active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(U172(X1, X2, X3)) → U172(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(U181(X1, X2)) → U181(active(X1), X2)
active(U182(X1, X2)) → U182(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
U172(mark(X1), X2, X3) → mark(U172(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))
U181(mark(X1), X2) → mark(U181(X1, X2))
U182(mark(X1), X2) → mark(U182(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(U172(X1, X2, X3)) → U172(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U182(X1, X2)) → U182(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2)) → ok(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U182(ok(X1), ok(X2)) → ok(U182(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Infered types.
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Heuristically decided to analyse the following defined symbols:
active', U102', isLNat', U12', snd', splitAt', U132', U142', U152', cons', natsFrom', s', U172', head', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U102' < active'
isLNat' < active'
U12' < active'
snd' < active'
splitAt' < active'
U132' < active'
U142' < active'
U152' < active'
cons' < active'
natsFrom' < active'
s' < active'
U172' < active'
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U102' < proper'
isLNat' < proper'
U12' < proper'
snd' < proper'
splitAt' < proper'
U132' < proper'
U142' < proper'
U152' < proper'
cons' < proper'
natsFrom' < proper'
s' < proper'
U172' < proper'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U102', active', isLNat', U12', snd', splitAt', U132', U142', U152', cons', natsFrom', s', U172', head', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U102' < active'
isLNat' < active'
U12' < active'
snd' < active'
splitAt' < active'
U132' < active'
U142' < active'
U152' < active'
cons' < active'
natsFrom' < active'
s' < active'
U172' < active'
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U102' < proper'
isLNat' < proper'
U12' < proper'
snd' < proper'
splitAt' < proper'
U132' < proper'
U142' < proper'
U152' < proper'
cons' < proper'
natsFrom' < proper'
s' < proper'
U172' < proper'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
Induction Base:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n6, 1)))) →RΩ(1)
mark'(U102'(_gen_tt':mark':nil':0':ok'3(+(1, _$n6)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
isLNat', active', U12', snd', splitAt', U132', U142', U152', cons', natsFrom', s', U172', head', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
isLNat' < active'
U12' < active'
snd' < active'
splitAt' < active'
U132' < active'
U142' < active'
U152' < active'
cons' < active'
natsFrom' < active'
s' < active'
U172' < active'
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
isLNat' < proper'
U12' < proper'
snd' < proper'
splitAt' < proper'
U132' < proper'
U142' < proper'
U152' < proper'
cons' < proper'
natsFrom' < proper'
s' < proper'
U172' < proper'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Could not prove a rewrite lemma for the defined symbol isLNat'.
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U12', active', snd', splitAt', U132', U142', U152', cons', natsFrom', s', U172', head', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U12' < active'
snd' < active'
splitAt' < active'
U132' < active'
U142' < active'
U152' < active'
cons' < active'
natsFrom' < active'
s' < active'
U172' < active'
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U12' < proper'
snd' < proper'
splitAt' < proper'
U132' < proper'
U142' < proper'
U152' < proper'
cons' < proper'
natsFrom' < proper'
s' < proper'
U172' < proper'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
Induction Base:
U12'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c))
Induction Step:
U12'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n9389, 1))), _gen_tt':mark':nil':0':ok'3(_b10725), _gen_tt':mark':nil':0':ok'3(_c10726)) →RΩ(1)
mark'(U12'(_gen_tt':mark':nil':0':ok'3(+(1, _$n9389)), _gen_tt':mark':nil':0':ok'3(_b10725), _gen_tt':mark':nil':0':ok'3(_c10726))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
snd', active', splitAt', U132', U142', U152', cons', natsFrom', s', U172', head', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
snd' < active'
splitAt' < active'
U132' < active'
U142' < active'
U152' < active'
cons' < active'
natsFrom' < active'
s' < active'
U172' < active'
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
snd' < proper'
splitAt' < proper'
U132' < proper'
U142' < proper'
U152' < proper'
cons' < proper'
natsFrom' < proper'
s' < proper'
U172' < proper'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
Induction Base:
snd'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
snd'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n29128, 1)))) →RΩ(1)
mark'(snd'(_gen_tt':mark':nil':0':ok'3(+(1, _$n29128)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
splitAt', active', U132', U142', U152', cons', natsFrom', s', U172', head', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
splitAt' < active'
U132' < active'
U142' < active'
U152' < active'
cons' < active'
natsFrom' < active'
s' < active'
U172' < active'
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
splitAt' < proper'
U132' < proper'
U142' < proper'
U152' < proper'
cons' < proper'
natsFrom' < proper'
s' < proper'
U172' < proper'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
Induction Base:
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n38881, 1))), _gen_tt':mark':nil':0':ok'3(_b40565)) →RΩ(1)
mark'(splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _$n38881)), _gen_tt':mark':nil':0':ok'3(_b40565))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U132', active', U142', U152', cons', natsFrom', s', U172', head', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U132' < active'
U142' < active'
U152' < active'
cons' < active'
natsFrom' < active'
s' < active'
U172' < active'
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U132' < proper'
U142' < proper'
U152' < proper'
cons' < proper'
natsFrom' < proper'
s' < proper'
U172' < proper'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
Induction Base:
U132'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
U132'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n54396, 1)))) →RΩ(1)
mark'(U132'(_gen_tt':mark':nil':0':ok'3(+(1, _$n54396)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U142', active', U152', cons', natsFrom', s', U172', head', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U142' < active'
U152' < active'
cons' < active'
natsFrom' < active'
s' < active'
U172' < active'
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U142' < proper'
U152' < proper'
cons' < proper'
natsFrom' < proper'
s' < proper'
U172' < proper'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
Induction Base:
U142'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
U142'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n64464, 1)))) →RΩ(1)
mark'(U142'(_gen_tt':mark':nil':0':ok'3(+(1, _$n64464)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U152', active', cons', natsFrom', s', U172', head', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U152' < active'
cons' < active'
natsFrom' < active'
s' < active'
U172' < active'
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U152' < proper'
cons' < proper'
natsFrom' < proper'
s' < proper'
U172' < proper'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
Induction Base:
U152'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
U152'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n74656, 1)))) →RΩ(1)
mark'(U152'(_gen_tt':mark':nil':0':ok'3(+(1, _$n74656)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
cons', active', natsFrom', s', U172', head', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
natsFrom' < active'
s' < active'
U172' < active'
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
cons' < proper'
natsFrom' < proper'
s' < proper'
U172' < proper'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
Induction Base:
cons'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
cons'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n84972, 1))), _gen_tt':mark':nil':0':ok'3(_b87412)) →RΩ(1)
mark'(cons'(_gen_tt':mark':nil':0':ok'3(+(1, _$n84972)), _gen_tt':mark':nil':0':ok'3(_b87412))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
natsFrom', active', s', U172', head', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
natsFrom' < active'
s' < active'
U172' < active'
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
natsFrom' < proper'
s' < proper'
U172' < proper'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
Induction Base:
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n101359, 1)))) →RΩ(1)
mark'(natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _$n101359)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
s', active', U172', head', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
s' < active'
U172' < active'
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
s' < proper'
U172' < proper'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
Induction Base:
s'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
s'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n111990, 1)))) →RΩ(1)
mark'(s'(_gen_tt':mark':nil':0':ok'3(+(1, _$n111990)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U172', active', head', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U172' < active'
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U172' < proper'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
Induction Base:
U172'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c))
Induction Step:
U172'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n122745, 1))), _gen_tt':mark':nil':0':ok'3(_b127645), _gen_tt':mark':nil':0':ok'3(_c127646)) →RΩ(1)
mark'(U172'(_gen_tt':mark':nil':0':ok'3(+(1, _$n122745)), _gen_tt':mark':nil':0':ok'3(_b127645), _gen_tt':mark':nil':0':ok'3(_c127646))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
head', active', afterNth', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
head' < active'
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
head' < proper'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
Induction Base:
head'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
head'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n146454, 1)))) →RΩ(1)
mark'(head'(_gen_tt':mark':nil':0':ok'3(+(1, _$n146454)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
afterNth', active', U182', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
afterNth' < active'
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
afterNth' < proper'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
Induction Base:
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n157591, 1))), _gen_tt':mark':nil':0':ok'3(_b161651)) →RΩ(1)
mark'(afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _$n157591)), _gen_tt':mark':nil':0':ok'3(_b161651))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U182', active', pair', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U182' < active'
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U182' < proper'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
Induction Base:
U182'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U182'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n175783, 1))), _gen_tt':mark':nil':0':ok'3(_b179951)) →RΩ(1)
mark'(U182'(_gen_tt':mark':nil':0':ok'3(+(1, _$n175783)), _gen_tt':mark':nil':0':ok'3(_b179951))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
pair', active', U202', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
pair' < active'
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
pair' < proper'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
Induction Base:
pair'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
pair'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n194124, 1))), _gen_tt':mark':nil':0':ok'3(_b198832)) →RΩ(1)
mark'(pair'(_gen_tt':mark':nil':0':ok'3(+(1, _$n194124)), _gen_tt':mark':nil':0':ok'3(_b198832))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U202', active', isNatural', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U202' < active'
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U202' < proper'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
Induction Base:
U202'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d))
Induction Step:
U202'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n213052, 1))), _gen_tt':mark':nil':0':ok'3(_b223112), _gen_tt':mark':nil':0':ok'3(_c223113), _gen_tt':mark':nil':0':ok'3(_d223114)) →RΩ(1)
mark'(U202'(_gen_tt':mark':nil':0':ok'3(+(1, _$n213052)), _gen_tt':mark':nil':0':ok'3(_b223112), _gen_tt':mark':nil':0':ok'3(_c223113), _gen_tt':mark':nil':0':ok'3(_d223114))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
isNatural', active', U203', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
isNatural' < active'
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
isNatural' < proper'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Could not prove a rewrite lemma for the defined symbol isNatural'.
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U203', active', U204', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U203' < active'
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U203' < proper'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
Induction Base:
U203'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d))
Induction Step:
U203'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n247221, 1))), _gen_tt':mark':nil':0':ok'3(_b258361), _gen_tt':mark':nil':0':ok'3(_c258362), _gen_tt':mark':nil':0':ok'3(_d258363)) →RΩ(1)
mark'(U203'(_gen_tt':mark':nil':0':ok'3(+(1, _$n247221)), _gen_tt':mark':nil':0':ok'3(_b258361), _gen_tt':mark':nil':0':ok'3(_c258362), _gen_tt':mark':nil':0':ok'3(_d258363))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U204', active', U22', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U204' < active'
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U204' < proper'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
Induction Base:
U204'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U204'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n282456, 1))), _gen_tt':mark':nil':0':ok'3(_b288352)) →RΩ(1)
mark'(U204'(_gen_tt':mark':nil':0':ok'3(+(1, _$n282456)), _gen_tt':mark':nil':0':ok'3(_b288352))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U22', active', U212', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U22' < active'
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U22' < proper'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
Induction Base:
U22'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U22'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n302777, 1))), _gen_tt':mark':nil':0':ok'3(_b308997)) →RΩ(1)
mark'(U22'(_gen_tt':mark':nil':0':ok'3(+(1, _$n302777)), _gen_tt':mark':nil':0':ok'3(_b308997))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U212', active', U222', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U212' < active'
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U212' < proper'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
Induction Base:
U212'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U212'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n323466, 1))), _gen_tt':mark':nil':0':ok'3(_b330010)) →RΩ(1)
mark'(U212'(_gen_tt':mark':nil':0':ok'3(+(1, _$n323466)), _gen_tt':mark':nil':0':ok'3(_b330010))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U222', active', fst', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U222' < active'
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U222' < proper'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
Induction Base:
U222'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c))
Induction Step:
U222'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n344523, 1))), _gen_tt':mark':nil':0':ok'3(_b354931), _gen_tt':mark':nil':0':ok'3(_c354932)) →RΩ(1)
mark'(U222'(_gen_tt':mark':nil':0':ok'3(+(1, _$n344523)), _gen_tt':mark':nil':0':ok'3(_b354931), _gen_tt':mark':nil':0':ok'3(_c354932))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
fst', active', U32', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
fst' < active'
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
fst' < proper'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
Induction Base:
fst'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
fst'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n374430, 1)))) →RΩ(1)
mark'(fst'(_gen_tt':mark':nil':0':ok'3(+(1, _$n374430)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U32', active', U42', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U32' < active'
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U32' < proper'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
Induction Base:
U32'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U32'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n387745, 1))), _gen_tt':mark':nil':0':ok'3(_b395261)) →RΩ(1)
mark'(U32'(_gen_tt':mark':nil':0':ok'3(+(1, _$n387745)), _gen_tt':mark':nil':0':ok'3(_b395261))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U42', active', U52', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U42' < active'
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U42' < proper'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
Induction Base:
U42'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
U42'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n409906, 1)))) →RΩ(1)
mark'(U42'(_gen_tt':mark':nil':0':ok'3(+(1, _$n409906)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U52', active', U11', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U52' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U52' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
Induction Base:
U52'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
U52'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n423536, 1)))) →RΩ(1)
mark'(U52'(_gen_tt':mark':nil':0':ok'3(+(1, _$n423536)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U11', active', U21', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
Induction Base:
U11'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c))
Induction Step:
U11'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n437290, 1))), _gen_tt':mark':nil':0':ok'3(_b449804), _gen_tt':mark':nil':0':ok'3(_c449805)) →RΩ(1)
mark'(U11'(_gen_tt':mark':nil':0':ok'3(+(1, _$n437290)), _gen_tt':mark':nil':0':ok'3(_b449804), _gen_tt':mark':nil':0':ok'3(_c449805))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U21', active', U31', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
Induction Base:
U21'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c))
Induction Step:
U21'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n469548, 1))), _gen_tt':mark':nil':0':ok'3(_b482710), _gen_tt':mark':nil':0':ok'3(_c482711)) →RΩ(1)
mark'(U21'(_gen_tt':mark':nil':0':ok'3(+(1, _$n469548)), _gen_tt':mark':nil':0':ok'3(_b482710), _gen_tt':mark':nil':0':ok'3(_c482711))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U31', active', U41', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U31' < active'
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
Induction Base:
U31'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c))
Induction Step:
U31'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n502538, 1))), _gen_tt':mark':nil':0':ok'3(_b516348), _gen_tt':mark':nil':0':ok'3(_c516349)) →RΩ(1)
mark'(U31'(_gen_tt':mark':nil':0':ok'3(+(1, _$n502538)), _gen_tt':mark':nil':0':ok'3(_b516348), _gen_tt':mark':nil':0':ok'3(_c516349))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U41', active', U51', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U41' < active'
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U41' < proper'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
Induction Base:
U41'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U41'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n536260, 1))), _gen_tt':mark':nil':0':ok'3(_b545828)) →RΩ(1)
mark'(U41'(_gen_tt':mark':nil':0':ok'3(+(1, _$n536260)), _gen_tt':mark':nil':0':ok'3(_b545828))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U51', active', U61', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U51' < active'
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U51' < proper'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
Induction Base:
U51'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U51'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n560756, 1))), _gen_tt':mark':nil':0':ok'3(_b570648)) →RΩ(1)
mark'(U51'(_gen_tt':mark':nil':0':ok'3(+(1, _$n560756)), _gen_tt':mark':nil':0':ok'3(_b570648))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U61', active', isPLNat', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U61' < active'
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U61' < proper'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
Induction Base:
U61'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
U61'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n585620, 1)))) →RΩ(1)
mark'(U61'(_gen_tt':mark':nil':0':ok'3(+(1, _$n585620)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
isPLNat', active', U71', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
isPLNat' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
isPLNat' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Could not prove a rewrite lemma for the defined symbol isPLNat'.
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U71', active', U81', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U71' < active'
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
Induction Base:
U71'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
U71'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n600962, 1)))) →RΩ(1)
mark'(U71'(_gen_tt':mark':nil':0':ok'3(+(1, _$n600962)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U81', active', U91', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U81' < active'
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U81' < proper'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
Induction Base:
U81'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
U81'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n616120, 1)))) →RΩ(1)
mark'(U81'(_gen_tt':mark':nil':0':ok'3(+(1, _$n616120)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U91', active', U101', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U91' < active'
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U91' < proper'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
Induction Base:
U91'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
U91'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n631402, 1)))) →RΩ(1)
mark'(U91'(_gen_tt':mark':nil':0':ok'3(+(1, _$n631402)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U101', active', U111', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U101' < active'
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U101' < proper'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
Induction Base:
U101'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U101'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n646808, 1))), _gen_tt':mark':nil':0':ok'3(_b657888)) →RΩ(1)
mark'(U101'(_gen_tt':mark':nil':0':ok'3(+(1, _$n646808)), _gen_tt':mark':nil':0':ok'3(_b657888))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U111', active', U121', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U111' < active'
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U111' < proper'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
Induction Base:
U111'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
U111'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n673004, 1)))) →RΩ(1)
mark'(U111'(_gen_tt':mark':nil':0':ok'3(+(1, _$n673004)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U121', active', U131', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U121' < active'
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U121' < proper'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
Induction Base:
U121'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
U121'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n688725, 1)))) →RΩ(1)
mark'(U121'(_gen_tt':mark':nil':0':ok'3(+(1, _$n688725)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U131', active', U141', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U131' < active'
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U131' < proper'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
Induction Base:
U131'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U131'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n704570, 1))), _gen_tt':mark':nil':0':ok'3(_b716406)) →RΩ(1)
mark'(U131'(_gen_tt':mark':nil':0':ok'3(+(1, _$n704570)), _gen_tt':mark':nil':0':ok'3(_b716406))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U141', active', U151', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U141' < active'
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U141' < proper'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U141'(_gen_tt':mark':nil':0':ok'3(+(1, _n731615)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n731615)
Induction Base:
U141'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U141'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n731616, 1))), _gen_tt':mark':nil':0':ok'3(_b743776)) →RΩ(1)
mark'(U141'(_gen_tt':mark':nil':0':ok'3(+(1, _$n731616)), _gen_tt':mark':nil':0':ok'3(_b743776))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
U141'(_gen_tt':mark':nil':0':ok'3(+(1, _n731615)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n731615)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U151', active', U161', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U151' < active'
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U151' < proper'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U151'(_gen_tt':mark':nil':0':ok'3(+(1, _n759029)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n759029)
Induction Base:
U151'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U151'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n759030, 1))), _gen_tt':mark':nil':0':ok'3(_b771514)) →RΩ(1)
mark'(U151'(_gen_tt':mark':nil':0':ok'3(+(1, _$n759030)), _gen_tt':mark':nil':0':ok'3(_b771514))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
U141'(_gen_tt':mark':nil':0':ok'3(+(1, _n731615)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n731615)
U151'(_gen_tt':mark':nil':0':ok'3(+(1, _n759029)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n759029)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U161', active', U171', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U161' < active'
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U161' < proper'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U161'(_gen_tt':mark':nil':0':ok'3(+(1, _n786811)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n786811)
Induction Base:
U161'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U161'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n786812, 1))), _gen_tt':mark':nil':0':ok'3(_b799620)) →RΩ(1)
mark'(U161'(_gen_tt':mark':nil':0':ok'3(+(1, _$n786812)), _gen_tt':mark':nil':0':ok'3(_b799620))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
U141'(_gen_tt':mark':nil':0':ok'3(+(1, _n731615)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n731615)
U151'(_gen_tt':mark':nil':0':ok'3(+(1, _n759029)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n759029)
U161'(_gen_tt':mark':nil':0':ok'3(+(1, _n786811)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n786811)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U171', active', U181', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U171' < active'
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U171' < proper'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U171'(_gen_tt':mark':nil':0':ok'3(+(1, _n814961)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n814961)
Induction Base:
U171'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c))
Induction Step:
U171'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n814962, 1))), _gen_tt':mark':nil':0':ok'3(_b834766), _gen_tt':mark':nil':0':ok'3(_c834767)) →RΩ(1)
mark'(U171'(_gen_tt':mark':nil':0':ok'3(+(1, _$n814962)), _gen_tt':mark':nil':0':ok'3(_b834766), _gen_tt':mark':nil':0':ok'3(_c834767))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
U141'(_gen_tt':mark':nil':0':ok'3(+(1, _n731615)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n731615)
U151'(_gen_tt':mark':nil':0':ok'3(+(1, _n759029)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n759029)
U161'(_gen_tt':mark':nil':0':ok'3(+(1, _n786811)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n786811)
U171'(_gen_tt':mark':nil':0':ok'3(+(1, _n814961)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n814961)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U181', active', U191', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U181' < active'
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U181' < proper'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U181'(_gen_tt':mark':nil':0':ok'3(+(1, _n855378)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n855378)
Induction Base:
U181'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U181'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n855379, 1))), _gen_tt':mark':nil':0':ok'3(_b868943)) →RΩ(1)
mark'(U181'(_gen_tt':mark':nil':0':ok'3(+(1, _$n855379)), _gen_tt':mark':nil':0':ok'3(_b868943))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
U141'(_gen_tt':mark':nil':0':ok'3(+(1, _n731615)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n731615)
U151'(_gen_tt':mark':nil':0':ok'3(+(1, _n759029)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n759029)
U161'(_gen_tt':mark':nil':0':ok'3(+(1, _n786811)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n786811)
U171'(_gen_tt':mark':nil':0':ok'3(+(1, _n814961)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n814961)
U181'(_gen_tt':mark':nil':0':ok'3(+(1, _n855378)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n855378)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U191', active', U201', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U191' < active'
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U191' < proper'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U191'(_gen_tt':mark':nil':0':ok'3(+(1, _n884391)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n884391)
Induction Base:
U191'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U191'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n884392, 1))), _gen_tt':mark':nil':0':ok'3(_b898280)) →RΩ(1)
mark'(U191'(_gen_tt':mark':nil':0':ok'3(+(1, _$n884392)), _gen_tt':mark':nil':0':ok'3(_b898280))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
U141'(_gen_tt':mark':nil':0':ok'3(+(1, _n731615)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n731615)
U151'(_gen_tt':mark':nil':0':ok'3(+(1, _n759029)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n759029)
U161'(_gen_tt':mark':nil':0':ok'3(+(1, _n786811)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n786811)
U171'(_gen_tt':mark':nil':0':ok'3(+(1, _n814961)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n814961)
U181'(_gen_tt':mark':nil':0':ok'3(+(1, _n855378)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n855378)
U191'(_gen_tt':mark':nil':0':ok'3(+(1, _n884391)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n884391)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U201', active', U211', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U201' < active'
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U201' < proper'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U201'(_gen_tt':mark':nil':0':ok'3(+(1, _n913772)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n913772)
Induction Base:
U201'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d))
Induction Step:
U201'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n913773, 1))), _gen_tt':mark':nil':0':ok'3(_b942625), _gen_tt':mark':nil':0':ok'3(_c942626), _gen_tt':mark':nil':0':ok'3(_d942627)) →RΩ(1)
mark'(U201'(_gen_tt':mark':nil':0':ok'3(+(1, _$n913773)), _gen_tt':mark':nil':0':ok'3(_b942625), _gen_tt':mark':nil':0':ok'3(_c942626), _gen_tt':mark':nil':0':ok'3(_d942627))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
U141'(_gen_tt':mark':nil':0':ok'3(+(1, _n731615)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n731615)
U151'(_gen_tt':mark':nil':0':ok'3(+(1, _n759029)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n759029)
U161'(_gen_tt':mark':nil':0':ok'3(+(1, _n786811)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n786811)
U171'(_gen_tt':mark':nil':0':ok'3(+(1, _n814961)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n814961)
U181'(_gen_tt':mark':nil':0':ok'3(+(1, _n855378)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n855378)
U191'(_gen_tt':mark':nil':0':ok'3(+(1, _n884391)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n884391)
U201'(_gen_tt':mark':nil':0':ok'3(+(1, _n913772)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n913772)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U211', active', U221', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U211' < active'
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U211' < proper'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U211'(_gen_tt':mark':nil':0':ok'3(+(1, _n968729)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n968729)
Induction Base:
U211'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
U211'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n968730, 1))), _gen_tt':mark':nil':0':ok'3(_b983482)) →RΩ(1)
mark'(U211'(_gen_tt':mark':nil':0':ok'3(+(1, _$n968730)), _gen_tt':mark':nil':0':ok'3(_b983482))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
U141'(_gen_tt':mark':nil':0':ok'3(+(1, _n731615)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n731615)
U151'(_gen_tt':mark':nil':0':ok'3(+(1, _n759029)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n759029)
U161'(_gen_tt':mark':nil':0':ok'3(+(1, _n786811)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n786811)
U171'(_gen_tt':mark':nil':0':ok'3(+(1, _n814961)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n814961)
U181'(_gen_tt':mark':nil':0':ok'3(+(1, _n855378)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n855378)
U191'(_gen_tt':mark':nil':0':ok'3(+(1, _n884391)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n884391)
U201'(_gen_tt':mark':nil':0':ok'3(+(1, _n913772)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n913772)
U211'(_gen_tt':mark':nil':0':ok'3(+(1, _n968729)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n968729)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
U221', active', tail', take', sel', proper', top'
They will be analysed ascendingly in the following order:
U221' < active'
tail' < active'
take' < active'
sel' < active'
active' < top'
U221' < proper'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
U221'(_gen_tt':mark':nil':0':ok'3(+(1, _n999100)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n999100)
Induction Base:
U221'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c))
Induction Step:
U221'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n999101, 1))), _gen_tt':mark':nil':0':ok'3(_b1021821), _gen_tt':mark':nil':0':ok'3(_c1021822)) →RΩ(1)
mark'(U221'(_gen_tt':mark':nil':0':ok'3(+(1, _$n999101)), _gen_tt':mark':nil':0':ok'3(_b1021821), _gen_tt':mark':nil':0':ok'3(_c1021822))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
U141'(_gen_tt':mark':nil':0':ok'3(+(1, _n731615)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n731615)
U151'(_gen_tt':mark':nil':0':ok'3(+(1, _n759029)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n759029)
U161'(_gen_tt':mark':nil':0':ok'3(+(1, _n786811)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n786811)
U171'(_gen_tt':mark':nil':0':ok'3(+(1, _n814961)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n814961)
U181'(_gen_tt':mark':nil':0':ok'3(+(1, _n855378)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n855378)
U191'(_gen_tt':mark':nil':0':ok'3(+(1, _n884391)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n884391)
U201'(_gen_tt':mark':nil':0':ok'3(+(1, _n913772)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n913772)
U211'(_gen_tt':mark':nil':0':ok'3(+(1, _n968729)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n968729)
U221'(_gen_tt':mark':nil':0':ok'3(+(1, _n999100)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n999100)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
tail', active', take', sel', proper', top'
They will be analysed ascendingly in the following order:
tail' < active'
take' < active'
sel' < active'
active' < top'
tail' < proper'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
tail'(_gen_tt':mark':nil':0':ok'3(+(1, _n1042803))) → _*4, rt ∈ Ω(n1042803)
Induction Base:
tail'(_gen_tt':mark':nil':0':ok'3(+(1, 0)))
Induction Step:
tail'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n1042804, 1)))) →RΩ(1)
mark'(tail'(_gen_tt':mark':nil':0':ok'3(+(1, _$n1042804)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
U141'(_gen_tt':mark':nil':0':ok'3(+(1, _n731615)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n731615)
U151'(_gen_tt':mark':nil':0':ok'3(+(1, _n759029)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n759029)
U161'(_gen_tt':mark':nil':0':ok'3(+(1, _n786811)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n786811)
U171'(_gen_tt':mark':nil':0':ok'3(+(1, _n814961)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n814961)
U181'(_gen_tt':mark':nil':0':ok'3(+(1, _n855378)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n855378)
U191'(_gen_tt':mark':nil':0':ok'3(+(1, _n884391)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n884391)
U201'(_gen_tt':mark':nil':0':ok'3(+(1, _n913772)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n913772)
U211'(_gen_tt':mark':nil':0':ok'3(+(1, _n968729)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n968729)
U221'(_gen_tt':mark':nil':0':ok'3(+(1, _n999100)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n999100)
tail'(_gen_tt':mark':nil':0':ok'3(+(1, _n1042803))) → _*4, rt ∈ Ω(n1042803)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
take', active', sel', proper', top'
They will be analysed ascendingly in the following order:
take' < active'
sel' < active'
active' < top'
take' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
take'(_gen_tt':mark':nil':0':ok'3(+(1, _n1060950)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n1060950)
Induction Base:
take'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
take'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n1060951, 1))), _gen_tt':mark':nil':0':ok'3(_b1076891)) →RΩ(1)
mark'(take'(_gen_tt':mark':nil':0':ok'3(+(1, _$n1060951)), _gen_tt':mark':nil':0':ok'3(_b1076891))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
U141'(_gen_tt':mark':nil':0':ok'3(+(1, _n731615)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n731615)
U151'(_gen_tt':mark':nil':0':ok'3(+(1, _n759029)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n759029)
U161'(_gen_tt':mark':nil':0':ok'3(+(1, _n786811)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n786811)
U171'(_gen_tt':mark':nil':0':ok'3(+(1, _n814961)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n814961)
U181'(_gen_tt':mark':nil':0':ok'3(+(1, _n855378)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n855378)
U191'(_gen_tt':mark':nil':0':ok'3(+(1, _n884391)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n884391)
U201'(_gen_tt':mark':nil':0':ok'3(+(1, _n913772)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n913772)
U211'(_gen_tt':mark':nil':0':ok'3(+(1, _n968729)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n968729)
U221'(_gen_tt':mark':nil':0':ok'3(+(1, _n999100)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n999100)
tail'(_gen_tt':mark':nil':0':ok'3(+(1, _n1042803))) → _*4, rt ∈ Ω(n1042803)
take'(_gen_tt':mark':nil':0':ok'3(+(1, _n1060950)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n1060950)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
sel', active', proper', top'
They will be analysed ascendingly in the following order:
sel' < active'
active' < top'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
sel'(_gen_tt':mark':nil':0':ok'3(+(1, _n1092644)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n1092644)
Induction Base:
sel'(_gen_tt':mark':nil':0':ok'3(+(1, 0)), _gen_tt':mark':nil':0':ok'3(b))
Induction Step:
sel'(_gen_tt':mark':nil':0':ok'3(+(1, +(_$n1092645, 1))), _gen_tt':mark':nil':0':ok'3(_b1108909)) →RΩ(1)
mark'(sel'(_gen_tt':mark':nil':0':ok'3(+(1, _$n1092645)), _gen_tt':mark':nil':0':ok'3(_b1108909))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U101'(tt', V2)) → mark'(U102'(isLNat'(V2)))
active'(U102'(tt')) → mark'(tt')
active'(U11'(tt', N, XS)) → mark'(U12'(isLNat'(XS), N, XS))
active'(U111'(tt')) → mark'(tt')
active'(U12'(tt', N, XS)) → mark'(snd'(splitAt'(N, XS)))
active'(U121'(tt')) → mark'(tt')
active'(U131'(tt', V2)) → mark'(U132'(isLNat'(V2)))
active'(U132'(tt')) → mark'(tt')
active'(U141'(tt', V2)) → mark'(U142'(isLNat'(V2)))
active'(U142'(tt')) → mark'(tt')
active'(U151'(tt', V2)) → mark'(U152'(isLNat'(V2)))
active'(U152'(tt')) → mark'(tt')
active'(U161'(tt', N)) → mark'(cons'(N, natsFrom'(s'(N))))
active'(U171'(tt', N, XS)) → mark'(U172'(isLNat'(XS), N, XS))
active'(U172'(tt', N, XS)) → mark'(head'(afterNth'(N, XS)))
active'(U181'(tt', Y)) → mark'(U182'(isLNat'(Y), Y))
active'(U182'(tt', Y)) → mark'(Y)
active'(U191'(tt', XS)) → mark'(pair'(nil', XS))
active'(U201'(tt', N, X, XS)) → mark'(U202'(isNatural'(X), N, X, XS))
active'(U202'(tt', N, X, XS)) → mark'(U203'(isLNat'(XS), N, X, XS))
active'(U203'(tt', N, X, XS)) → mark'(U204'(splitAt'(N, XS), X))
active'(U204'(pair'(YS, ZS), X)) → mark'(pair'(cons'(X, YS), ZS))
active'(U21'(tt', X, Y)) → mark'(U22'(isLNat'(Y), X))
active'(U211'(tt', XS)) → mark'(U212'(isLNat'(XS), XS))
active'(U212'(tt', XS)) → mark'(XS)
active'(U22'(tt', X)) → mark'(X)
active'(U221'(tt', N, XS)) → mark'(U222'(isLNat'(XS), N, XS))
active'(U222'(tt', N, XS)) → mark'(fst'(splitAt'(N, XS)))
active'(U31'(tt', N, XS)) → mark'(U32'(isLNat'(XS), N))
active'(U32'(tt', N)) → mark'(N)
active'(U41'(tt', V2)) → mark'(U42'(isLNat'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isLNat'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt')) → mark'(tt')
active'(U91'(tt')) → mark'(tt')
active'(afterNth'(N, XS)) → mark'(U11'(isNatural'(N), N, XS))
active'(fst'(pair'(X, Y))) → mark'(U21'(isLNat'(X), X, Y))
active'(head'(cons'(N, XS))) → mark'(U31'(isNatural'(N), N, XS))
active'(isLNat'(nil')) → mark'(tt')
active'(isLNat'(afterNth'(V1, V2))) → mark'(U41'(isNatural'(V1), V2))
active'(isLNat'(cons'(V1, V2))) → mark'(U51'(isNatural'(V1), V2))
active'(isLNat'(fst'(V1))) → mark'(U61'(isPLNat'(V1)))
active'(isLNat'(natsFrom'(V1))) → mark'(U71'(isNatural'(V1)))
active'(isLNat'(snd'(V1))) → mark'(U81'(isPLNat'(V1)))
active'(isLNat'(tail'(V1))) → mark'(U91'(isLNat'(V1)))
active'(isLNat'(take'(V1, V2))) → mark'(U101'(isNatural'(V1), V2))
active'(isNatural'(0')) → mark'(tt')
active'(isNatural'(head'(V1))) → mark'(U111'(isLNat'(V1)))
active'(isNatural'(s'(V1))) → mark'(U121'(isNatural'(V1)))
active'(isNatural'(sel'(V1, V2))) → mark'(U131'(isNatural'(V1), V2))
active'(isPLNat'(pair'(V1, V2))) → mark'(U141'(isLNat'(V1), V2))
active'(isPLNat'(splitAt'(V1, V2))) → mark'(U151'(isNatural'(V1), V2))
active'(natsFrom'(N)) → mark'(U161'(isNatural'(N), N))
active'(sel'(N, XS)) → mark'(U171'(isNatural'(N), N, XS))
active'(snd'(pair'(X, Y))) → mark'(U181'(isLNat'(X), Y))
active'(splitAt'(0', XS)) → mark'(U191'(isLNat'(XS), XS))
active'(splitAt'(s'(N), cons'(X, XS))) → mark'(U201'(isNatural'(N), N, X, XS))
active'(tail'(cons'(N, XS))) → mark'(U211'(isNatural'(N), XS))
active'(take'(N, XS)) → mark'(U221'(isNatural'(N), N, XS))
active'(U101'(X1, X2)) → U101'(active'(X1), X2)
active'(U102'(X)) → U102'(active'(X))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(U111'(X)) → U111'(active'(X))
active'(snd'(X)) → snd'(active'(X))
active'(splitAt'(X1, X2)) → splitAt'(active'(X1), X2)
active'(splitAt'(X1, X2)) → splitAt'(X1, active'(X2))
active'(U121'(X)) → U121'(active'(X))
active'(U131'(X1, X2)) → U131'(active'(X1), X2)
active'(U132'(X)) → U132'(active'(X))
active'(U141'(X1, X2)) → U141'(active'(X1), X2)
active'(U142'(X)) → U142'(active'(X))
active'(U151'(X1, X2)) → U151'(active'(X1), X2)
active'(U152'(X)) → U152'(active'(X))
active'(U161'(X1, X2)) → U161'(active'(X1), X2)
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(natsFrom'(X)) → natsFrom'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(U171'(X1, X2, X3)) → U171'(active'(X1), X2, X3)
active'(U172'(X1, X2, X3)) → U172'(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'(U181'(X1, X2)) → U181'(active'(X1), X2)
active'(U182'(X1, X2)) → U182'(active'(X1), X2)
active'(U191'(X1, X2)) → U191'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(active'(X1), X2)
active'(pair'(X1, X2)) → pair'(X1, active'(X2))
active'(U201'(X1, X2, X3, X4)) → U201'(active'(X1), X2, X3, X4)
active'(U202'(X1, X2, X3, X4)) → U202'(active'(X1), X2, X3, X4)
active'(U203'(X1, X2, X3, X4)) → U203'(active'(X1), X2, X3, X4)
active'(U204'(X1, X2)) → U204'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U211'(X1, X2)) → U211'(active'(X1), X2)
active'(U212'(X1, X2)) → U212'(active'(X1), X2)
active'(U221'(X1, X2, X3)) → U221'(active'(X1), X2, X3)
active'(U222'(X1, X2, X3)) → U222'(active'(X1), X2, X3)
active'(fst'(X)) → fst'(active'(X))
active'(U31'(X1, X2, X3)) → U31'(active'(X1), X2, X3)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X)) → U91'(active'(X))
active'(tail'(X)) → tail'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
U101'(mark'(X1), X2) → mark'(U101'(X1, X2))
U102'(mark'(X)) → mark'(U102'(X))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
U111'(mark'(X)) → mark'(U111'(X))
snd'(mark'(X)) → mark'(snd'(X))
splitAt'(mark'(X1), X2) → mark'(splitAt'(X1, X2))
splitAt'(X1, mark'(X2)) → mark'(splitAt'(X1, X2))
U121'(mark'(X)) → mark'(U121'(X))
U131'(mark'(X1), X2) → mark'(U131'(X1, X2))
U132'(mark'(X)) → mark'(U132'(X))
U141'(mark'(X1), X2) → mark'(U141'(X1, X2))
U142'(mark'(X)) → mark'(U142'(X))
U151'(mark'(X1), X2) → mark'(U151'(X1, X2))
U152'(mark'(X)) → mark'(U152'(X))
U161'(mark'(X1), X2) → mark'(U161'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
natsFrom'(mark'(X)) → mark'(natsFrom'(X))
s'(mark'(X)) → mark'(s'(X))
U171'(mark'(X1), X2, X3) → mark'(U171'(X1, X2, X3))
U172'(mark'(X1), X2, X3) → mark'(U172'(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))
U181'(mark'(X1), X2) → mark'(U181'(X1, X2))
U182'(mark'(X1), X2) → mark'(U182'(X1, X2))
U191'(mark'(X1), X2) → mark'(U191'(X1, X2))
pair'(mark'(X1), X2) → mark'(pair'(X1, X2))
pair'(X1, mark'(X2)) → mark'(pair'(X1, X2))
U201'(mark'(X1), X2, X3, X4) → mark'(U201'(X1, X2, X3, X4))
U202'(mark'(X1), X2, X3, X4) → mark'(U202'(X1, X2, X3, X4))
U203'(mark'(X1), X2, X3, X4) → mark'(U203'(X1, X2, X3, X4))
U204'(mark'(X1), X2) → mark'(U204'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U211'(mark'(X1), X2) → mark'(U211'(X1, X2))
U212'(mark'(X1), X2) → mark'(U212'(X1, X2))
U221'(mark'(X1), X2, X3) → mark'(U221'(X1, X2, X3))
U222'(mark'(X1), X2, X3) → mark'(U222'(X1, X2, X3))
fst'(mark'(X)) → mark'(fst'(X))
U31'(mark'(X1), X2, X3) → mark'(U31'(X1, X2, X3))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X)) → mark'(U91'(X))
tail'(mark'(X)) → mark'(tail'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(U101'(X1, X2)) → U101'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U102'(X)) → U102'(proper'(X))
proper'(isLNat'(X)) → isLNat'(proper'(X))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(U111'(X)) → U111'(proper'(X))
proper'(snd'(X)) → snd'(proper'(X))
proper'(splitAt'(X1, X2)) → splitAt'(proper'(X1), proper'(X2))
proper'(U121'(X)) → U121'(proper'(X))
proper'(U131'(X1, X2)) → U131'(proper'(X1), proper'(X2))
proper'(U132'(X)) → U132'(proper'(X))
proper'(U141'(X1, X2)) → U141'(proper'(X1), proper'(X2))
proper'(U142'(X)) → U142'(proper'(X))
proper'(U151'(X1, X2)) → U151'(proper'(X1), proper'(X2))
proper'(U152'(X)) → U152'(proper'(X))
proper'(U161'(X1, X2)) → U161'(proper'(X1), proper'(X2))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(natsFrom'(X)) → natsFrom'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(U171'(X1, X2, X3)) → U171'(proper'(X1), proper'(X2), proper'(X3))
proper'(U172'(X1, X2, X3)) → U172'(proper'(X1), proper'(X2), proper'(X3))
proper'(head'(X)) → head'(proper'(X))
proper'(afterNth'(X1, X2)) → afterNth'(proper'(X1), proper'(X2))
proper'(U181'(X1, X2)) → U181'(proper'(X1), proper'(X2))
proper'(U182'(X1, X2)) → U182'(proper'(X1), proper'(X2))
proper'(U191'(X1, X2)) → U191'(proper'(X1), proper'(X2))
proper'(pair'(X1, X2)) → pair'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U201'(X1, X2, X3, X4)) → U201'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U202'(X1, X2, X3, X4)) → U202'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(isNatural'(X)) → isNatural'(proper'(X))
proper'(U203'(X1, X2, X3, X4)) → U203'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U204'(X1, X2)) → U204'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(U211'(X1, X2)) → U211'(proper'(X1), proper'(X2))
proper'(U212'(X1, X2)) → U212'(proper'(X1), proper'(X2))
proper'(U221'(X1, X2, X3)) → U221'(proper'(X1), proper'(X2), proper'(X3))
proper'(U222'(X1, X2, X3)) → U222'(proper'(X1), proper'(X2), proper'(X3))
proper'(fst'(X)) → fst'(proper'(X))
proper'(U31'(X1, X2, X3)) → U31'(proper'(X1), proper'(X2), proper'(X3))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(U91'(X)) → U91'(proper'(X))
proper'(isPLNat'(X)) → isPLNat'(proper'(X))
proper'(tail'(X)) → tail'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
U101'(ok'(X1), ok'(X2)) → ok'(U101'(X1, X2))
U102'(ok'(X)) → ok'(U102'(X))
isLNat'(ok'(X)) → ok'(isLNat'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
U111'(ok'(X)) → ok'(U111'(X))
snd'(ok'(X)) → ok'(snd'(X))
splitAt'(ok'(X1), ok'(X2)) → ok'(splitAt'(X1, X2))
U121'(ok'(X)) → ok'(U121'(X))
U131'(ok'(X1), ok'(X2)) → ok'(U131'(X1, X2))
U132'(ok'(X)) → ok'(U132'(X))
U141'(ok'(X1), ok'(X2)) → ok'(U141'(X1, X2))
U142'(ok'(X)) → ok'(U142'(X))
U151'(ok'(X1), ok'(X2)) → ok'(U151'(X1, X2))
U152'(ok'(X)) → ok'(U152'(X))
U161'(ok'(X1), ok'(X2)) → ok'(U161'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
natsFrom'(ok'(X)) → ok'(natsFrom'(X))
s'(ok'(X)) → ok'(s'(X))
U171'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U171'(X1, X2, X3))
U172'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U172'(X1, X2, X3))
head'(ok'(X)) → ok'(head'(X))
afterNth'(ok'(X1), ok'(X2)) → ok'(afterNth'(X1, X2))
U181'(ok'(X1), ok'(X2)) → ok'(U181'(X1, X2))
U182'(ok'(X1), ok'(X2)) → ok'(U182'(X1, X2))
U191'(ok'(X1), ok'(X2)) → ok'(U191'(X1, X2))
pair'(ok'(X1), ok'(X2)) → ok'(pair'(X1, X2))
U201'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U201'(X1, X2, X3, X4))
U202'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U202'(X1, X2, X3, X4))
isNatural'(ok'(X)) → ok'(isNatural'(X))
U203'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U203'(X1, X2, X3, X4))
U204'(ok'(X1), ok'(X2)) → ok'(U204'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
U211'(ok'(X1), ok'(X2)) → ok'(U211'(X1, X2))
U212'(ok'(X1), ok'(X2)) → ok'(U212'(X1, X2))
U221'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U221'(X1, X2, X3))
U222'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U222'(X1, X2, X3))
fst'(ok'(X)) → ok'(fst'(X))
U31'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U31'(X1, X2, X3))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X)) → ok'(U91'(X))
isPLNat'(ok'(X)) → ok'(isPLNat'(X))
tail'(ok'(X)) → ok'(tail'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U101' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tt' :: tt':mark':nil':0':ok'
mark' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U102' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U11' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U12' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U111' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
snd' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
splitAt' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U121' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U131' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U132' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U141' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U142' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U151' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U152' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U161' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
cons' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
natsFrom' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
s' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U171' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U172' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
head' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
afterNth' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U181' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U182' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U191' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
pair' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
nil' :: tt':mark':nil':0':ok'
U201' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U202' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isNatural' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U203' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U204' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U21' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U22' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U211' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U212' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U221' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U222' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
fst' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U31' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U32' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U41' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U42' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U51' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U52' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U61' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U71' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U81' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
U91' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
isPLNat' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
tail' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
take' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
0' :: tt':mark':nil':0':ok'
sel' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
proper' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
ok' :: tt':mark':nil':0':ok' → tt':mark':nil':0':ok'
top' :: tt':mark':nil':0':ok' → top'
_hole_tt':mark':nil':0':ok'1 :: tt':mark':nil':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':nil':0':ok'3 :: Nat → tt':mark':nil':0':ok'
Lemmas:
U102'(_gen_tt':mark':nil':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
U12'(_gen_tt':mark':nil':0':ok'3(+(1, _n9388)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n9388)
snd'(_gen_tt':mark':nil':0':ok'3(+(1, _n29127))) → _*4, rt ∈ Ω(n29127)
splitAt'(_gen_tt':mark':nil':0':ok'3(+(1, _n38880)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n38880)
U132'(_gen_tt':mark':nil':0':ok'3(+(1, _n54395))) → _*4, rt ∈ Ω(n54395)
U142'(_gen_tt':mark':nil':0':ok'3(+(1, _n64463))) → _*4, rt ∈ Ω(n64463)
U152'(_gen_tt':mark':nil':0':ok'3(+(1, _n74655))) → _*4, rt ∈ Ω(n74655)
cons'(_gen_tt':mark':nil':0':ok'3(+(1, _n84971)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n84971)
natsFrom'(_gen_tt':mark':nil':0':ok'3(+(1, _n101358))) → _*4, rt ∈ Ω(n101358)
s'(_gen_tt':mark':nil':0':ok'3(+(1, _n111989))) → _*4, rt ∈ Ω(n111989)
U172'(_gen_tt':mark':nil':0':ok'3(+(1, _n122744)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n122744)
head'(_gen_tt':mark':nil':0':ok'3(+(1, _n146453))) → _*4, rt ∈ Ω(n146453)
afterNth'(_gen_tt':mark':nil':0':ok'3(+(1, _n157590)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n157590)
U182'(_gen_tt':mark':nil':0':ok'3(+(1, _n175782)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n175782)
pair'(_gen_tt':mark':nil':0':ok'3(+(1, _n194123)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n194123)
U202'(_gen_tt':mark':nil':0':ok'3(+(1, _n213051)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n213051)
U203'(_gen_tt':mark':nil':0':ok'3(+(1, _n247220)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n247220)
U204'(_gen_tt':mark':nil':0':ok'3(+(1, _n282455)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n282455)
U22'(_gen_tt':mark':nil':0':ok'3(+(1, _n302776)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n302776)
U212'(_gen_tt':mark':nil':0':ok'3(+(1, _n323465)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n323465)
U222'(_gen_tt':mark':nil':0':ok'3(+(1, _n344522)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n344522)
fst'(_gen_tt':mark':nil':0':ok'3(+(1, _n374429))) → _*4, rt ∈ Ω(n374429)
U32'(_gen_tt':mark':nil':0':ok'3(+(1, _n387744)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n387744)
U42'(_gen_tt':mark':nil':0':ok'3(+(1, _n409905))) → _*4, rt ∈ Ω(n409905)
U52'(_gen_tt':mark':nil':0':ok'3(+(1, _n423535))) → _*4, rt ∈ Ω(n423535)
U11'(_gen_tt':mark':nil':0':ok'3(+(1, _n437289)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n437289)
U21'(_gen_tt':mark':nil':0':ok'3(+(1, _n469547)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n469547)
U31'(_gen_tt':mark':nil':0':ok'3(+(1, _n502537)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n502537)
U41'(_gen_tt':mark':nil':0':ok'3(+(1, _n536259)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n536259)
U51'(_gen_tt':mark':nil':0':ok'3(+(1, _n560755)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n560755)
U61'(_gen_tt':mark':nil':0':ok'3(+(1, _n585619))) → _*4, rt ∈ Ω(n585619)
U71'(_gen_tt':mark':nil':0':ok'3(+(1, _n600961))) → _*4, rt ∈ Ω(n600961)
U81'(_gen_tt':mark':nil':0':ok'3(+(1, _n616119))) → _*4, rt ∈ Ω(n616119)
U91'(_gen_tt':mark':nil':0':ok'3(+(1, _n631401))) → _*4, rt ∈ Ω(n631401)
U101'(_gen_tt':mark':nil':0':ok'3(+(1, _n646807)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n646807)
U111'(_gen_tt':mark':nil':0':ok'3(+(1, _n673003))) → _*4, rt ∈ Ω(n673003)
U121'(_gen_tt':mark':nil':0':ok'3(+(1, _n688724))) → _*4, rt ∈ Ω(n688724)
U131'(_gen_tt':mark':nil':0':ok'3(+(1, _n704569)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n704569)
U141'(_gen_tt':mark':nil':0':ok'3(+(1, _n731615)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n731615)
U151'(_gen_tt':mark':nil':0':ok'3(+(1, _n759029)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n759029)
U161'(_gen_tt':mark':nil':0':ok'3(+(1, _n786811)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n786811)
U171'(_gen_tt':mark':nil':0':ok'3(+(1, _n814961)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n814961)
U181'(_gen_tt':mark':nil':0':ok'3(+(1, _n855378)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n855378)
U191'(_gen_tt':mark':nil':0':ok'3(+(1, _n884391)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n884391)
U201'(_gen_tt':mark':nil':0':ok'3(+(1, _n913772)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c), _gen_tt':mark':nil':0':ok'3(d)) → _*4, rt ∈ Ω(n913772)
U211'(_gen_tt':mark':nil':0':ok'3(+(1, _n968729)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n968729)
U221'(_gen_tt':mark':nil':0':ok'3(+(1, _n999100)), _gen_tt':mark':nil':0':ok'3(b), _gen_tt':mark':nil':0':ok'3(c)) → _*4, rt ∈ Ω(n999100)
tail'(_gen_tt':mark':nil':0':ok'3(+(1, _n1042803))) → _*4, rt ∈ Ω(n1042803)
take'(_gen_tt':mark':nil':0':ok'3(+(1, _n1060950)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n1060950)
sel'(_gen_tt':mark':nil':0':ok'3(+(1, _n1092644)), _gen_tt':mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n1092644)
Generator Equations:
_gen_tt':mark':nil':0':ok'3(0) ⇔ tt'
_gen_tt':mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
active', proper', top'
They will be analysed ascendingly in the following order:
active' < top'
proper' < top'