Runtime Complexity TRS:
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

Runtime Complexity TRS:
The TRS R consists of the following rules:

active'(U11'(tt', N, XS)) → mark'(U12'(tt', N, XS))
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U21'(tt', X)) → mark'(U22'(tt', X))
active'(U22'(tt', X)) → mark'(X)
active'(U31'(tt', N)) → mark'(U32'(tt', N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', N, XS)) → mark'(U42'(tt', N, XS))
active'(U42'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U51'(tt', Y)) → mark'(U52'(tt', Y))
active'(U52'(tt', Y)) → mark'(Y)
active'(U61'(tt', N, X, XS)) → mark'(U62'(tt', N, X, XS))
active'(U62'(tt', N, X, XS)) → mark'(U63'(tt', N, X, XS))
active'(U63'(tt', N, X, XS)) → mark'(U64'(splitAt'(N, XS), X))
active'(U64'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U71'(tt', XS)) → mark'(U72'(tt', XS))
active'(U72'(tt', XS)) → mark'(XS)
active'(U81'(tt', N, XS)) → mark'(U82'(tt', N, XS))
active'(U82'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(afterNth'(N, XS)) → mark'(U11'(tt', N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(tt', X))
active'(head'(cons'(N, XS))) → mark'(U31'(tt', N))
active'(natsFrom'(N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(sel'(N, XS)) → mark'(U41'(tt', N, XS))
active'(snd'(pair'(X, Y))) → mark'(U51'(tt', Y))
active'(splitAt'(0', XS)) → mark'(pair'(nil', XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U61'(tt', N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U71'(tt', XS))
active'(take'(N, XS)) → mark'(U81'(tt', N, XS))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U21'(X1, X2)) → U21'(active'(X1), X2)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U31'(X1, X2)) → U31'(active'(X1), X2)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2, X3)) → U41'(active'(X1), X2, X3)
active'(U42'(X1, X2, X3)) → U42'(active'(X1), X2, X3)
active'(head'(X)) → head'(active'(X))
active'(afterNth'(X1, X2)) → afterNth'(active'(X1), X2)
active'(afterNth'(X1, X2)) → afterNth'(X1, active'(X2))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X1, X2)) → U52'(active'(X1), X2)
active'(U61'(X1, X2, X3, X4)) → U61'(active'(X1), X2, X3, X4)
active'(U62'(X1, X2, X3, X4)) → U62'(active'(X1), X2, X3, X4)
active'(U63'(X1, X2, X3, X4)) → U63'(active'(X1), X2, X3, X4)
active'(U64'(X1, X2)) → U64'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U71'(X1, X2)) → U71'(active'(X1), X2)
active'(U72'(X1, X2)) → U72'(active'(X1), X2)
active'(U81'(X1, X2, X3)) → U81'(active'(X1), X2, X3)
active'(U82'(X1, X2, X3)) → U82'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U21'(mark'(X1), X2) → mark'(U21'(X1, X2))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U31'(mark'(X1), X2) → mark'(U31'(X1, X2))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2, X3) → mark'(U41'(X1, X2, X3))
U42'(mark'(X1), X2, X3) → mark'(U42'(X1, X2, X3))
head'(mark'(X)) → mark'(head'(X))
afterNth'(mark'(X1), X2) → mark'(afterNth'(X1, X2))
afterNth'(X1, mark'(X2)) → mark'(afterNth'(X1, X2))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X1), X2) → mark'(U52'(X1, X2))
U61'(mark'(X1), X2, X3, X4) → mark'(U61'(X1, X2, X3, X4))
U62'(mark'(X1), X2, X3, X4) → mark'(U62'(X1, X2, X3, X4))
U63'(mark'(X1), X2, X3, X4) → mark'(U63'(X1, X2, X3, X4))
U64'(mark'(X1), X2) → mark'(U64'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U71'(mark'(X1), X2) → mark'(U71'(X1, X2))
U72'(mark'(X1), X2) → mark'(U72'(X1, X2))
U81'(mark'(X1), X2, X3) → mark'(U81'(X1, X2, X3))
U82'(mark'(X1), X2, X3) → mark'(U82'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2)) → U21'(proper'(X1), proper'(X2))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U31'(X1, X2)) → U31'(proper'(X1), proper'(X2))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2, X3)) → U41'(proper'(X1), proper'(X2), proper'(X3))
proper'(U42'(X1, X2, X3)) → U42'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X1, X2)) → U52'(proper'(X1), proper'(X2))
proper'(U61'(X1, X2, X3, X4)) → U61'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U62'(X1, X2, X3, X4)) → U62'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U63'(X1, X2, X3, X4)) → U63'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U64'(X1, X2)) → U64'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(U71'(X1, X2)) → U71'(proper'(X1), proper'(X2))
proper'(U72'(X1, X2)) → U72'(proper'(X1), proper'(X2))
proper'(U81'(X1, X2, X3)) → U81'(proper'(X1), proper'(X2), proper'(X3))
proper'(U82'(X1, X2, X3)) → U82'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(nil') → ok'(nil')
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U21'(ok'(X1), ok'(X2)) → ok'(U21'(X1, X2))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U31'(ok'(X1), ok'(X2)) → ok'(U31'(X1, X2))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U41'(X1, X2, X3))
U42'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U42'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X1), ok'(X2)) → ok'(U52'(X1, X2))
U61'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U61'(X1, X2, X3, X4))
U62'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U62'(X1, X2, X3, X4))
U63'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U63'(X1, X2, X3, X4))
U64'(ok'(X1), ok'(X2)) → ok'(U64'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U71'(ok'(X1), ok'(X2)) → ok'(U71'(X1, X2))
U72'(ok'(X1), ok'(X2)) → ok'(U72'(X1, X2))
U81'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U81'(X1, X2, X3))
U82'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U82'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(U11'(tt', N, XS)) → mark'(U12'(tt', N, XS))
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U21'(tt', X)) → mark'(U22'(tt', X))
active'(U22'(tt', X)) → mark'(X)
active'(U31'(tt', N)) → mark'(U32'(tt', N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', N, XS)) → mark'(U42'(tt', N, XS))
active'(U42'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U51'(tt', Y)) → mark'(U52'(tt', Y))
active'(U52'(tt', Y)) → mark'(Y)
active'(U61'(tt', N, X, XS)) → mark'(U62'(tt', N, X, XS))
active'(U62'(tt', N, X, XS)) → mark'(U63'(tt', N, X, XS))
active'(U63'(tt', N, X, XS)) → mark'(U64'(splitAt'(N, XS), X))
active'(U64'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U71'(tt', XS)) → mark'(U72'(tt', XS))
active'(U72'(tt', XS)) → mark'(XS)
active'(U81'(tt', N, XS)) → mark'(U82'(tt', N, XS))
active'(U82'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(afterNth'(N, XS)) → mark'(U11'(tt', N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(tt', X))
active'(head'(cons'(N, XS))) → mark'(U31'(tt', N))
active'(natsFrom'(N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(sel'(N, XS)) → mark'(U41'(tt', N, XS))
active'(snd'(pair'(X, Y))) → mark'(U51'(tt', Y))
active'(splitAt'(0', XS)) → mark'(pair'(nil', XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U61'(tt', N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U71'(tt', XS))
active'(take'(N, XS)) → mark'(U81'(tt', N, XS))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U21'(X1, X2)) → U21'(active'(X1), X2)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U31'(X1, X2)) → U31'(active'(X1), X2)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2, X3)) → U41'(active'(X1), X2, X3)
active'(U42'(X1, X2, X3)) → U42'(active'(X1), X2, X3)
active'(head'(X)) → head'(active'(X))
active'(afterNth'(X1, X2)) → afterNth'(active'(X1), X2)
active'(afterNth'(X1, X2)) → afterNth'(X1, active'(X2))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X1, X2)) → U52'(active'(X1), X2)
active'(U61'(X1, X2, X3, X4)) → U61'(active'(X1), X2, X3, X4)
active'(U62'(X1, X2, X3, X4)) → U62'(active'(X1), X2, X3, X4)
active'(U63'(X1, X2, X3, X4)) → U63'(active'(X1), X2, X3, X4)
active'(U64'(X1, X2)) → U64'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U71'(X1, X2)) → U71'(active'(X1), X2)
active'(U72'(X1, X2)) → U72'(active'(X1), X2)
active'(U81'(X1, X2, X3)) → U81'(active'(X1), X2, X3)
active'(U82'(X1, X2, X3)) → U82'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U21'(mark'(X1), X2) → mark'(U21'(X1, X2))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U31'(mark'(X1), X2) → mark'(U31'(X1, X2))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2, X3) → mark'(U41'(X1, X2, X3))
U42'(mark'(X1), X2, X3) → mark'(U42'(X1, X2, X3))
head'(mark'(X)) → mark'(head'(X))
afterNth'(mark'(X1), X2) → mark'(afterNth'(X1, X2))
afterNth'(X1, mark'(X2)) → mark'(afterNth'(X1, X2))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X1), X2) → mark'(U52'(X1, X2))
U61'(mark'(X1), X2, X3, X4) → mark'(U61'(X1, X2, X3, X4))
U62'(mark'(X1), X2, X3, X4) → mark'(U62'(X1, X2, X3, X4))
U63'(mark'(X1), X2, X3, X4) → mark'(U63'(X1, X2, X3, X4))
U64'(mark'(X1), X2) → mark'(U64'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U71'(mark'(X1), X2) → mark'(U71'(X1, X2))
U72'(mark'(X1), X2) → mark'(U72'(X1, X2))
U81'(mark'(X1), X2, X3) → mark'(U81'(X1, X2, X3))
U82'(mark'(X1), X2, X3) → mark'(U82'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2)) → U21'(proper'(X1), proper'(X2))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U31'(X1, X2)) → U31'(proper'(X1), proper'(X2))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2, X3)) → U41'(proper'(X1), proper'(X2), proper'(X3))
proper'(U42'(X1, X2, X3)) → U42'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X1, X2)) → U52'(proper'(X1), proper'(X2))
proper'(U61'(X1, X2, X3, X4)) → U61'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U62'(X1, X2, X3, X4)) → U62'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U63'(X1, X2, X3, X4)) → U63'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U64'(X1, X2)) → U64'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(U71'(X1, X2)) → U71'(proper'(X1), proper'(X2))
proper'(U72'(X1, X2)) → U72'(proper'(X1), proper'(X2))
proper'(U81'(X1, X2, X3)) → U81'(proper'(X1), proper'(X2), proper'(X3))
proper'(U82'(X1, X2, X3)) → U82'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(nil') → ok'(nil')
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U21'(ok'(X1), ok'(X2)) → ok'(U21'(X1, X2))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U31'(ok'(X1), ok'(X2)) → ok'(U31'(X1, X2))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U41'(X1, X2, X3))
U42'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U42'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X1), ok'(X2)) → ok'(U52'(X1, X2))
U61'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U61'(X1, X2, X3, X4))
U62'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U62'(X1, X2, X3, X4))
U63'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U63'(X1, X2, X3, X4))
U64'(ok'(X1), ok'(X2)) → ok'(U64'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U71'(ok'(X1), ok'(X2)) → ok'(U71'(X1, X2))
U72'(ok'(X1), ok'(X2)) → ok'(U72'(X1, X2))
U81'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U81'(X1, X2, X3))
U82'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U82'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U11' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
tt' :: tt':mark':0':nil':ok'
mark' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U12' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
snd' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
splitAt' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U21' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U22' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U31' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U32' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U41' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U42' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
head' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
afterNth' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U51' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U52' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U61' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U62' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U63' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U64' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
pair' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
cons' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U71' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U72' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U81' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
U82' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
fst' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
natsFrom' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
s' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
sel' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
0' :: tt':mark':0':nil':ok'
nil' :: tt':mark':0':nil':ok'
tail' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
take' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
proper' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
ok' :: tt':mark':0':nil':ok' → tt':mark':0':nil':ok'
top' :: tt':mark':0':nil':ok' → top'
_hole_tt':mark':0':nil':ok'1 :: tt':mark':0':nil':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':nil':ok'3 :: Nat → tt':mark':0':nil':ok'

Heuristically decided to analyse the following defined symbols:
active', U12', snd', splitAt', U22', U32', U42', head', afterNth', U52', U62', U63', U64', pair', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
U12' < active'
snd' < active'
splitAt' < active'
U22' < active'
U32' < active'
U42' < active'
head' < active'
afterNth' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
pair' < active'
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
U12' < proper'
snd' < proper'
splitAt' < proper'
U22' < proper'
U32' < proper'
U42' < proper'
head' < proper'
afterNth' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
pair' < proper'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
U12', active', snd', splitAt', U22', U32', U42', head', afterNth', U52', U62', U63', U64', pair', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

Proved the following rewrite lemma:
U12'(_gen_tt':mark':0':nil':ok'3(+(1, _n5)), _gen_tt':mark':0':nil':ok'3(b), _gen_tt':mark':0':nil':ok'3(c)) → _*4, rt ∈ Ω(_n5)

Induction Base:
U12'(_gen_tt':mark':0':nil':ok'3(+(1, 0)), _gen_tt':mark':0':nil':ok'3(b), _gen_tt':mark':0':nil':ok'3(c))

Induction Step:
U12'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n6, 1))), _gen_tt':mark':0':nil':ok'3(_b1018), _gen_tt':mark':0':nil':ok'3(_c1019)) →_R^Ω(1)
mark'(U12'(_gen_tt':mark':0':nil':ok'3(+(1, _$n6)), _gen_tt':mark':0':nil':ok'3(_b1018), _gen_tt':mark':0':nil':ok'3(_c1019))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Lemmas:
U12'(_gen_tt':mark':0':nil':ok'3(+(1, _n5)), _gen_tt':mark':0':nil':ok'3(b), _gen_tt':mark':0':nil':ok'3(c)) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
snd', active', splitAt', U22', U32', U42', head', afterNth', U52', U62', U63', U64', pair', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
snd' < active'
splitAt' < active'
U22' < active'
U32' < active'
U42' < active'
head' < active'
afterNth' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
pair' < active'
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
snd' < proper'
splitAt' < proper'
U22' < proper'
U32' < proper'
U42' < proper'
head' < proper'
afterNth' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
pair' < proper'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Proved the following rewrite lemma:
snd'(_gen_tt':mark':0':nil':ok'3(+(1, _n13068))) → _*4, rt ∈ Ω(_n13068)

Induction Base:
snd'(_gen_tt':mark':0':nil':ok'3(+(1, 0)))

Induction Step:
snd'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n13069, 1)))) →_R^Ω(1)
mark'(snd'(_gen_tt':mark':0':nil':ok'3(+(1, _$n13069)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
splitAt', active', U22', U32', U42', head', afterNth', U52', U62', U63', U64', pair', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
splitAt' < active'
U22' < active'
U32' < active'
U42' < active'
head' < active'
afterNth' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
pair' < active'
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
splitAt' < proper'
U22' < proper'
U32' < proper'
U42' < proper'
head' < proper'
afterNth' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
pair' < proper'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Proved the following rewrite lemma:
splitAt'(_gen_tt':mark':0':nil':ok'3(+(1, _n19667)), _gen_tt':mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n19667)

Induction Base:
splitAt'(_gen_tt':mark':0':nil':ok'3(+(1, 0)), _gen_tt':mark':0':nil':ok'3(b))

Induction Step:
splitAt'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n19668, 1))), _gen_tt':mark':0':nil':ok'3(_b21136)) →_R^Ω(1)
mark'(splitAt'(_gen_tt':mark':0':nil':ok'3(+(1, _$n19668)), _gen_tt':mark':0':nil':ok'3(_b21136))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
U22', active', U32', U42', head', afterNth', U52', U62', U63', U64', pair', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
U22' < active'
U32' < active'
U42' < active'
head' < active'
afterNth' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
pair' < active'
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
U22' < proper'
U32' < proper'
U42' < proper'
head' < proper'
afterNth' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
pair' < proper'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Proved the following rewrite lemma:
U22'(_gen_tt':mark':0':nil':ok'3(+(1, _n30267)), _gen_tt':mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n30267)

Induction Base:
U22'(_gen_tt':mark':0':nil':ok'3(+(1, 0)), _gen_tt':mark':0':nil':ok'3(b))

Induction Step:
U22'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n30268, 1))), _gen_tt':mark':0':nil':ok'3(_b31844)) →_R^Ω(1)
mark'(U22'(_gen_tt':mark':0':nil':ok'3(+(1, _$n30268)), _gen_tt':mark':0':nil':ok'3(_b31844))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
U32', active', U42', head', afterNth', U52', U62', U63', U64', pair', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
U32' < active'
U42' < active'
head' < active'
afterNth' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
pair' < active'
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
U32' < proper'
U42' < proper'
head' < proper'
afterNth' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
pair' < proper'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Proved the following rewrite lemma:
U32'(_gen_tt':mark':0':nil':ok'3(+(1, _n41016)), _gen_tt':mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n41016)

Induction Base:
U32'(_gen_tt':mark':0':nil':ok'3(+(1, 0)), _gen_tt':mark':0':nil':ok'3(b))

Induction Step:
U32'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n41017, 1))), _gen_tt':mark':0':nil':ok'3(_b42917)) →_R^Ω(1)
mark'(U32'(_gen_tt':mark':0':nil':ok'3(+(1, _$n41017)), _gen_tt':mark':0':nil':ok'3(_b42917))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
U42', active', head', afterNth', U52', U62', U63', U64', pair', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
U42' < active'
head' < active'
afterNth' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
pair' < active'
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
U42' < proper'
head' < proper'
afterNth' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
pair' < proper'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Proved the following rewrite lemma:
U42'(_gen_tt':mark':0':nil':ok'3(+(1, _n52133)), _gen_tt':mark':0':nil':ok'3(b), _gen_tt':mark':0':nil':ok'3(c)) → _*4, rt ∈ Ω(_n52133)

Induction Base:
U42'(_gen_tt':mark':0':nil':ok'3(+(1, 0)), _gen_tt':mark':0':nil':ok'3(b), _gen_tt':mark':0':nil':ok'3(c))

Induction Step:
U42'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n52134, 1))), _gen_tt':mark':0':nil':ok'3(_b55576), _gen_tt':mark':0':nil':ok'3(_c55577)) →_R^Ω(1)
mark'(U42'(_gen_tt':mark':0':nil':ok'3(+(1, _$n52134)), _gen_tt':mark':0':nil':ok'3(_b55576), _gen_tt':mark':0':nil':ok'3(_c55577))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
head', active', afterNth', U52', U62', U63', U64', pair', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
head' < active'
afterNth' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
pair' < active'
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
head' < proper'
afterNth' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
pair' < proper'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Proved the following rewrite lemma:
head'(_gen_tt':mark':0':nil':ok'3(+(1, _n67921))) → _*4, rt ∈ Ω(_n67921)

Induction Base:
head'(_gen_tt':mark':0':nil':ok'3(+(1, 0)))

Induction Step:
head'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n67922, 1)))) →_R^Ω(1)
mark'(head'(_gen_tt':mark':0':nil':ok'3(+(1, _$n67922)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
afterNth', active', U52', U62', U63', U64', pair', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
afterNth' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
pair' < active'
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
afterNth' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
pair' < proper'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Proved the following rewrite lemma:
afterNth'(_gen_tt':mark':0':nil':ok'3(+(1, _n75475)), _gen_tt':mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n75475)

Induction Base:
afterNth'(_gen_tt':mark':0':nil':ok'3(+(1, 0)), _gen_tt':mark':0':nil':ok'3(b))

Induction Step:
afterNth'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n75476, 1))), _gen_tt':mark':0':nil':ok'3(_b78564)) →_R^Ω(1)
mark'(afterNth'(_gen_tt':mark':0':nil':ok'3(+(1, _$n75476)), _gen_tt':mark':0':nil':ok'3(_b78564))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
U52', active', U62', U63', U64', pair', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
U52' < active'
U62' < active'
U63' < active'
U64' < active'
pair' < active'
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
pair' < proper'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Proved the following rewrite lemma:
U52'(_gen_tt':mark':0':nil':ok'3(+(1, _n87915)), _gen_tt':mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n87915)

Induction Base:
U52'(_gen_tt':mark':0':nil':ok'3(+(1, 0)), _gen_tt':mark':0':nil':ok'3(b))

Induction Step:
U52'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n87916, 1))), _gen_tt':mark':0':nil':ok'3(_b91112)) →_R^Ω(1)
mark'(U52'(_gen_tt':mark':0':nil':ok'3(+(1, _$n87916)), _gen_tt':mark':0':nil':ok'3(_b91112))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
U62', active', U63', U64', pair', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
U62' < active'
U63' < active'
U64' < active'
pair' < active'
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
U62' < proper'
U63' < proper'
U64' < proper'
pair' < proper'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Proved the following rewrite lemma:
U62'(_gen_tt':mark':0':nil':ok'3(+(1, _n100504)), _gen_tt':mark':0':nil':ok'3(b), _gen_tt':mark':0':nil':ok'3(c), _gen_tt':mark':0':nil':ok'3(d)) → _*4, rt ∈ Ω(_n100504)

Induction Base:
U62'(_gen_tt':mark':0':nil':ok'3(+(1, 0)), _gen_tt':mark':0':nil':ok'3(b), _gen_tt':mark':0':nil':ok'3(c), _gen_tt':mark':0':nil':ok'3(d))

Induction Step:
U62'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n100505, 1))), _gen_tt':mark':0':nil':ok'3(_b107973), _gen_tt':mark':0':nil':ok'3(_c107974), _gen_tt':mark':0':nil':ok'3(_d107975)) →_R^Ω(1)
mark'(U62'(_gen_tt':mark':0':nil':ok'3(+(1, _$n100505)), _gen_tt':mark':0':nil':ok'3(_b107973), _gen_tt':mark':0':nil':ok'3(_c107974), _gen_tt':mark':0':nil':ok'3(_d107975))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Lemmas:
U12'(_gen_tt':mark':0':nil':ok'3(+(1, _n5)), _gen_tt':mark':0':nil':ok'3(b), _gen_tt':mark':0':nil':ok'3(c)) → _*4, rt ∈ Ω(_n5)
snd'(_gen_tt':mark':0':nil':ok'3(+(1, _n13068))) → _*4, rt ∈ Ω(_n13068)
splitAt'(_gen_tt':mark':0':nil':ok'3(+(1, _n19667)), _gen_tt':mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n19667)
U22'(_gen_tt':mark':0':nil':ok'3(+(1, _n30267)), _gen_tt':mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n30267)
U32'(_gen_tt':mark':0':nil':ok'3(+(1, _n41016)), _gen_tt':mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n41016)
U42'(_gen_tt':mark':0':nil':ok'3(+(1, _n52133)), _gen_tt':mark':0':nil':ok'3(b), _gen_tt':mark':0':nil':ok'3(c)) → _*4, rt ∈ Ω(_n52133)
head'(_gen_tt':mark':0':nil':ok'3(+(1, _n67921))) → _*4, rt ∈ Ω(_n67921)
afterNth'(_gen_tt':mark':0':nil':ok'3(+(1, _n75475)), _gen_tt':mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n75475)
U52'(_gen_tt':mark':0':nil':ok'3(+(1, _n87915)), _gen_tt':mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n87915)
U62'(_gen_tt':mark':0':nil':ok'3(+(1, _n100504)), _gen_tt':mark':0':nil':ok'3(b), _gen_tt':mark':0':nil':ok'3(c), _gen_tt':mark':0':nil':ok'3(d)) → _*4, rt ∈ Ω(_n100504)

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
U63', active', U64', pair', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
U63' < active'
U64' < active'
pair' < active'
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
U63' < proper'
U64' < proper'
pair' < proper'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Proved the following rewrite lemma:
U63'(_gen_tt':mark':0':nil':ok'3(+(1, _n123711)), _gen_tt':mark':0':nil':ok'3(b), _gen_tt':mark':0':nil':ok'3(c), _gen_tt':mark':0':nil':ok'3(d)) → _*4, rt ∈ Ω(_n123711)

Induction Base:
U63'(_gen_tt':mark':0':nil':ok'3(+(1, 0)), _gen_tt':mark':0':nil':ok'3(b), _gen_tt':mark':0':nil':ok'3(c), _gen_tt':mark':0':nil':ok'3(d))

Induction Step:
U63'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n123712, 1))), _gen_tt':mark':0':nil':ok'3(_b132260), _gen_tt':mark':0':nil':ok'3(_c132261), _gen_tt':mark':0':nil':ok'3(_d132262)) →_R^Ω(1)
mark'(U63'(_gen_tt':mark':0':nil':ok'3(+(1, _$n123712)), _gen_tt':mark':0':nil':ok'3(_b132260), _gen_tt':mark':0':nil':ok'3(_c132261), _gen_tt':mark':0':nil':ok'3(_d132262))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
U64', active', pair', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
U64' < active'
pair' < active'
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
U64' < proper'
pair' < proper'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Proved the following rewrite lemma:
U64'(_gen_tt':mark':0':nil':ok'3(+(1, _n148134)), _gen_tt':mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n148134)

Induction Base:
U64'(_gen_tt':mark':0':nil':ok'3(+(1, 0)), _gen_tt':mark':0':nil':ok'3(b))

Induction Step:
U64'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n148135, 1))), _gen_tt':mark':0':nil':ok'3(_b152735)) →_R^Ω(1)
mark'(U64'(_gen_tt':mark':0':nil':ok'3(+(1, _$n148135)), _gen_tt':mark':0':nil':ok'3(_b152735))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
pair', active', cons', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
pair' < active'
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
pair' < proper'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Proved the following rewrite lemma:
pair'(_gen_tt':mark':0':nil':ok'3(+(1, _n162335)), _gen_tt':mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n162335)

Induction Base:
pair'(_gen_tt':mark':0':nil':ok'3(+(1, 0)), _gen_tt':mark':0':nil':ok'3(b))

Induction Step:
pair'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n162336, 1))), _gen_tt':mark':0':nil':ok'3(_b167476)) →_R^Ω(1)
mark'(pair'(_gen_tt':mark':0':nil':ok'3(+(1, _$n162336)), _gen_tt':mark':0':nil':ok'3(_b167476))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':nil':ok'3(0) ⇔ tt'
_gen_tt':mark':0':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':nil':ok'3(x))

The following defined symbols remain to be analysed:
cons', active', U72', U82', fst', U11', U21', U31', natsFrom', s', U41', U51', U61', U71', U81', sel', tail', take', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
U72' < active'
U82' < active'
fst' < active'
U11' < active'
U21' < active'
U31' < active'
natsFrom' < active'
s' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
sel' < active'
tail' < active'
take' < active'
active' < top'
cons' < proper'
U72' < proper'
U82' < proper'
fst' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
natsFrom' < proper'
s' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
sel' < proper'
tail' < proper'
take' < proper'
proper' < top'

Proved the following rewrite lemma:
cons'(_gen_tt':mark':0':nil':ok'3(+(1, _n177123)), _gen_tt':mark':0':nil':ok'3(b)) → _*4, rt ∈ Ω(_n177123)

Induction Base:
cons'(_gen_tt':mark':0':nil':ok'3(+(1, 0)), _gen_tt':mark':0':nil':ok'3(b))

Induction Step:
cons'(_gen_tt':mark':0':nil':ok'3(+(1, +(_$n177124, 1))), _gen_tt':mark':0':nil':ok'3(_b182372)) →_R^Ω(1)
mark'(cons'(_gen_tt':mark':0':nil':ok'3(+(1, _$n177124)), _gen_tt':mark':0':nil':ok'3(_b182372))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).