KILLEDRuntime Complexity (full) proof of /tmp/tmp8WGIrL/ExSec4_2_DLMMU04_C.xml
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).0 CpxTRS↳1 DecreasingLoopProof (⇔, 7711 ms)↳2 BOUNDS(n^1, INF)↳3 RenamingProof (⇔, 0 ms)↳4 CpxRelTRS↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)↳6 typed CpxTrs↳7 OrderProof (LOWER BOUND(ID), 0 ms)↳8 typed CpxTrs↳9 RewriteLemmaProof (LOWER BOUND(ID), 430 ms)↳10 BEST↳11 typed CpxTrs↳12 RewriteLemmaProof (LOWER BOUND(ID), 67 ms)↳13 BEST↳14 typed CpxTrs↳15 RewriteLemmaProof (LOWER BOUND(ID), 59 ms)↳16 BEST↳17 typed CpxTrs↳18 RewriteLemmaProof (LOWER BOUND(ID), 55 ms)↳19 BEST↳20 typed CpxTrs↳21 RewriteLemmaProof (LOWER BOUND(ID), 129 ms)↳22 BEST↳23 typed CpxTrs↳24 RewriteLemmaProof (LOWER BOUND(ID), 84 ms)↳25 BEST↳26 typed CpxTrs↳27 RewriteLemmaProof (LOWER BOUND(ID), 51 ms)↳28 BEST↳29 typed CpxTrs↳30 RewriteLemmaProof (LOWER BOUND(ID), 67 ms)↳31 BEST↳32 typed CpxTrs↳33 RewriteLemmaProof (LOWER BOUND(ID), 47 ms)↳34 BEST↳35 typed CpxTrs↳36 RewriteLemmaProof (LOWER BOUND(ID), 69 ms)↳37 BEST↳38 typed CpxTrs↳39 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳40 BEST↳41 typed CpxTrs↳42 RewriteLemmaProof (LOWER BOUND(ID), 26 ms)↳43 BEST↳44 typed CpxTrs↳45 RewriteLemmaProof (LOWER BOUND(ID), 109 ms)↳46 BEST↳47 typed CpxTrs↳48 NoRewriteLemmaProof (LOWER BOUND(ID), 202.1 s)↳49 typed CpxTrs↳50 NoRewriteLemmaProof (LOWER BOUND(ID), 743 ms)↳51 typed CpxTrs↳52 typed CpxTrs↳53 typed CpxTrs↳54 typed CpxTrs↳55 typed CpxTrs↳56 typed CpxTrs↳57 typed CpxTrs↳58 typed CpxTrs↳59 typed CpxTrs↳60 typed CpxTrs↳61 typed CpxTrs↳62 typed CpxTrs↳63 typed CpxTrs↳64 typed CpxTrs(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: FULL(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
natsFrom(mark(X)) →+ mark(natsFrom(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / mark(X)].
The result substitution is [ ].(2) BOUNDS(n^1, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S is empty.
Rewrite Strategy: FULL(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.(6) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active, cons, natsFrom, s, pair, u, splitAt, head, afterNth, fst, snd, tail, sel, take, proper, topThey will be analysed ascendingly in the following order:
cons < active
natsFrom < active
s < active
pair < active
u < active
splitAt < active
head < active
afterNth < active
fst < active
snd < active
tail < active
sel < active
take < active
active < top
cons < proper
natsFrom < proper
s < proper
pair < proper
u < proper
splitAt < proper
head < proper
afterNth < proper
fst < proper
snd < proper
tail < proper
sel < proper
take < proper
proper < top(8) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okGenerator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
cons, active, natsFrom, s, pair, u, splitAt, head, afterNth, fst, snd, tail, sel, take, proper, topThey will be analysed ascendingly in the following order:
cons < active
natsFrom < active
s < active
pair < active
u < active
splitAt < active
head < active
afterNth < active
fst < active
snd < active
tail < active
sel < active
take < active
active < top
cons < proper
natsFrom < proper
s < proper
pair < proper
u < proper
splitAt < proper
head < proper
afterNth < proper
fst < proper
snd < proper
tail < proper
sel < proper
take < proper
proper < top(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)Induction Base:
cons(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))Induction Step:
cons(gen_mark:0':nil:ok3_0(+(1, +(n5_0, 1))), gen_mark:0':nil:ok3_0(b)) →RΩ(1)
mark(cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
natsFrom, active, s, pair, u, splitAt, head, afterNth, fst, snd, tail, sel, take, proper, topThey will be analysed ascendingly in the following order:
natsFrom < active
s < active
pair < active
u < active
splitAt < active
head < active
afterNth < active
fst < active
snd < active
tail < active
sel < active
take < active
active < top
natsFrom < proper
s < proper
pair < proper
u < proper
splitAt < proper
head < proper
afterNth < proper
fst < proper
snd < proper
tail < proper
sel < proper
take < proper
proper < top(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)Induction Base:
natsFrom(gen_mark:0':nil:ok3_0(+(1, 0)))Induction Step:
natsFrom(gen_mark:0':nil:ok3_0(+(1, +(n1466_0, 1)))) →RΩ(1)
mark(natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(13) Complex Obligation (BEST)
(14) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
s, active, pair, u, splitAt, head, afterNth, fst, snd, tail, sel, take, proper, topThey will be analysed ascendingly in the following order:
s < active
pair < active
u < active
splitAt < active
head < active
afterNth < active
fst < active
snd < active
tail < active
sel < active
take < active
active < top
s < proper
pair < proper
u < proper
splitAt < proper
head < proper
afterNth < proper
fst < proper
snd < proper
tail < proper
sel < proper
take < proper
proper < top(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)Induction Base:
s(gen_mark:0':nil:ok3_0(+(1, 0)))Induction Step:
s(gen_mark:0':nil:ok3_0(+(1, +(n2114_0, 1)))) →RΩ(1)
mark(s(gen_mark:0':nil:ok3_0(+(1, n2114_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(16) Complex Obligation (BEST)
(17) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
pair, active, u, splitAt, head, afterNth, fst, snd, tail, sel, take, proper, topThey will be analysed ascendingly in the following order:
pair < active
u < active
splitAt < active
head < active
afterNth < active
fst < active
snd < active
tail < active
sel < active
take < active
active < top
pair < proper
u < proper
splitAt < proper
head < proper
afterNth < proper
fst < proper
snd < proper
tail < proper
sel < proper
take < proper
proper < top(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)Induction Base:
pair(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))Induction Step:
pair(gen_mark:0':nil:ok3_0(+(1, +(n2863_0, 1))), gen_mark:0':nil:ok3_0(b)) →RΩ(1)
mark(pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(19) Complex Obligation (BEST)
(20) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
u, active, splitAt, head, afterNth, fst, snd, tail, sel, take, proper, topThey will be analysed ascendingly in the following order:
u < active
splitAt < active
head < active
afterNth < active
fst < active
snd < active
tail < active
sel < active
take < active
active < top
u < proper
splitAt < proper
head < proper
afterNth < proper
fst < proper
snd < proper
tail < proper
sel < proper
take < proper
proper < top(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)Induction Base:
u(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d))Induction Step:
u(gen_mark:0':nil:ok3_0(+(1, +(n5239_0, 1))), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) →RΩ(1)
mark(u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(22) Complex Obligation (BEST)
(23) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
splitAt, active, head, afterNth, fst, snd, tail, sel, take, proper, topThey will be analysed ascendingly in the following order:
splitAt < active
head < active
afterNth < active
fst < active
snd < active
tail < active
sel < active
take < active
active < top
splitAt < proper
head < proper
afterNth < proper
fst < proper
snd < proper
tail < proper
sel < proper
take < proper
proper < top(24) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)Induction Base:
splitAt(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))Induction Step:
splitAt(gen_mark:0':nil:ok3_0(+(1, +(n11098_0, 1))), gen_mark:0':nil:ok3_0(b)) →RΩ(1)
mark(splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(25) Complex Obligation (BEST)
(26) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
head, active, afterNth, fst, snd, tail, sel, take, proper, topThey will be analysed ascendingly in the following order:
head < active
afterNth < active
fst < active
snd < active
tail < active
sel < active
take < active
active < top
head < proper
afterNth < proper
fst < proper
snd < proper
tail < proper
sel < proper
take < proper
proper < top(27) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)Induction Base:
head(gen_mark:0':nil:ok3_0(+(1, 0)))Induction Step:
head(gen_mark:0':nil:ok3_0(+(1, +(n14278_0, 1)))) →RΩ(1)
mark(head(gen_mark:0':nil:ok3_0(+(1, n14278_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(28) Complex Obligation (BEST)
(29) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
afterNth, active, fst, snd, tail, sel, take, proper, topThey will be analysed ascendingly in the following order:
afterNth < active
fst < active
snd < active
tail < active
sel < active
take < active
active < top
afterNth < proper
fst < proper
snd < proper
tail < proper
sel < proper
take < proper
proper < top(30) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)Induction Base:
afterNth(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))Induction Step:
afterNth(gen_mark:0':nil:ok3_0(+(1, +(n15676_0, 1))), gen_mark:0':nil:ok3_0(b)) →RΩ(1)
mark(afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(31) Complex Obligation (BEST)
(32) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
fst, active, snd, tail, sel, take, proper, topThey will be analysed ascendingly in the following order:
fst < active
snd < active
tail < active
sel < active
take < active
active < top
fst < proper
snd < proper
tail < proper
sel < proper
take < proper
proper < top(33) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
fst(gen_mark:0':nil:ok3_0(+(1, n19366_0))) → *4_0, rt ∈ Ω(n193660)Induction Base:
fst(gen_mark:0':nil:ok3_0(+(1, 0)))Induction Step:
fst(gen_mark:0':nil:ok3_0(+(1, +(n19366_0, 1)))) →RΩ(1)
mark(fst(gen_mark:0':nil:ok3_0(+(1, n19366_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(34) Complex Obligation (BEST)
(35) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)
fst(gen_mark:0':nil:ok3_0(+(1, n19366_0))) → *4_0, rt ∈ Ω(n193660)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
snd, active, tail, sel, take, proper, topThey will be analysed ascendingly in the following order:
snd < active
tail < active
sel < active
take < active
active < top
snd < proper
tail < proper
sel < proper
take < proper
proper < top(36) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
snd(gen_mark:0':nil:ok3_0(+(1, n21015_0))) → *4_0, rt ∈ Ω(n210150)Induction Base:
snd(gen_mark:0':nil:ok3_0(+(1, 0)))Induction Step:
snd(gen_mark:0':nil:ok3_0(+(1, +(n21015_0, 1)))) →RΩ(1)
mark(snd(gen_mark:0':nil:ok3_0(+(1, n21015_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(37) Complex Obligation (BEST)
(38) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)
fst(gen_mark:0':nil:ok3_0(+(1, n19366_0))) → *4_0, rt ∈ Ω(n193660)
snd(gen_mark:0':nil:ok3_0(+(1, n21015_0))) → *4_0, rt ∈ Ω(n210150)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
tail, active, sel, take, proper, topThey will be analysed ascendingly in the following order:
tail < active
sel < active
take < active
active < top
tail < proper
sel < proper
take < proper
proper < top(39) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
tail(gen_mark:0':nil:ok3_0(+(1, n22765_0))) → *4_0, rt ∈ Ω(n227650)Induction Base:
tail(gen_mark:0':nil:ok3_0(+(1, 0)))Induction Step:
tail(gen_mark:0':nil:ok3_0(+(1, +(n22765_0, 1)))) →RΩ(1)
mark(tail(gen_mark:0':nil:ok3_0(+(1, n22765_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(40) Complex Obligation (BEST)
(41) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)
fst(gen_mark:0':nil:ok3_0(+(1, n19366_0))) → *4_0, rt ∈ Ω(n193660)
snd(gen_mark:0':nil:ok3_0(+(1, n21015_0))) → *4_0, rt ∈ Ω(n210150)
tail(gen_mark:0':nil:ok3_0(+(1, n22765_0))) → *4_0, rt ∈ Ω(n227650)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
sel, active, take, proper, topThey will be analysed ascendingly in the following order:
sel < active
take < active
active < top
sel < proper
take < proper
proper < top(42) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
sel(gen_mark:0':nil:ok3_0(+(1, n24616_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n246160)Induction Base:
sel(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))Induction Step:
sel(gen_mark:0':nil:ok3_0(+(1, +(n24616_0, 1))), gen_mark:0':nil:ok3_0(b)) →RΩ(1)
mark(sel(gen_mark:0':nil:ok3_0(+(1, n24616_0)), gen_mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(43) Complex Obligation (BEST)
(44) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)
fst(gen_mark:0':nil:ok3_0(+(1, n19366_0))) → *4_0, rt ∈ Ω(n193660)
snd(gen_mark:0':nil:ok3_0(+(1, n21015_0))) → *4_0, rt ∈ Ω(n210150)
tail(gen_mark:0':nil:ok3_0(+(1, n22765_0))) → *4_0, rt ∈ Ω(n227650)
sel(gen_mark:0':nil:ok3_0(+(1, n24616_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n246160)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
take, active, proper, topThey will be analysed ascendingly in the following order:
take < active
active < top
take < proper
proper < top(45) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
take(gen_mark:0':nil:ok3_0(+(1, n29228_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n292280)Induction Base:
take(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))Induction Step:
take(gen_mark:0':nil:ok3_0(+(1, +(n29228_0, 1))), gen_mark:0':nil:ok3_0(b)) →RΩ(1)
mark(take(gen_mark:0':nil:ok3_0(+(1, n29228_0)), gen_mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(46) Complex Obligation (BEST)
(47) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)
fst(gen_mark:0':nil:ok3_0(+(1, n19366_0))) → *4_0, rt ∈ Ω(n193660)
snd(gen_mark:0':nil:ok3_0(+(1, n21015_0))) → *4_0, rt ∈ Ω(n210150)
tail(gen_mark:0':nil:ok3_0(+(1, n22765_0))) → *4_0, rt ∈ Ω(n227650)
sel(gen_mark:0':nil:ok3_0(+(1, n24616_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n246160)
take(gen_mark:0':nil:ok3_0(+(1, n29228_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n292280)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_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(48) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.(49) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)
fst(gen_mark:0':nil:ok3_0(+(1, n19366_0))) → *4_0, rt ∈ Ω(n193660)
snd(gen_mark:0':nil:ok3_0(+(1, n21015_0))) → *4_0, rt ∈ Ω(n210150)
tail(gen_mark:0':nil:ok3_0(+(1, n22765_0))) → *4_0, rt ∈ Ω(n227650)
sel(gen_mark:0':nil:ok3_0(+(1, n24616_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n246160)
take(gen_mark:0':nil:ok3_0(+(1, n29228_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n292280)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
proper, topThey will be analysed ascendingly in the following order:
proper < top(50) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.(51) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)
fst(gen_mark:0':nil:ok3_0(+(1, n19366_0))) → *4_0, rt ∈ Ω(n193660)
snd(gen_mark:0':nil:ok3_0(+(1, n21015_0))) → *4_0, rt ∈ Ω(n210150)
tail(gen_mark:0':nil:ok3_0(+(1, n22765_0))) → *4_0, rt ∈ Ω(n227650)
sel(gen_mark:0':nil:ok3_0(+(1, n24616_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n246160)
take(gen_mark:0':nil:ok3_0(+(1, n29228_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n292280)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
top(52) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)
fst(gen_mark:0':nil:ok3_0(+(1, n19366_0))) → *4_0, rt ∈ Ω(n193660)
snd(gen_mark:0':nil:ok3_0(+(1, n21015_0))) → *4_0, rt ∈ Ω(n210150)
tail(gen_mark:0':nil:ok3_0(+(1, n22765_0))) → *4_0, rt ∈ Ω(n227650)
sel(gen_mark:0':nil:ok3_0(+(1, n24616_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n246160)
take(gen_mark:0':nil:ok3_0(+(1, n29228_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n292280)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(53) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)
fst(gen_mark:0':nil:ok3_0(+(1, n19366_0))) → *4_0, rt ∈ Ω(n193660)
snd(gen_mark:0':nil:ok3_0(+(1, n21015_0))) → *4_0, rt ∈ Ω(n210150)
tail(gen_mark:0':nil:ok3_0(+(1, n22765_0))) → *4_0, rt ∈ Ω(n227650)
sel(gen_mark:0':nil:ok3_0(+(1, n24616_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n246160)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(54) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)
fst(gen_mark:0':nil:ok3_0(+(1, n19366_0))) → *4_0, rt ∈ Ω(n193660)
snd(gen_mark:0':nil:ok3_0(+(1, n21015_0))) → *4_0, rt ∈ Ω(n210150)
tail(gen_mark:0':nil:ok3_0(+(1, n22765_0))) → *4_0, rt ∈ Ω(n227650)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(55) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)
fst(gen_mark:0':nil:ok3_0(+(1, n19366_0))) → *4_0, rt ∈ Ω(n193660)
snd(gen_mark:0':nil:ok3_0(+(1, n21015_0))) → *4_0, rt ∈ Ω(n210150)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(56) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)
fst(gen_mark:0':nil:ok3_0(+(1, n19366_0))) → *4_0, rt ∈ Ω(n193660)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(57) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)
afterNth(gen_mark:0':nil:ok3_0(+(1, n15676_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n156760)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(58) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)
head(gen_mark:0':nil:ok3_0(+(1, n14278_0))) → *4_0, rt ∈ Ω(n142780)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(59) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)
splitAt(gen_mark:0':nil:ok3_0(+(1, n11098_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n110980)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(60) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)
u(gen_mark:0':nil:ok3_0(+(1, n5239_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n52390)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(61) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)
pair(gen_mark:0':nil:ok3_0(+(1, n2863_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28630)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(62) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)
s(gen_mark:0':nil:ok3_0(+(1, n2114_0))) → *4_0, rt ∈ Ω(n21140)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(63) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
natsFrom(gen_mark:0':nil:ok3_0(+(1, n1466_0))) → *4_0, rt ∈ Ω(n14660)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(64) Obligation:
TRS:
Rules:
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:ok → mark:0':nil:ok
natsFrom :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
fst :: mark:0':nil:ok → mark:0':nil:ok
pair :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
snd :: mark:0':nil:ok → mark:0':nil:ok
splitAt :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
nil :: mark:0':nil:ok
u :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
head :: mark:0':nil:ok → mark:0':nil:ok
tail :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
afterNth :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
take :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:okLemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:ok3_0(x))No more defined symbols left to analyse.