KILLEDRuntime Complexity (innermost) proof of /tmp/tmpsI3158/LISTUTILITIES_nosorts_noand_C.xml
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).0 CpxTRS↳1 DecreasingLoopProof (⇔, 6554 ms)↳2 BOUNDS(n^1, INF)↳3 RenamingProof (⇔, 0 ms)↳4 CpxRelTRS↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)↳6 typed CpxTrs↳7 OrderProof (LOWER BOUND(ID), 0 ms)↳8 typed CpxTrs↳9 RewriteLemmaProof (LOWER BOUND(ID), 552 ms)↳10 BEST↳11 typed CpxTrs↳12 RewriteLemmaProof (LOWER BOUND(ID), 106 ms)↳13 BEST↳14 typed CpxTrs↳15 RewriteLemmaProof (LOWER BOUND(ID), 159 ms)↳16 BEST↳17 typed CpxTrs↳18 RewriteLemmaProof (LOWER BOUND(ID), 157 ms)↳19 BEST↳20 typed CpxTrs↳21 RewriteLemmaProof (LOWER BOUND(ID), 74 ms)↳22 BEST↳23 typed CpxTrs↳24 RewriteLemmaProof (LOWER BOUND(ID), 205 ms)↳25 BEST↳26 typed CpxTrs↳27 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳28 BEST↳29 typed CpxTrs↳30 RewriteLemmaProof (LOWER BOUND(ID), 106 ms)↳31 BEST↳32 typed CpxTrs↳33 RewriteLemmaProof (LOWER BOUND(ID), 66 ms)↳34 BEST↳35 typed CpxTrs↳36 RewriteLemmaProof (LOWER BOUND(ID), 210 ms)↳37 BEST↳38 typed CpxTrs↳39 RewriteLemmaProof (LOWER BOUND(ID), 178 ms)↳40 BEST↳41 typed CpxTrs↳42 RewriteLemmaProof (LOWER BOUND(ID), 145 ms)↳43 BEST↳44 typed CpxTrs↳45 RewriteLemmaProof (LOWER BOUND(ID), 95 ms)↳46 BEST↳47 typed CpxTrs↳48 RewriteLemmaProof (LOWER BOUND(ID), 63 ms)↳49 BEST↳50 typed CpxTrs↳51 RewriteLemmaProof (LOWER BOUND(ID), 36 ms)↳52 BEST↳53 typed CpxTrs↳54 RewriteLemmaProof (LOWER BOUND(ID), 110 ms)↳55 BEST↳56 typed CpxTrs↳57 RewriteLemmaProof (LOWER BOUND(ID), 33 ms)↳58 BEST↳59 typed CpxTrs↳60 RewriteLemmaProof (LOWER BOUND(ID), 151 ms)↳61 BEST↳62 typed CpxTrs↳63 RewriteLemmaProof (LOWER BOUND(ID), 122 ms)↳64 BEST↳65 typed CpxTrs↳66 RewriteLemmaProof (LOWER BOUND(ID), 89 ms)↳67 BEST↳68 typed CpxTrs↳69 RewriteLemmaProof (LOWER BOUND(ID), 56 ms)↳70 BEST↳71 typed CpxTrs↳72 RewriteLemmaProof (LOWER BOUND(ID), 76 ms)↳73 BEST↳74 typed CpxTrs↳75 RewriteLemmaProof (LOWER BOUND(ID), 92 ms)↳76 BEST↳77 typed CpxTrs↳78 RewriteLemmaProof (LOWER BOUND(ID), 114 ms)↳79 BEST↳80 typed CpxTrs↳81 RewriteLemmaProof (LOWER BOUND(ID), 185 ms)↳82 BEST↳83 typed CpxTrs↳84 RewriteLemmaProof (LOWER BOUND(ID), 46 ms)↳85 BEST↳86 typed CpxTrs↳87 RewriteLemmaProof (LOWER BOUND(ID), 192 ms)↳88 BEST↳89 typed CpxTrs↳90 RewriteLemmaProof (LOWER BOUND(ID), 97 ms)↳91 BEST↳92 typed CpxTrs↳93 RewriteLemmaProof (LOWER BOUND(ID), 59 ms)↳94 BEST↳95 typed CpxTrs↳96 RewriteLemmaProof (LOWER BOUND(ID), 67 ms)↳97 BEST↳98 typed CpxTrs↳99 typed CpxTrs↳100 typed CpxTrs↳101 typed CpxTrs↳102 typed CpxTrs↳103 typed CpxTrs↳104 typed CpxTrs↳105 typed CpxTrs↳106 typed CpxTrs↳107 typed CpxTrs↳108 typed CpxTrs↳109 typed CpxTrs↳110 typed CpxTrs↳111 typed CpxTrs↳112 typed CpxTrs↳113 typed CpxTrs↳114 typed CpxTrs↳115 typed CpxTrs↳116 typed CpxTrs↳117 typed CpxTrs↳118 typed CpxTrs↳119 typed CpxTrs↳120 typed CpxTrs↳121 typed CpxTrs↳122 typed CpxTrs↳123 typed CpxTrs↳124 typed CpxTrs↳125 typed CpxTrs↳126 typed CpxTrs↳127 typed CpxTrs↳128 typed CpxTrs(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: INNERMOST(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
U11(mark(X1), X2, X3) →+ mark(U11(X1, X2, X3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].(2) BOUNDS(n^1, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S is empty.
Rewrite Strategy: INNERMOST(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.(6) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active, U12, snd, splitAt, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U12 < active
snd < active
splitAt < active
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U12 < proper
snd < proper
splitAt < proper
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(8) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okGenerator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U12, active, snd, splitAt, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U12 < active
snd < active
splitAt < active
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U12 < proper
snd < proper
splitAt < proper
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)Induction Base:
U12(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))Induction Step:
U12(gen_tt:mark:0':nil:ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) →RΩ(1)
mark(U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
snd, active, splitAt, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
snd < active
splitAt < active
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
snd < proper
splitAt < proper
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)Induction Base:
snd(gen_tt:mark:0':nil:ok3_0(+(1, 0)))Induction Step:
snd(gen_tt:mark:0':nil:ok3_0(+(1, +(n5788_0, 1)))) →RΩ(1)
mark(snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(13) Complex Obligation (BEST)
(14) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
splitAt, active, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
splitAt < active
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
splitAt < proper
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)Induction Base:
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, +(n7226_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(16) Complex Obligation (BEST)
(17) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U22, active, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)Induction Base:
U22(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
U22(gen_tt:mark:0':nil:ok3_0(+(1, +(n11492_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(19) Complex Obligation (BEST)
(20) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U32, active, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)Induction Base:
U32(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
U32(gen_tt:mark:0':nil:ok3_0(+(1, +(n15863_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(22) Complex Obligation (BEST)
(23) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U42, active, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(24) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)Induction Base:
U42(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))Induction Step:
U42(gen_tt:mark:0':nil:ok3_0(+(1, +(n20538_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) →RΩ(1)
mark(U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(25) Complex Obligation (BEST)
(26) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
head, active, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(27) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)Induction Base:
head(gen_tt:mark:0':nil:ok3_0(+(1, 0)))Induction Step:
head(gen_tt:mark:0':nil:ok3_0(+(1, +(n28611_0, 1)))) →RΩ(1)
mark(head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(28) Complex Obligation (BEST)
(29) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
afterNth, active, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(30) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)Induction Base:
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, +(n30799_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(31) Complex Obligation (BEST)
(32) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U52, active, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(33) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)Induction Base:
U52(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
U52(gen_tt:mark:0':nil:ok3_0(+(1, +(n36585_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(34) Complex Obligation (BEST)
(35) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U62, active, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(36) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)Induction Base:
U62(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))Induction Step:
U62(gen_tt:mark:0':nil:ok3_0(+(1, +(n42476_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) →RΩ(1)
mark(U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(37) Complex Obligation (BEST)
(38) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U63, active, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(39) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)Induction Base:
U63(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))Induction Step:
U63(gen_tt:mark:0':nil:ok3_0(+(1, +(n56299_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) →RΩ(1)
mark(U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(40) Complex Obligation (BEST)
(41) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U64, active, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(42) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)Induction Base:
U64(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
U64(gen_tt:mark:0':nil:ok3_0(+(1, +(n71126_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(43) Complex Obligation (BEST)
(44) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
pair, active, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(45) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)Induction Base:
pair(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
pair(gen_tt:mark:0':nil:ok3_0(+(1, +(n78321_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(46) Complex Obligation (BEST)
(47) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
cons, active, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(48) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)Induction Base:
cons(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
cons(gen_tt:mark:0':nil:ok3_0(+(1, +(n86019_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(49) Complex Obligation (BEST)
(50) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U72, active, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(51) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)Induction Base:
U72(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
U72(gen_tt:mark:0':nil:ok3_0(+(1, +(n93822_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(52) Complex Obligation (BEST)
(53) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U82, active, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(54) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)Induction Base:
U82(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))Induction Step:
U82(gen_tt:mark:0':nil:ok3_0(+(1, +(n101929_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) →RΩ(1)
mark(U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(55) Complex Obligation (BEST)
(56) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
fst, active, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(57) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)Induction Base:
fst(gen_tt:mark:0':nil:ok3_0(+(1, 0)))Induction Step:
fst(gen_tt:mark:0':nil:ok3_0(+(1, +(n115170_0, 1)))) →RΩ(1)
mark(fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(58) Complex Obligation (BEST)
(59) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U11, active, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(60) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)Induction Base:
U11(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))Induction Step:
U11(gen_tt:mark:0':nil:ok3_0(+(1, +(n119054_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) →RΩ(1)
mark(U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(61) Complex Obligation (BEST)
(62) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U21, active, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(63) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)Induction Base:
U21(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
U21(gen_tt:mark:0':nil:ok3_0(+(1, +(n133211_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(64) Complex Obligation (BEST)
(65) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U31, active, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(66) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)Induction Base:
U31(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
U31(gen_tt:mark:0':nil:ok3_0(+(1, +(n142632_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(67) Complex Obligation (BEST)
(68) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
natsFrom, active, s, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(69) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)Induction Base:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, 0)))Induction Step:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, +(n152357_0, 1)))) →RΩ(1)
mark(natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(70) Complex Obligation (BEST)
(71) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
s, active, U41, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(72) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)Induction Base:
s(gen_tt:mark:0':nil:ok3_0(+(1, 0)))Induction Step:
s(gen_tt:mark:0':nil:ok3_0(+(1, +(n156841_0, 1)))) →RΩ(1)
mark(s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(73) Complex Obligation (BEST)
(74) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U41, active, U51, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(75) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)Induction Base:
U41(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))Induction Step:
U41(gen_tt:mark:0':nil:ok3_0(+(1, +(n161426_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) →RΩ(1)
mark(U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(76) Complex Obligation (BEST)
(77) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U51, active, U61, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(78) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)Induction Base:
U51(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
U51(gen_tt:mark:0':nil:ok3_0(+(1, +(n177726_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(79) Complex Obligation (BEST)
(80) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U61, active, U71, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(81) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n1885690)Induction Base:
U61(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))Induction Step:
U61(gen_tt:mark:0':nil:ok3_0(+(1, +(n188569_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) →RΩ(1)
mark(U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(82) Complex Obligation (BEST)
(83) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n1885690)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U71, active, U81, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(84) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U71(gen_tt:mark:0':nil:ok3_0(+(1, n212356_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2123560)Induction Base:
U71(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
U71(gen_tt:mark:0':nil:ok3_0(+(1, +(n212356_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U71(gen_tt:mark:0':nil:ok3_0(+(1, n212356_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(85) Complex Obligation (BEST)
(86) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n1885690)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n212356_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2123560)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
U81, active, sel, tail, take, proper, topThey will be analysed ascendingly in the following order:
U81 < active
sel < active
tail < active
take < active
active < top
U81 < proper
sel < proper
tail < proper
take < proper
proper < top(87) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U81(gen_tt:mark:0':nil:ok3_0(+(1, n224003_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n2240030)Induction Base:
U81(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))Induction Step:
U81(gen_tt:mark:0':nil:ok3_0(+(1, +(n224003_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) →RΩ(1)
mark(U81(gen_tt:mark:0':nil:ok3_0(+(1, n224003_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(88) Complex Obligation (BEST)
(89) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n1885690)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n212356_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2123560)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n224003_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n2240030)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
sel, active, tail, take, proper, topThey will be analysed ascendingly in the following order:
sel < active
tail < active
take < active
active < top
sel < proper
tail < proper
take < proper
proper < top(90) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
sel(gen_tt:mark:0':nil:ok3_0(+(1, n242576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2425760)Induction Base:
sel(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
sel(gen_tt:mark:0':nil:ok3_0(+(1, +(n242576_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(sel(gen_tt:mark:0':nil:ok3_0(+(1, n242576_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(91) Complex Obligation (BEST)
(92) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n1885690)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n212356_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2123560)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n224003_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n2240030)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n242576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2425760)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
tail, active, take, proper, topThey will be analysed ascendingly in the following order:
tail < active
take < active
active < top
tail < proper
take < proper
proper < top(93) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
tail(gen_tt:mark:0':nil:ok3_0(+(1, n255128_0))) → *4_0, rt ∈ Ω(n2551280)Induction Base:
tail(gen_tt:mark:0':nil:ok3_0(+(1, 0)))Induction Step:
tail(gen_tt:mark:0':nil:ok3_0(+(1, +(n255128_0, 1)))) →RΩ(1)
mark(tail(gen_tt:mark:0':nil:ok3_0(+(1, n255128_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(94) Complex Obligation (BEST)
(95) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n1885690)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n212356_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2123560)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n224003_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n2240030)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n242576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2425760)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n255128_0))) → *4_0, rt ∈ Ω(n2551280)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
take, active, proper, topThey will be analysed ascendingly in the following order:
take < active
active < top
take < proper
proper < top(96) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
take(gen_tt:mark:0':nil:ok3_0(+(1, n260910_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2609100)Induction Base:
take(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))Induction Step:
take(gen_tt:mark:0':nil:ok3_0(+(1, +(n260910_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(take(gen_tt:mark:0':nil:ok3_0(+(1, n260910_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(97) Complex Obligation (BEST)
(98) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n1885690)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n212356_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2123560)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n224003_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n2240030)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n242576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2425760)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n255128_0))) → *4_0, rt ∈ Ω(n2551280)
take(gen_tt:mark:0':nil:ok3_0(+(1, n260910_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2609100)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))The following defined symbols remain to be analysed:
active, proper, topThey will be analysed ascendingly in the following order:
active < top
proper < top(99) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n1885690)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n212356_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2123560)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n224003_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n2240030)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n242576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2425760)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n255128_0))) → *4_0, rt ∈ Ω(n2551280)
take(gen_tt:mark:0':nil:ok3_0(+(1, n260910_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2609100)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(100) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n1885690)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n212356_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2123560)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n224003_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n2240030)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n242576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2425760)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n255128_0))) → *4_0, rt ∈ Ω(n2551280)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(101) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n1885690)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n212356_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2123560)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n224003_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n2240030)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n242576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2425760)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(102) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n1885690)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n212356_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2123560)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n224003_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n2240030)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(103) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n1885690)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n212356_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2123560)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(104) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n188569_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n1885690)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(105) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n177726_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1777260)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(106) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n161426_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1614260)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(107) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)
s(gen_tt:mark:0':nil:ok3_0(+(1, n156841_0))) → *4_0, rt ∈ Ω(n1568410)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(108) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n152357_0))) → *4_0, rt ∈ Ω(n1523570)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(109) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n142632_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1426320)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(110) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n133211_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1332110)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(111) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n119054_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1190540)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(112) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n115170_0))) → *4_0, rt ∈ Ω(n1151700)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(113) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n101929_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1019290)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(114) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n93822_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n938220)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(115) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n86019_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n860190)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(116) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n78321_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783210)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(117) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n71126_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n711260)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(118) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n56299_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n562990)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(119) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n42476_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n424760)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(120) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n36585_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n365850)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(121) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n30799_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n307990)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(122) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)
head(gen_tt:mark:0':nil:ok3_0(+(1, n28611_0))) → *4_0, rt ∈ Ω(n286110)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(123) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n20538_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n205380)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(124) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n15863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n158630)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(125) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n11492_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n114920)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(126) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n7226_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n72260)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(127) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n5788_0))) → *4_0, rt ∈ Ω(n57880)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.
(128) Obligation:
Innermost TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:okLemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n50)Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))No more defined symbols left to analyse.