KILLEDRuntime Complexity (full) proof of /tmp/tmpfmuWHw/LISTUTILITIES_nosorts_noand_C.xml
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).0 CpxTRS↳1 RenamingProof (⇔, 0 ms)↳2 CpxRelTRS↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 4 ms)↳4 typed CpxTrs↳5 OrderProof (LOWER BOUND(ID), 0 ms)↳6 typed CpxTrs↳7 RewriteLemmaProof (LOWER BOUND(ID), 657 ms)↳8 BEST↳9 typed CpxTrs↳10 RewriteLemmaProof (LOWER BOUND(ID), 76 ms)↳11 BEST↳12 typed CpxTrs↳13 RewriteLemmaProof (LOWER BOUND(ID), 126 ms)↳14 BEST↳15 typed CpxTrs↳16 RewriteLemmaProof (LOWER BOUND(ID), 118 ms)↳17 BEST↳18 typed CpxTrs↳19 RewriteLemmaProof (LOWER BOUND(ID), 135 ms)↳20 BEST↳21 typed CpxTrs↳22 RewriteLemmaProof (LOWER BOUND(ID), 156 ms)↳23 BEST↳24 typed CpxTrs↳25 RewriteLemmaProof (LOWER BOUND(ID), 105 ms)↳26 BEST↳27 typed CpxTrs↳28 RewriteLemmaProof (LOWER BOUND(ID), 94 ms)↳29 BEST↳30 typed CpxTrs↳31 RewriteLemmaProof (LOWER BOUND(ID), 103 ms)↳32 BEST↳33 typed CpxTrs↳34 RewriteLemmaProof (LOWER BOUND(ID), 221 ms)↳35 BEST↳36 typed CpxTrs↳37 RewriteLemmaProof (LOWER BOUND(ID), 145 ms)↳38 BEST↳39 typed CpxTrs↳40 RewriteLemmaProof (LOWER BOUND(ID), 107 ms)↳41 BEST↳42 typed CpxTrs↳43 RewriteLemmaProof (LOWER BOUND(ID), 70 ms)↳44 BEST↳45 typed CpxTrs↳46 RewriteLemmaProof (LOWER BOUND(ID), 26 ms)↳47 BEST↳48 typed CpxTrs↳49 RewriteLemmaProof (LOWER BOUND(ID), 100 ms)↳50 BEST↳51 typed CpxTrs↳52 RewriteLemmaProof (LOWER BOUND(ID), 152 ms)↳53 BEST↳54 typed CpxTrs↳55 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳56 BEST↳57 typed CpxTrs↳58 RewriteLemmaProof (LOWER BOUND(ID), 99 ms)↳59 BEST↳60 typed CpxTrs↳61 RewriteLemmaProof (LOWER BOUND(ID), 27 ms)↳62 BEST↳63 typed CpxTrs↳64 RewriteLemmaProof (LOWER BOUND(ID), 101 ms)↳65 BEST↳66 typed CpxTrs↳67 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳68 BEST↳69 typed CpxTrs↳70 RewriteLemmaProof (LOWER BOUND(ID), 5 ms)↳71 BEST↳72 typed CpxTrs↳73 RewriteLemmaProof (LOWER BOUND(ID), 133 ms)↳74 BEST↳75 typed CpxTrs↳76 RewriteLemmaProof (LOWER BOUND(ID), 76 ms)↳77 BEST↳78 typed CpxTrs↳79 RewriteLemmaProof (LOWER BOUND(ID), 142 ms)↳80 BEST↳81 typed CpxTrs↳82 RewriteLemmaProof (LOWER BOUND(ID), 17 ms)↳83 BEST↳84 typed CpxTrs↳85 RewriteLemmaProof (LOWER BOUND(ID), 103 ms)↳86 BEST↳87 typed CpxTrs↳88 RewriteLemmaProof (LOWER BOUND(ID), 45 ms)↳89 BEST↳90 typed CpxTrs↳91 RewriteLemmaProof (LOWER BOUND(ID), 25 ms)↳92 BEST↳93 typed CpxTrs↳94 RewriteLemmaProof (LOWER BOUND(ID), 55 ms)↳95 BEST↳96 typed CpxTrs↳97 typed CpxTrs↳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(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: FULL(1) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.(2) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
active(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: FULL(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.(4) Obligation:
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(5) 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(6) Obligation:
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(7) 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).
(8) Complex Obligation (BEST)
(9) Obligation:
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(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n43390)Induction Base:
snd(gen_tt:mark:0':nil:ok3_0(+(1, 0)))Induction Step:
snd(gen_tt:mark:0':nil:ok3_0(+(1, +(n4339_0, 1)))) →RΩ(1)
mark(snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(11) Complex Obligation (BEST)
(12) Obligation:
TRS:
Rules:
active(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, n4339_0))) → *4_0, rt ∈ Ω(n43390)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(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)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, +(n5294_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_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).
(14) Complex Obligation (BEST)
(15) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)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(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)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, +(n8594_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_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).
(17) Complex Obligation (BEST)
(18) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)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(19) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)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, +(n11999_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_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).
(20) Complex Obligation (BEST)
(21) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)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(22) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)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, +(n15708_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, n15708_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).
(23) Complex Obligation (BEST)
(24) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)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(25) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)Induction Base:
head(gen_tt:mark:0':nil:ok3_0(+(1, 0)))Induction Step:
head(gen_tt:mark:0':nil:ok3_0(+(1, +(n22332_0, 1)))) →RΩ(1)
mark(head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(26) Complex Obligation (BEST)
(27) Obligation:
TRS:
Rules:
active(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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)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(28) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)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, +(n24037_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_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).
(29) Complex Obligation (BEST)
(30) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)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(31) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)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, +(n28857_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_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).
(32) Complex Obligation (BEST)
(33) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)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(34) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)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, +(n33782_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, n33782_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).
(35) Complex Obligation (BEST)
(36) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)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(37) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)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, +(n45673_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, n45673_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).
(38) Complex Obligation (BEST)
(39) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)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(40) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)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, +(n58568_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_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).
(41) Complex Obligation (BEST)
(42) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)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(43) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)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, +(n64797_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_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).
(44) Complex Obligation (BEST)
(45) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)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(46) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)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, +(n71529_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_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).
(47) Complex Obligation (BEST)
(48) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)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(49) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)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, +(n78366_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_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).
(50) Complex Obligation (BEST)
(51) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)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(52) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)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, +(n85507_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, n85507_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).
(53) Complex Obligation (BEST)
(54) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)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(55) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)Induction Base:
fst(gen_tt:mark:0':nil:ok3_0(+(1, 0)))Induction Step:
fst(gen_tt:mark:0':nil:ok3_0(+(1, +(n97299_0, 1)))) →RΩ(1)
mark(fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(56) Complex Obligation (BEST)
(57) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)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(58) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)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, +(n100700_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, n100700_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).
(59) Complex Obligation (BEST)
(60) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)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(61) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)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, +(n113408_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_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).
(62) Complex Obligation (BEST)
(63) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)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(64) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)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, +(n121863_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_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).
(65) Complex Obligation (BEST)
(66) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)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(67) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)Induction Base:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, 0)))Induction Step:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, +(n130622_0, 1)))) →RΩ(1)
mark(natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(68) Complex Obligation (BEST)
(69) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)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(70) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)Induction Base:
s(gen_tt:mark:0':nil:ok3_0(+(1, 0)))Induction Step:
s(gen_tt:mark:0':nil:ok3_0(+(1, +(n134623_0, 1)))) →RΩ(1)
mark(s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(71) Complex Obligation (BEST)
(72) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)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(73) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)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, +(n138725_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, n138725_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).
(74) Complex Obligation (BEST)
(75) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)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(76) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)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, +(n153576_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_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).
(77) Complex Obligation (BEST)
(78) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)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(79) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_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(n1634530)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, +(n163453_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, n163453_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).
(80) Complex Obligation (BEST)
(81) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_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(n1634530)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(82) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1853080)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, +(n185308_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_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).
(83) Complex Obligation (BEST)
(84) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_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(n1634530)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1853080)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(85) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1959890)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, +(n195989_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, n195989_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).
(86) Complex Obligation (BEST)
(87) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_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(n1634530)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1853080)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1959890)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(88) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2131130)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, +(n213113_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_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).
(89) Complex Obligation (BEST)
(90) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_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(n1634530)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1853080)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1959890)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2131130)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(91) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
tail(gen_tt:mark:0':nil:ok3_0(+(1, n224699_0))) → *4_0, rt ∈ Ω(n2246990)Induction Base:
tail(gen_tt:mark:0':nil:ok3_0(+(1, 0)))Induction Step:
tail(gen_tt:mark:0':nil:ok3_0(+(1, +(n224699_0, 1)))) →RΩ(1)
mark(tail(gen_tt:mark:0':nil:ok3_0(+(1, n224699_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(92) Complex Obligation (BEST)
(93) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_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(n1634530)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1853080)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1959890)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2131130)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n224699_0))) → *4_0, rt ∈ Ω(n2246990)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(94) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
take(gen_tt:mark:0':nil:ok3_0(+(1, n229998_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2299980)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, +(n229998_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(take(gen_tt:mark:0':nil:ok3_0(+(1, n229998_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).
(95) Complex Obligation (BEST)
(96) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_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(n1634530)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1853080)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1959890)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2131130)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n224699_0))) → *4_0, rt ∈ Ω(n2246990)
take(gen_tt:mark:0':nil:ok3_0(+(1, n229998_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2299980)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(97) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_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(n1634530)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1853080)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1959890)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2131130)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n224699_0))) → *4_0, rt ∈ Ω(n2246990)
take(gen_tt:mark:0':nil:ok3_0(+(1, n229998_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2299980)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.
(98) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_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(n1634530)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1853080)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1959890)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2131130)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n224699_0))) → *4_0, rt ∈ Ω(n2246990)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.
(99) Obligation:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_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(n1634530)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1853080)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1959890)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2131130)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_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(n1634530)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1853080)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1959890)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_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(n1634530)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1853080)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_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(n1634530)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1535760)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1387250)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n1346230)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n1306220)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1218630)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1134080)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n1007000)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n972990)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n855070)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n783660)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n715290)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n647970)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n585680)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_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(n456730)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_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(n337820)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n288570)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n240370)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n223320)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n157080)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n119990)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n85940)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n52940)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:
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, n4339_0))) → *4_0, rt ∈ Ω(n43390)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:
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.