLemmaProof (LOWER BOUND(ID), 31 ms)↳32 BEST↳33 typed CpxTrs↳34 RewriteLemmaProof (LOWER BOUND(ID), 44 ms)↳35 BEST↳36 typed CpxTrs↳37 RewriteLemmaProof (LOWER BOUND(ID), 103 ms)↳38 BEST↳39 typed CpxTrs↳40 RewriteLemmaProof (LOWER BOUND(ID), 61 ms)↳41 BEST↳42 typed CpxTrs↳43 RewriteLemmaProof (LOWER BOUND(ID), 66 ms)↳44 BEST↳45 typed CpxTrs↳46 typed CpxTrs↳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(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) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.(2) 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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.(4) 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(5) 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(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:nilGenerator 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, 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(7) 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).
(8) Complex Obligation (BEST)
(9) 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:nilLemmas:
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, topThey 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(10) 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).
(11) Complex Obligation (BEST)
(12) 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:nilLemmas:
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, topThey 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(13) 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).
(14) Complex Obligation (BEST)
(15) 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:nilLemmas:
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, topThey 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(16) 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).
(17) Complex Obligation (BEST)
(18) 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:nilLemmas:
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, topThey 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(19) 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).
(20) Complex Obligation (BEST)
(21) 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:nilLemmas:
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, topThey 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(22) 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).
(23) Complex Obligation (BEST)
(24) 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:nilLemmas:
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, topThey 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(25) 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).
(26) Complex Obligation (BEST)
(27) 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:nilLemmas:
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, topThey 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(28) 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).
(29) Complex Obligation (BEST)
(30) 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:nilLemmas:
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, topThey 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(31) 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).
(32) Complex Obligation (BEST)
(33) 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:nilLemmas:
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, topThey 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(34) 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).
(35) Complex Obligation (BEST)
(36) 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:nilLemmas:
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, topThey will be analysed ascendingly in the following order:
times < active
pi < active
square < active
active < top
times < proper
pi < proper
square < proper
proper < top(37) 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).
(38) Complex Obligation (BEST)
(39) 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:nilLemmas:
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, topThey will be analysed ascendingly in the following order:
pi < active
square < active
active < top
pi < proper
square < proper
proper < top(40) 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).
(41) Complex Obligation (BEST)
(42) 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:nilLemmas:
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, topThey will be analysed ascendingly in the following order:
square < active
active < top
square < proper
proper < top(43) 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).
(44) Complex Obligation (BEST)
(45) 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:nilLemmas:
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, topThey will be analysed ascendingly in the following order:
active < top
proper < top(46) 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:nilLemmas:
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.
(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:nilLemmas:
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.
(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:nilLemmas:
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.
(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:nilLemmas:
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.
(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:nilLemmas:
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.
(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:nilLemmas:
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.
(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:nilLemmas:
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.
(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:nilLemmas:
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.
(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:nilLemmas:
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.
(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:nilLemmas:
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.
(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:nilLemmas:
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.
(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:nilLemmas:
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.
(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:nilLemmas:
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.