KILLEDRuntime Complexity (full) proof of /tmp/tmpVY_ZXv/LISTUTILITIES_nokinds_C.xml
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).0 CpxTRS↳1 DecreasingLoopProof (⇔, 26.8 s)↳2 BOUNDS(n^1, INF)↳3 RenamingProof (⇔, 0 ms)↳4 CpxRelTRS↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)↳6 typed CpxTrs↳7 OrderProof (LOWER BOUND(ID), 0 ms)↳8 typed CpxTrs↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 2 ms)↳10 typed CpxTrs↳11 NoRewriteLemmaProof (LOWER BOUND(ID), 17 ms)↳12 typed CpxTrs↳13 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳14 typed CpxTrs↳15 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳16 typed CpxTrs↳17 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳18 typed CpxTrs↳19 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳20 typed CpxTrs↳21 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳22 typed CpxTrs↳23 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳24 typed CpxTrs↳25 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳26 typed CpxTrs↳27 NoRewriteLemmaProof (LOWER BOUND(ID), 5 ms)↳28 typed CpxTrs↳29 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳30 typed CpxTrs↳31 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳32 typed CpxTrs↳33 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳34 typed CpxTrs↳35 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳36 typed CpxTrs↳37 NoRewriteLemmaProof (LOWER BOUND(ID), 13 ms)↳38 typed CpxTrs↳39 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳40 typed CpxTrs↳41 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳42 typed CpxTrs↳43 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳44 typed CpxTrs↳45 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳46 typed CpxTrs↳47 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳48 typed CpxTrs↳49 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳50 typed CpxTrs↳51 NoRewriteLemmaProof (LOWER BOUND(ID), 1 ms)↳52 typed CpxTrs↳53 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳54 typed CpxTrs↳55 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳56 typed CpxTrs↳57 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳58 typed CpxTrs↳59 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳60 typed CpxTrs↳61 NoRewriteLemmaProof (LOWER BOUND(ID), 9 ms)↳62 typed CpxTrs(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: FULL(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
U101(mark(X1), X2, X3) →+ mark(U101(X1, X2, X3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].(2) BOUNDS(n^1, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S is empty.
Rewrite Strategy: FULL(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.(6) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active, fst, splitAt, snd, cons, natsFrom, s, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
fst < active
splitAt < active
snd < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U82 < active
U11 < active
and < active
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
fst < proper
splitAt < proper
snd < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U82 < proper
U11 < proper
and < proper
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(8) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
fst, active, splitAt, snd, cons, natsFrom, s, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
fst < active
splitAt < active
snd < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U82 < active
U11 < active
and < active
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
fst < proper
splitAt < proper
snd < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U82 < proper
U11 < proper
and < proper
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol fst.(10) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
splitAt, active, snd, cons, natsFrom, s, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
splitAt < active
snd < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U82 < active
U11 < active
and < active
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
splitAt < proper
snd < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U82 < proper
U11 < proper
and < proper
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol splitAt.(12) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
snd, active, cons, natsFrom, s, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
snd < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U82 < active
U11 < active
and < active
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
snd < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U82 < proper
U11 < proper
and < proper
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol snd.(14) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
cons, active, natsFrom, s, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U82 < active
U11 < active
and < active
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U82 < proper
U11 < proper
and < proper
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol cons.(16) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
natsFrom, active, s, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U82 < active
U11 < active
and < active
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U82 < proper
U11 < proper
and < proper
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol natsFrom.(18) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
s, active, head, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
s < active
head < active
afterNth < active
pair < active
U82 < active
U11 < active
and < active
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
s < proper
head < proper
afterNth < proper
pair < proper
U82 < proper
U11 < proper
and < proper
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol s.(20) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
head, active, afterNth, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
head < active
afterNth < active
pair < active
U82 < active
U11 < active
and < active
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
head < proper
afterNth < proper
pair < proper
U82 < proper
U11 < proper
and < proper
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol head.(22) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
afterNth, active, pair, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
afterNth < active
pair < active
U82 < active
U11 < active
and < active
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
afterNth < proper
pair < proper
U82 < proper
U11 < proper
and < proper
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol afterNth.(24) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
pair, active, U82, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
pair < active
U82 < active
U11 < active
and < active
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
pair < proper
U82 < proper
U11 < proper
and < proper
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol pair.(26) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U82, active, U11, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U82 < active
U11 < active
and < active
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
U82 < proper
U11 < proper
and < proper
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U82.(28) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U11, active, and, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U11 < active
and < active
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
U11 < proper
and < proper
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U11.(30) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
and, active, isNatural, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
and < active
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
and < proper
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(31) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol and.(32) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
isNatural, active, isLNat, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isNatural < active
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
isNatural < proper
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(33) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isNatural.(34) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
isLNat, active, U21, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isLNat < active
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
isLNat < proper
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(35) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isLNat.(36) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U21, active, U31, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U21 < active
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
U21 < proper
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(37) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U21.(38) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U31, active, isPLNat, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U31 < active
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
U31 < proper
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(39) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U31.(40) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
isPLNat, active, U41, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isPLNat < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
isPLNat < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(41) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isPLNat.(42) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U41, active, U51, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(43) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U41.(44) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U51, active, U61, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(45) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U51.(46) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U61, active, U71, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(47) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U61.(48) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U71, active, U81, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U71 < active
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
U71 < proper
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(49) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U71.(50) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U81, active, U91, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U81 < active
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
U81 < proper
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(51) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U81.(52) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U91, active, U101, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U91 < active
U101 < active
tail < active
take < active
sel < active
active < top
U91 < proper
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(53) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U91.(54) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U101, active, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U101 < active
tail < active
take < active
sel < active
active < top
U101 < proper
tail < proper
take < proper
sel < proper
proper < top(55) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U101.(56) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
tail, active, take, sel, proper, topThey will be analysed ascendingly in the following order:
tail < active
take < active
sel < active
active < top
tail < proper
take < proper
sel < proper
proper < top(57) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol tail.(58) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
take, active, sel, proper, topThey will be analysed ascendingly in the following order:
take < active
sel < active
active < top
take < proper
sel < proper
proper < top(59) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol take.(60) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
sel, active, proper, topThey will be analysed ascendingly in the following order:
sel < active
active < top
sel < proper
proper < top(61) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol sel.(62) Obligation:
TRS:
Rules:
active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0', XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':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