The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

natsFrom(N) → cons(N, n__natsFrom(s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
activate(n__natsFrom(X)) → natsFrom(X)
activate(X) → X

Rewrite Strategy: INNERMOST

(1) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to weighted TRS

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

natsFrom(N) → cons(N, n__natsFrom(s(N))) [1]
fst(pair(XS, YS)) → XS [1]
snd(pair(XS, YS)) → YS [1]
splitAt(0, XS) → pair(nil, XS) [1]
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS)) [1]
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS) [1]
head(cons(N, XS)) → N [1]
tail(cons(N, XS)) → activate(XS) [1]
sel(N, XS) → head(afterNth(N, XS)) [1]
take(N, XS) → fst(splitAt(N, XS)) [1]
afterNth(N, XS) → snd(splitAt(N, XS)) [1]
natsFrom(X) → n__natsFrom(X) [1]
activate(n__natsFrom(X)) → natsFrom(X) [1]
activate(X) → X [1]

Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

natsFrom(N) → cons(N, n__natsFrom(s(N))) [1]
fst(pair(XS, YS)) → XS [1]
snd(pair(XS, YS)) → YS [1]
splitAt(0, XS) → pair(nil, XS) [1]
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS)) [1]
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS) [1]
head(cons(N, XS)) → N [1]
tail(cons(N, XS)) → activate(XS) [1]
sel(N, XS) → head(afterNth(N, XS)) [1]
take(N, XS) → fst(splitAt(N, XS)) [1]
afterNth(N, XS) → snd(splitAt(N, XS)) [1]
natsFrom(X) → n__natsFrom(X) [1]
activate(n__natsFrom(X)) → natsFrom(X) [1]
activate(X) → X [1]

The TRS has the following type information:

natsFrom :: s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil cons :: s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil n__natsFrom :: s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil s :: s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil fst :: pair → s:n__natsFrom:cons:0:nil pair :: s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil → pair snd :: pair → s:n__natsFrom:cons:0:nil splitAt :: s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil → pair 0 :: s:n__natsFrom:cons:0:nil nil :: s:n__natsFrom:cons:0:nil u :: pair → s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil → pair activate :: s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil head :: s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil tail :: s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil sel :: s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil afterNth :: s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil take :: s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil → s:n__natsFrom:cons:0:nil

Rewrite Strategy: INNERMOST

(5) CompletionProof (UPPER BOUND(ID) transformation)

The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

fst
head
tail
sel
take

(c) The following functions are completely defined:

splitAt
activate
afterNth
natsFrom
snd
u

Due to the following rules being added:

splitAt(v0, v1) → const [0]
snd(v0) → null_snd [0]
u(v0, v1, v2, v3) → const [0]

And the following fresh constants:

const, null_snd

(6) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: natsFrom(N) → cons(N, n__natsFrom(s(N))) [1] fst(pair(XS, YS)) → XS [1] snd(pair(XS, YS)) → YS [1] splitAt(0, XS) → pair(nil, XS) [1] splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS)) [1] u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS) [1] head(cons(N, XS)) → N [1] tail(cons(N, XS)) → activate(XS) [1] sel(N, XS) → head(afterNth(N, XS)) [1] take(N, XS) → fst(splitAt(N, XS)) [1] afterNth(N, XS) → snd(splitAt(N, XS)) [1] natsFrom(X) → n__natsFrom(X) [1] activate(n__natsFrom(X)) → natsFrom(X) [1] activate(X) → X [1] splitAt(v0, v1) → const [0] snd(v0) → null_snd [0] u(v0, v1, v2, v3) → const [0] The TRS has the following type information: natsFrom :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd cons :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd n__natsFrom :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd s :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd fst :: pair:const → s:n__natsFrom:cons:0:nil:null_snd pair :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → pair:const snd :: pair:const → s:n__natsFrom:cons:0:nil:null_snd splitAt :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → pair:const 0 :: s:n__natsFrom:cons:0:nil:null_snd nil :: s:n__natsFrom:cons:0:nil:null_snd u :: pair:const → s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → pair:const activate :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd head :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd tail :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd sel :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd afterNth :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd take :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd const :: pair:const null_snd :: s:n__natsFrom:cons:0:nil:null_snd Rewrite Strategy: INNERMOST

(7) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(8) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: natsFrom(N) → cons(N, n__natsFrom(s(N))) [1] fst(pair(XS, YS)) → XS [1] snd(pair(XS, YS)) → YS [1] splitAt(0, XS) → pair(nil, XS) [1] splitAt(s(N), cons(X, n__natsFrom(X'))) → u(splitAt(N, natsFrom(X')), N, X, natsFrom(X')) [3] splitAt(s(N), cons(X, n__natsFrom(X'))) → u(splitAt(N, natsFrom(X')), N, X, n__natsFrom(X')) [3] splitAt(s(N), cons(X, n__natsFrom(X''))) → u(splitAt(N, n__natsFrom(X'')), N, X, natsFrom(X'')) [3] splitAt(s(N), cons(X, XS)) → u(splitAt(N, XS), N, X, XS) [3] u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS) [1] head(cons(N, XS)) → N [1] tail(cons(N, XS)) → activate(XS) [1] sel(N, XS) → head(snd(splitAt(N, XS))) [2] take(0, XS) → fst(pair(nil, XS)) [2] take(s(N'), cons(X1, XS')) → fst(u(splitAt(N', activate(XS')), N', X1, activate(XS'))) [2] take(N, XS) → fst(const) [1] afterNth(0, XS) → snd(pair(nil, XS)) [2] afterNth(s(N''), cons(X2, XS'')) → snd(u(splitAt(N'', activate(XS'')), N'', X2, activate(XS''))) [2] afterNth(N, XS) → snd(const) [1] natsFrom(X) → n__natsFrom(X) [1] activate(n__natsFrom(X)) → natsFrom(X) [1] activate(X) → X [1] splitAt(v0, v1) → const [0] snd(v0) → null_snd [0] u(v0, v1, v2, v3) → const [0] The TRS has the following type information: natsFrom :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd cons :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd n__natsFrom :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd s :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd fst :: pair:const → s:n__natsFrom:cons:0:nil:null_snd pair :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → pair:const snd :: pair:const → s:n__natsFrom:cons:0:nil:null_snd splitAt :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → pair:const 0 :: s:n__natsFrom:cons:0:nil:null_snd nil :: s:n__natsFrom:cons:0:nil:null_snd u :: pair:const → s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → pair:const activate :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd head :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd tail :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd sel :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd afterNth :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd take :: s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd → s:n__natsFrom:cons:0:nil:null_snd const :: pair:const null_snd :: s:n__natsFrom:cons:0:nil:null_snd Rewrite Strategy: INNERMOST

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

0 => 0
nil => 1
const => 0
null_snd => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 1 }→ natsFrom(X) :|: z = 1 + X, X >= 0
afterNth(z, z') -{ 2 }→ snd(u(splitAt(N'', activate(XS'')), N'', X2, activate(XS''))) :|: z' = 1 + X2 + XS'', XS'' >= 0, N'' >= 0, X2 >= 0, z = 1 + N''
afterNth(z, z') -{ 1 }→ snd(0) :|: z' = XS, z = N, XS >= 0, N >= 0
afterNth(z, z') -{ 2 }→ snd(1 + 1 + XS) :|: z' = XS, z = 0, XS >= 0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
natsFrom(z) -{ 1 }→ 1 + N + (1 + (1 + N)) :|: z = N, N >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(N, XS))) :|: z' = XS, z = N, XS >= 0, N >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
splitAt(z, z') -{ 3 }→ u(splitAt(N, XS), N, X, XS) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(N, natsFrom(X')), N, X, natsFrom(X')) :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(N, natsFrom(X')), N, X, 1 + X') :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(N, 1 + X''), N, X, natsFrom(X'')) :|: z' = 1 + X + (1 + X''), z = 1 + N, X >= 0, X'' >= 0, N >= 0
splitAt(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
splitAt(z, z') -{ 1 }→ 1 + 1 + XS :|: z' = XS, z = 0, XS >= 0
tail(z) -{ 1 }→ activate(XS) :|: z = 1 + N + XS, XS >= 0, N >= 0
take(z, z') -{ 2 }→ fst(u(splitAt(N', activate(XS')), N', X1, activate(XS'))) :|: X1 >= 0, z = 1 + N', XS' >= 0, z' = 1 + X1 + XS', N' >= 0
take(z, z') -{ 1 }→ fst(0) :|: z' = XS, z = N, XS >= 0, N >= 0
take(z, z') -{ 2 }→ fst(1 + 1 + XS) :|: z' = XS, z = 0, XS >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
u(z, z', z'', z1) -{ 1 }→ 1 + (1 + activate(X) + YS) + ZS :|: z = 1 + YS + ZS, z'' = X, YS >= 0, X >= 0, z' = N, z1 = XS, ZS >= 0, XS >= 0, N >= 0

(11) InliningProof (UPPER BOUND(ID) transformation)

Inlined the following terminating rules on right-hand sides where appropriate:

natsFrom(z) -{ 1 }→ 1 + N + (1 + (1 + N)) :|: z = N, N >= 0
natsFrom(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
activate(z) -{ 1 }→ natsFrom(X) :|: z = 1 + X, X >= 0
activate(z) -{ 1 }→ X :|: X >= 0, z = X

(12) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z = 1 + X, X >= 0, X = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z' = XS, z = 0, XS >= 0, 1 + 1 + XS = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(N'', X), N'', X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, N'' >= 0, X2 >= 0, z = 1 + N'', X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 4 }→ snd(u(splitAt(N'', X), N'', X2, natsFrom(X'))) :|: z' = 1 + X2 + XS'', XS'' >= 0, N'' >= 0, X2 >= 0, z = 1 + N'', X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(N'', natsFrom(X)), N'', X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, N'' >= 0, X2 >= 0, z = 1 + N'', XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 4 }→ snd(u(splitAt(N'', natsFrom(X)), N'', X2, natsFrom(X'))) :|: z' = 1 + X2 + XS'', XS'' >= 0, N'' >= 0, X2 >= 0, z = 1 + N'', XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 2 }→ 0 :|: z' = XS, z = 0, XS >= 0, v0 >= 0, 1 + 1 + XS = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' = XS, z = N, XS >= 0, N >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
natsFrom(z) -{ 1 }→ 1 + N + (1 + (1 + N)) :|: z = N, N >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(N, XS))) :|: z' = XS, z = N, XS >= 0, N >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
splitAt(z, z') -{ 3 }→ u(splitAt(N, XS), N, X, XS) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(N, 1 + X''), N, X, 1 + X') :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(N, 1 + X''), N, X, 1 + X') :|: z' = 1 + X + (1 + X''), z = 1 + N, X >= 0, X'' >= 0, N >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(N, 1 + X''), N, X, 1 + X1) :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(N, 1 + X''), N, X, 1 + N' + (1 + (1 + N'))) :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(N, 1 + X''), N, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), z = 1 + N, X >= 0, X'' >= 0, N >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(N, 1 + N' + (1 + (1 + N'))), N, X, 1 + X') :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(N, 1 + N' + (1 + (1 + N'))), N, X, 1 + X'') :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(N, 1 + N' + (1 + (1 + N'))), N, X, 1 + N'' + (1 + (1 + N''))) :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
splitAt(z, z') -{ 1 }→ 1 + 1 + XS :|: z' = XS, z = 0, XS >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 2 }→ natsFrom(X) :|: z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z' = XS, z = 0, XS >= 0, 1 + 1 + XS = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(N', X), N', X1, X')) :|: X1 >= 0, z = 1 + N', XS' >= 0, z' = 1 + X1 + XS', N' >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 4 }→ fst(u(splitAt(N', X), N', X1, natsFrom(X'))) :|: X1 >= 0, z = 1 + N', XS' >= 0, z' = 1 + X1 + XS', N' >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(N', natsFrom(X)), N', X1, X')) :|: X1 >= 0, z = 1 + N', XS' >= 0, z' = 1 + X1 + XS', N' >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ 4 }→ fst(u(splitAt(N', natsFrom(X)), N', X1, natsFrom(X'))) :|: X1 >= 0, z = 1 + N', XS' >= 0, z' = 1 + X1 + XS', N' >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 1 }→ fst(0) :|: z' = XS, z = N, XS >= 0, N >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + X' + YS) + ZS :|: z = 1 + YS + ZS, z'' = X, YS >= 0, X >= 0, z' = N, z1 = XS, ZS >= 0, XS >= 0, N >= 0, X' >= 0, X = X'
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + natsFrom(X') + YS) + ZS :|: z = 1 + YS + ZS, z'' = X, YS >= 0, X >= 0, z' = N, z1 = XS, ZS >= 0, XS >= 0, N >= 0, X = 1 + X', X' >= 0

(13) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(14) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, natsFrom(X'))) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, natsFrom(X)), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, natsFrom(X)), z - 1, X2, natsFrom(X'))) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 2 }→ natsFrom(X) :|: z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, natsFrom(X'))) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, natsFrom(X)), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, natsFrom(X)), z - 1, X1, natsFrom(X'))) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 1 }→ fst(0) :|: z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + natsFrom(z'' - 1) + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0

(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

{ natsFrom } { snd } { fst } { activate } { head } { tail } { u } { splitAt } { take } { afterNth } { sel }

(16) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, natsFrom(X'))) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, natsFrom(X)), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, natsFrom(X)), z - 1, X2, natsFrom(X'))) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 2 }→ natsFrom(X) :|: z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, natsFrom(X'))) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, natsFrom(X)), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, natsFrom(X)), z - 1, X1, natsFrom(X'))) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 1 }→ fst(0) :|: z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + natsFrom(z'' - 1) + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {natsFrom}, {snd}, {fst}, {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: natsFrom
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3 + 2·z

(18) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, natsFrom(X'))) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, natsFrom(X)), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, natsFrom(X)), z - 1, X2, natsFrom(X'))) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 2 }→ natsFrom(X) :|: z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, natsFrom(X'))) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, natsFrom(X)), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, natsFrom(X)), z - 1, X1, natsFrom(X'))) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 1 }→ fst(0) :|: z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + natsFrom(z'' - 1) + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {natsFrom}, {snd}, {fst}, {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: ?, size: O(n1) [3 + 2·z]

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: natsFrom
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(20) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, natsFrom(X'))) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, natsFrom(X)), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, natsFrom(X)), z - 1, X2, natsFrom(X'))) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 2 }→ natsFrom(X) :|: z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, natsFrom(X'))) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, natsFrom(X)), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, natsFrom(X)), z - 1, X1, natsFrom(X'))) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 1 }→ fst(0) :|: z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + natsFrom(z'' - 1) + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {snd}, {fst}, {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z]

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(22) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ 1 }→ fst(0) :|: z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {snd}, {fst}, {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z]

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: snd
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z

(24) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ 1 }→ fst(0) :|: z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {snd}, {fst}, {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: ?, size: O(n1) [z]

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: snd
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(26) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ 1 }→ fst(0) :|: z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {fst}, {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z]

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(28) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ 1 }→ fst(0) :|: z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {fst}, {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z]

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: fst
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z

(30) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ 1 }→ fst(0) :|: z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {fst}, {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: ?, size: O(n1) [z]

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: fst
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(32) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ 1 }→ fst(0) :|: z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z]

(33) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(34) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z]

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + 2·z

(36) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: ?, size: O(n1) [1 + 2·z]

(37) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 5

(38) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z]

(39) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(40) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z]

(41) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: head
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z

(42) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: ?, size: O(n1) [z]

(43) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: head
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(44) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z]

(45) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(46) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z]

(47) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: tail
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 2·z

(48) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: ?, size: O(n1) [2·z]

(49) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: tail
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

(50) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z]

(51) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(52) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z]

(53) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: u
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 2 + z + 2·z''

(54) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {u}, {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: ?, size: O(n1) [2 + z + 2·z'']

(55) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: u
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

(56) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: O(1) [3], size: O(n1) [2 + z + 2·z'']

(57) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(58) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: O(1) [3], size: O(n1) [2 + z + 2·z'']

(59) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: splitAt
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

(60) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {splitAt}, {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: O(1) [3], size: O(n1) [2 + z + 2·z''] splitAt: runtime: ?, size: EXP

(61) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: splitAt
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + 66·z

(62) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ 4 }→ snd(u(splitAt(z - 1, X), z - 1, X2, X')) :|: z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, X), z - 1, X2, s7)) :|: s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 6 }→ snd(u(splitAt(z - 1, s4), z - 1, X2, s5)) :|: s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ 5 }→ snd(u(splitAt(z - 1, s6), z - 1, X2, X')) :|: s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 2 }→ head(snd(splitAt(z, z'))) :|: z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ 3 }→ u(splitAt(z - 1, XS), z - 1, X, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + X1) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + X''), z - 1, X, 1 + N' + (1 + (1 + N'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ 4 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ 5 }→ u(splitAt(z - 1, 1 + N' + (1 + (1 + N'))), z - 1, X, 1 + N'' + (1 + (1 + N''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
take(z, z') -{ 4 }→ fst(u(splitAt(z - 1, X), z - 1, X1, X')) :|: X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, X), z - 1, X1, s3)) :|: s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ 6 }→ fst(u(splitAt(z - 1, s''), z - 1, X1, s1)) :|: s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ 5 }→ fst(u(splitAt(z - 1, s2), z - 1, X1, X')) :|: s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: O(1) [3], size: O(n1) [2 + z + 2·z''] splitAt: runtime: O(n1) [1 + 66·z], size: EXP

(63) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(64) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ -55 + 66·z }→ s44 :|: s42 >= 0, s42 <= inf12, s43 >= 0, s43 <= 1 * s42 + 2 * X2 + 2, s44 >= 0, s44 <= 1 * s43, s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -56 + 66·z }→ s47 :|: s45 >= 0, s45 <= inf13, s46 >= 0, s46 <= 1 * s45 + 2 * X2 + 2, s47 >= 0, s47 <= 1 * s46, s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ -56 + 66·z }→ s50 :|: s48 >= 0, s48 <= inf14, s49 >= 0, s49 <= 1 * s48 + 2 * X2 + 2, s50 >= 0, s50 <= 1 * s49, s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -57 + 66·z }→ s53 :|: s51 >= 0, s51 <= inf15, s52 >= 0, s52 <= 1 * s51 + 2 * X2 + 2, s53 >= 0, s53 <= 1 * s52, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 5 + 66·z }→ s29 :|: s27 >= 0, s27 <= inf7, s28 >= 0, s28 <= 1 * s27, s29 >= 0, s29 <= 1 * s28, z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ -59 + 66·z }→ s10 :|: s9 >= 0, s9 <= inf, s10 >= 0, s10 <= 1 * s9 + 2 * X + 2, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ -57 + 66·z }→ s12 :|: s11 >= 0, s11 <= inf', s12 >= 0, s12 <= 1 * s11 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= 1 * s13 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -57 + 66·z }→ s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= 1 * s15 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= 1 * s17 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ -58 + 66·z }→ s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= 1 * s19 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= 1 * s21 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -58 + 66·z }→ s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= 1 * s23 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= 1 * s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ -55 + 66·z }→ s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= 1 * s30 + 2 * X1 + 2, s32 >= 0, s32 <= 1 * s31, s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ -56 + 66·z }→ s35 :|: s33 >= 0, s33 <= inf9, s34 >= 0, s34 <= 1 * s33 + 2 * X1 + 2, s35 >= 0, s35 <= 1 * s34, s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ -56 + 66·z }→ s38 :|: s36 >= 0, s36 <= inf10, s37 >= 0, s37 <= 1 * s36 + 2 * X1 + 2, s38 >= 0, s38 <= 1 * s37, s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ -57 + 66·z }→ s41 :|: s39 >= 0, s39 <= inf11, s40 >= 0, s40 <= 1 * s39 + 2 * X1 + 2, s41 >= 0, s41 <= 1 * s40, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: O(1) [3], size: O(n1) [2 + z + 2·z''] splitAt: runtime: O(n1) [1 + 66·z], size: EXP

(65) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: take
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

(66) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ -55 + 66·z }→ s44 :|: s42 >= 0, s42 <= inf12, s43 >= 0, s43 <= 1 * s42 + 2 * X2 + 2, s44 >= 0, s44 <= 1 * s43, s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -56 + 66·z }→ s47 :|: s45 >= 0, s45 <= inf13, s46 >= 0, s46 <= 1 * s45 + 2 * X2 + 2, s47 >= 0, s47 <= 1 * s46, s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ -56 + 66·z }→ s50 :|: s48 >= 0, s48 <= inf14, s49 >= 0, s49 <= 1 * s48 + 2 * X2 + 2, s50 >= 0, s50 <= 1 * s49, s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -57 + 66·z }→ s53 :|: s51 >= 0, s51 <= inf15, s52 >= 0, s52 <= 1 * s51 + 2 * X2 + 2, s53 >= 0, s53 <= 1 * s52, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 5 + 66·z }→ s29 :|: s27 >= 0, s27 <= inf7, s28 >= 0, s28 <= 1 * s27, s29 >= 0, s29 <= 1 * s28, z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ -59 + 66·z }→ s10 :|: s9 >= 0, s9 <= inf, s10 >= 0, s10 <= 1 * s9 + 2 * X + 2, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ -57 + 66·z }→ s12 :|: s11 >= 0, s11 <= inf', s12 >= 0, s12 <= 1 * s11 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= 1 * s13 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -57 + 66·z }→ s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= 1 * s15 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= 1 * s17 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ -58 + 66·z }→ s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= 1 * s19 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= 1 * s21 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -58 + 66·z }→ s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= 1 * s23 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= 1 * s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ -55 + 66·z }→ s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= 1 * s30 + 2 * X1 + 2, s32 >= 0, s32 <= 1 * s31, s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ -56 + 66·z }→ s35 :|: s33 >= 0, s33 <= inf9, s34 >= 0, s34 <= 1 * s33 + 2 * X1 + 2, s35 >= 0, s35 <= 1 * s34, s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ -56 + 66·z }→ s38 :|: s36 >= 0, s36 <= inf10, s37 >= 0, s37 <= 1 * s36 + 2 * X1 + 2, s38 >= 0, s38 <= 1 * s37, s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ -57 + 66·z }→ s41 :|: s39 >= 0, s39 <= inf11, s40 >= 0, s40 <= 1 * s39 + 2 * X1 + 2, s41 >= 0, s41 <= 1 * s40, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {take}, {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: O(1) [3], size: O(n1) [2 + z + 2·z''] splitAt: runtime: O(n1) [1 + 66·z], size: EXP take: runtime: ?, size: INF

(67) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: take
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 5 + 264·z

(68) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ -55 + 66·z }→ s44 :|: s42 >= 0, s42 <= inf12, s43 >= 0, s43 <= 1 * s42 + 2 * X2 + 2, s44 >= 0, s44 <= 1 * s43, s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -56 + 66·z }→ s47 :|: s45 >= 0, s45 <= inf13, s46 >= 0, s46 <= 1 * s45 + 2 * X2 + 2, s47 >= 0, s47 <= 1 * s46, s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ -56 + 66·z }→ s50 :|: s48 >= 0, s48 <= inf14, s49 >= 0, s49 <= 1 * s48 + 2 * X2 + 2, s50 >= 0, s50 <= 1 * s49, s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -57 + 66·z }→ s53 :|: s51 >= 0, s51 <= inf15, s52 >= 0, s52 <= 1 * s51 + 2 * X2 + 2, s53 >= 0, s53 <= 1 * s52, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 5 + 66·z }→ s29 :|: s27 >= 0, s27 <= inf7, s28 >= 0, s28 <= 1 * s27, s29 >= 0, s29 <= 1 * s28, z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ -59 + 66·z }→ s10 :|: s9 >= 0, s9 <= inf, s10 >= 0, s10 <= 1 * s9 + 2 * X + 2, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ -57 + 66·z }→ s12 :|: s11 >= 0, s11 <= inf', s12 >= 0, s12 <= 1 * s11 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= 1 * s13 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -57 + 66·z }→ s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= 1 * s15 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= 1 * s17 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ -58 + 66·z }→ s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= 1 * s19 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= 1 * s21 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -58 + 66·z }→ s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= 1 * s23 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= 1 * s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ -55 + 66·z }→ s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= 1 * s30 + 2 * X1 + 2, s32 >= 0, s32 <= 1 * s31, s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ -56 + 66·z }→ s35 :|: s33 >= 0, s33 <= inf9, s34 >= 0, s34 <= 1 * s33 + 2 * X1 + 2, s35 >= 0, s35 <= 1 * s34, s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ -56 + 66·z }→ s38 :|: s36 >= 0, s36 <= inf10, s37 >= 0, s37 <= 1 * s36 + 2 * X1 + 2, s38 >= 0, s38 <= 1 * s37, s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ -57 + 66·z }→ s41 :|: s39 >= 0, s39 <= inf11, s40 >= 0, s40 <= 1 * s39 + 2 * X1 + 2, s41 >= 0, s41 <= 1 * s40, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: O(1) [3], size: O(n1) [2 + z + 2·z''] splitAt: runtime: O(n1) [1 + 66·z], size: EXP take: runtime: O(n1) [5 + 264·z], size: INF

(69) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(70) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ -55 + 66·z }→ s44 :|: s42 >= 0, s42 <= inf12, s43 >= 0, s43 <= 1 * s42 + 2 * X2 + 2, s44 >= 0, s44 <= 1 * s43, s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -56 + 66·z }→ s47 :|: s45 >= 0, s45 <= inf13, s46 >= 0, s46 <= 1 * s45 + 2 * X2 + 2, s47 >= 0, s47 <= 1 * s46, s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ -56 + 66·z }→ s50 :|: s48 >= 0, s48 <= inf14, s49 >= 0, s49 <= 1 * s48 + 2 * X2 + 2, s50 >= 0, s50 <= 1 * s49, s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -57 + 66·z }→ s53 :|: s51 >= 0, s51 <= inf15, s52 >= 0, s52 <= 1 * s51 + 2 * X2 + 2, s53 >= 0, s53 <= 1 * s52, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 5 + 66·z }→ s29 :|: s27 >= 0, s27 <= inf7, s28 >= 0, s28 <= 1 * s27, s29 >= 0, s29 <= 1 * s28, z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ -59 + 66·z }→ s10 :|: s9 >= 0, s9 <= inf, s10 >= 0, s10 <= 1 * s9 + 2 * X + 2, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ -57 + 66·z }→ s12 :|: s11 >= 0, s11 <= inf', s12 >= 0, s12 <= 1 * s11 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= 1 * s13 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -57 + 66·z }→ s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= 1 * s15 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= 1 * s17 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ -58 + 66·z }→ s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= 1 * s19 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= 1 * s21 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -58 + 66·z }→ s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= 1 * s23 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= 1 * s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ -55 + 66·z }→ s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= 1 * s30 + 2 * X1 + 2, s32 >= 0, s32 <= 1 * s31, s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ -56 + 66·z }→ s35 :|: s33 >= 0, s33 <= inf9, s34 >= 0, s34 <= 1 * s33 + 2 * X1 + 2, s35 >= 0, s35 <= 1 * s34, s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ -56 + 66·z }→ s38 :|: s36 >= 0, s36 <= inf10, s37 >= 0, s37 <= 1 * s36 + 2 * X1 + 2, s38 >= 0, s38 <= 1 * s37, s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ -57 + 66·z }→ s41 :|: s39 >= 0, s39 <= inf11, s40 >= 0, s40 <= 1 * s39 + 2 * X1 + 2, s41 >= 0, s41 <= 1 * s40, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: O(1) [3], size: O(n1) [2 + z + 2·z''] splitAt: runtime: O(n1) [1 + 66·z], size: EXP take: runtime: O(n1) [5 + 264·z], size: INF

(71) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: afterNth
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

(72) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ -55 + 66·z }→ s44 :|: s42 >= 0, s42 <= inf12, s43 >= 0, s43 <= 1 * s42 + 2 * X2 + 2, s44 >= 0, s44 <= 1 * s43, s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -56 + 66·z }→ s47 :|: s45 >= 0, s45 <= inf13, s46 >= 0, s46 <= 1 * s45 + 2 * X2 + 2, s47 >= 0, s47 <= 1 * s46, s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ -56 + 66·z }→ s50 :|: s48 >= 0, s48 <= inf14, s49 >= 0, s49 <= 1 * s48 + 2 * X2 + 2, s50 >= 0, s50 <= 1 * s49, s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -57 + 66·z }→ s53 :|: s51 >= 0, s51 <= inf15, s52 >= 0, s52 <= 1 * s51 + 2 * X2 + 2, s53 >= 0, s53 <= 1 * s52, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 5 + 66·z }→ s29 :|: s27 >= 0, s27 <= inf7, s28 >= 0, s28 <= 1 * s27, s29 >= 0, s29 <= 1 * s28, z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ -59 + 66·z }→ s10 :|: s9 >= 0, s9 <= inf, s10 >= 0, s10 <= 1 * s9 + 2 * X + 2, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ -57 + 66·z }→ s12 :|: s11 >= 0, s11 <= inf', s12 >= 0, s12 <= 1 * s11 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= 1 * s13 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -57 + 66·z }→ s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= 1 * s15 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= 1 * s17 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ -58 + 66·z }→ s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= 1 * s19 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= 1 * s21 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -58 + 66·z }→ s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= 1 * s23 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= 1 * s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ -55 + 66·z }→ s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= 1 * s30 + 2 * X1 + 2, s32 >= 0, s32 <= 1 * s31, s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ -56 + 66·z }→ s35 :|: s33 >= 0, s33 <= inf9, s34 >= 0, s34 <= 1 * s33 + 2 * X1 + 2, s35 >= 0, s35 <= 1 * s34, s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ -56 + 66·z }→ s38 :|: s36 >= 0, s36 <= inf10, s37 >= 0, s37 <= 1 * s36 + 2 * X1 + 2, s38 >= 0, s38 <= 1 * s37, s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ -57 + 66·z }→ s41 :|: s39 >= 0, s39 <= inf11, s40 >= 0, s40 <= 1 * s39 + 2 * X1 + 2, s41 >= 0, s41 <= 1 * s40, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {afterNth}, {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: O(1) [3], size: O(n1) [2 + z + 2·z''] splitAt: runtime: O(n1) [1 + 66·z], size: EXP take: runtime: O(n1) [5 + 264·z], size: INF afterNth: runtime: ?, size: INF

(73) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: afterNth
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 6 + 264·z

(74) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ -55 + 66·z }→ s44 :|: s42 >= 0, s42 <= inf12, s43 >= 0, s43 <= 1 * s42 + 2 * X2 + 2, s44 >= 0, s44 <= 1 * s43, s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -56 + 66·z }→ s47 :|: s45 >= 0, s45 <= inf13, s46 >= 0, s46 <= 1 * s45 + 2 * X2 + 2, s47 >= 0, s47 <= 1 * s46, s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ -56 + 66·z }→ s50 :|: s48 >= 0, s48 <= inf14, s49 >= 0, s49 <= 1 * s48 + 2 * X2 + 2, s50 >= 0, s50 <= 1 * s49, s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -57 + 66·z }→ s53 :|: s51 >= 0, s51 <= inf15, s52 >= 0, s52 <= 1 * s51 + 2 * X2 + 2, s53 >= 0, s53 <= 1 * s52, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 5 + 66·z }→ s29 :|: s27 >= 0, s27 <= inf7, s28 >= 0, s28 <= 1 * s27, s29 >= 0, s29 <= 1 * s28, z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ -59 + 66·z }→ s10 :|: s9 >= 0, s9 <= inf, s10 >= 0, s10 <= 1 * s9 + 2 * X + 2, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ -57 + 66·z }→ s12 :|: s11 >= 0, s11 <= inf', s12 >= 0, s12 <= 1 * s11 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= 1 * s13 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -57 + 66·z }→ s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= 1 * s15 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= 1 * s17 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ -58 + 66·z }→ s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= 1 * s19 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= 1 * s21 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -58 + 66·z }→ s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= 1 * s23 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= 1 * s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ -55 + 66·z }→ s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= 1 * s30 + 2 * X1 + 2, s32 >= 0, s32 <= 1 * s31, s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ -56 + 66·z }→ s35 :|: s33 >= 0, s33 <= inf9, s34 >= 0, s34 <= 1 * s33 + 2 * X1 + 2, s35 >= 0, s35 <= 1 * s34, s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ -56 + 66·z }→ s38 :|: s36 >= 0, s36 <= inf10, s37 >= 0, s37 <= 1 * s36 + 2 * X1 + 2, s38 >= 0, s38 <= 1 * s37, s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ -57 + 66·z }→ s41 :|: s39 >= 0, s39 <= inf11, s40 >= 0, s40 <= 1 * s39 + 2 * X1 + 2, s41 >= 0, s41 <= 1 * s40, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: O(1) [3], size: O(n1) [2 + z + 2·z''] splitAt: runtime: O(n1) [1 + 66·z], size: EXP take: runtime: O(n1) [5 + 264·z], size: INF afterNth: runtime: O(n1) [6 + 264·z], size: INF

(75) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(76) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ -55 + 66·z }→ s44 :|: s42 >= 0, s42 <= inf12, s43 >= 0, s43 <= 1 * s42 + 2 * X2 + 2, s44 >= 0, s44 <= 1 * s43, s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -56 + 66·z }→ s47 :|: s45 >= 0, s45 <= inf13, s46 >= 0, s46 <= 1 * s45 + 2 * X2 + 2, s47 >= 0, s47 <= 1 * s46, s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ -56 + 66·z }→ s50 :|: s48 >= 0, s48 <= inf14, s49 >= 0, s49 <= 1 * s48 + 2 * X2 + 2, s50 >= 0, s50 <= 1 * s49, s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -57 + 66·z }→ s53 :|: s51 >= 0, s51 <= inf15, s52 >= 0, s52 <= 1 * s51 + 2 * X2 + 2, s53 >= 0, s53 <= 1 * s52, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 5 + 66·z }→ s29 :|: s27 >= 0, s27 <= inf7, s28 >= 0, s28 <= 1 * s27, s29 >= 0, s29 <= 1 * s28, z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ -59 + 66·z }→ s10 :|: s9 >= 0, s9 <= inf, s10 >= 0, s10 <= 1 * s9 + 2 * X + 2, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ -57 + 66·z }→ s12 :|: s11 >= 0, s11 <= inf', s12 >= 0, s12 <= 1 * s11 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= 1 * s13 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -57 + 66·z }→ s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= 1 * s15 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= 1 * s17 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ -58 + 66·z }→ s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= 1 * s19 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= 1 * s21 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -58 + 66·z }→ s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= 1 * s23 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= 1 * s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ -55 + 66·z }→ s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= 1 * s30 + 2 * X1 + 2, s32 >= 0, s32 <= 1 * s31, s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ -56 + 66·z }→ s35 :|: s33 >= 0, s33 <= inf9, s34 >= 0, s34 <= 1 * s33 + 2 * X1 + 2, s35 >= 0, s35 <= 1 * s34, s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ -56 + 66·z }→ s38 :|: s36 >= 0, s36 <= inf10, s37 >= 0, s37 <= 1 * s36 + 2 * X1 + 2, s38 >= 0, s38 <= 1 * s37, s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ -57 + 66·z }→ s41 :|: s39 >= 0, s39 <= inf11, s40 >= 0, s40 <= 1 * s39 + 2 * X1 + 2, s41 >= 0, s41 <= 1 * s40, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: O(1) [3], size: O(n1) [2 + z + 2·z''] splitAt: runtime: O(n1) [1 + 66·z], size: EXP take: runtime: O(n1) [5 + 264·z], size: INF afterNth: runtime: O(n1) [6 + 264·z], size: INF

(77) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: sel
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

(78) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ -55 + 66·z }→ s44 :|: s42 >= 0, s42 <= inf12, s43 >= 0, s43 <= 1 * s42 + 2 * X2 + 2, s44 >= 0, s44 <= 1 * s43, s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -56 + 66·z }→ s47 :|: s45 >= 0, s45 <= inf13, s46 >= 0, s46 <= 1 * s45 + 2 * X2 + 2, s47 >= 0, s47 <= 1 * s46, s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ -56 + 66·z }→ s50 :|: s48 >= 0, s48 <= inf14, s49 >= 0, s49 <= 1 * s48 + 2 * X2 + 2, s50 >= 0, s50 <= 1 * s49, s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -57 + 66·z }→ s53 :|: s51 >= 0, s51 <= inf15, s52 >= 0, s52 <= 1 * s51 + 2 * X2 + 2, s53 >= 0, s53 <= 1 * s52, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 5 + 66·z }→ s29 :|: s27 >= 0, s27 <= inf7, s28 >= 0, s28 <= 1 * s27, s29 >= 0, s29 <= 1 * s28, z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ -59 + 66·z }→ s10 :|: s9 >= 0, s9 <= inf, s10 >= 0, s10 <= 1 * s9 + 2 * X + 2, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ -57 + 66·z }→ s12 :|: s11 >= 0, s11 <= inf', s12 >= 0, s12 <= 1 * s11 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= 1 * s13 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -57 + 66·z }→ s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= 1 * s15 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= 1 * s17 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ -58 + 66·z }→ s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= 1 * s19 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= 1 * s21 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -58 + 66·z }→ s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= 1 * s23 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= 1 * s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ -55 + 66·z }→ s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= 1 * s30 + 2 * X1 + 2, s32 >= 0, s32 <= 1 * s31, s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ -56 + 66·z }→ s35 :|: s33 >= 0, s33 <= inf9, s34 >= 0, s34 <= 1 * s33 + 2 * X1 + 2, s35 >= 0, s35 <= 1 * s34, s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ -56 + 66·z }→ s38 :|: s36 >= 0, s36 <= inf10, s37 >= 0, s37 <= 1 * s36 + 2 * X1 + 2, s38 >= 0, s38 <= 1 * s37, s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ -57 + 66·z }→ s41 :|: s39 >= 0, s39 <= inf11, s40 >= 0, s40 <= 1 * s39 + 2 * X1 + 2, s41 >= 0, s41 <= 1 * s40, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed: {sel}
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: O(1) [3], size: O(n1) [2 + z + 2·z''] splitAt: runtime: O(n1) [1 + 66·z], size: EXP take: runtime: O(n1) [5 + 264·z], size: INF afterNth: runtime: O(n1) [6 + 264·z], size: INF sel: runtime: ?, size: INF

(79) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: sel
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 5 + 66·z

(80) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + N + (1 + (1 + N)) :|: z - 1 >= 0, z - 1 = N, N >= 0
afterNth(z, z') -{ 3 }→ YS :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
afterNth(z, z') -{ -55 + 66·z }→ s44 :|: s42 >= 0, s42 <= inf12, s43 >= 0, s43 <= 1 * s42 + 2 * X2 + 2, s44 >= 0, s44 <= 1 * s43, s4 >= 0, s4 <= 2 * X + 3, s5 >= 0, s5 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -56 + 66·z }→ s47 :|: s45 >= 0, s45 <= inf13, s46 >= 0, s46 <= 1 * s45 + 2 * X2 + 2, s47 >= 0, s47 <= 1 * s46, s6 >= 0, s6 <= 2 * X + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, XS'' = 1 + X, X >= 0, X' >= 0, XS'' = X'
afterNth(z, z') -{ -56 + 66·z }→ s50 :|: s48 >= 0, s48 <= inf14, s49 >= 0, s49 <= 1 * s48 + 2 * X2 + 2, s50 >= 0, s50 <= 1 * s49, s7 >= 0, s7 <= 2 * X' + 3, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, XS'' = 1 + X', X' >= 0
afterNth(z, z') -{ -57 + 66·z }→ s53 :|: s51 >= 0, s51 <= inf15, s52 >= 0, s52 <= 1 * s51 + 2 * X2 + 2, s53 >= 0, s53 <= 1 * s52, z' = 1 + X2 + XS'', XS'' >= 0, z - 1 >= 0, X2 >= 0, X >= 0, XS'' = X, X' >= 0, XS'' = X'
afterNth(z, z') -{ 2 }→ 0 :|: z = 0, z' >= 0, v0 >= 0, 1 + 1 + z' = v0
afterNth(z, z') -{ 1 }→ 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0
fst(z) -{ 1 }→ XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
head(z) -{ 1 }→ N :|: z = 1 + N + XS, XS >= 0, N >= 0
natsFrom(z) -{ 1 }→ 1 + z :|: z >= 0
natsFrom(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 5 + 66·z }→ s29 :|: s27 >= 0, s27 <= inf7, s28 >= 0, s28 <= 1 * s27, s29 >= 0, s29 <= 1 * s28, z' >= 0, z >= 0
snd(z) -{ 1 }→ YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0
snd(z) -{ 0 }→ 0 :|: z >= 0
splitAt(z, z') -{ -59 + 66·z }→ s10 :|: s9 >= 0, s9 <= inf, s10 >= 0, s10 <= 1 * s9 + 2 * X + 2, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
splitAt(z, z') -{ -57 + 66·z }→ s12 :|: s11 >= 0, s11 <= inf', s12 >= 0, s12 <= 1 * s11 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X' = N'', N'' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= 1 * s13 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -57 + 66·z }→ s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= 1 * s15 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X' = N', N' >= 0
splitAt(z, z') -{ -57 + 66·z }→ s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= 1 * s17 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'', X1 >= 0, X' = X1
splitAt(z, z') -{ -58 + 66·z }→ s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= 1 * s19 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= 1 * s21 + 2 * X + 2, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
splitAt(z, z') -{ -58 + 66·z }→ s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= 1 * s23 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X'' = N', N' >= 0
splitAt(z, z') -{ -58 + 66·z }→ s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= 1 * s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
splitAt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
splitAt(z, z') -{ 1 }→ 1 + 1 + z' :|: z = 0, z' >= 0
tail(z) -{ 2 }→ X :|: z = 1 + N + XS, XS >= 0, N >= 0, X >= 0, XS = X
tail(z) -{ 3 }→ s' :|: s' >= 0, s' <= 2 * X + 3, z = 1 + N + XS, XS >= 0, N >= 0, XS = 1 + X, X >= 0
take(z, z') -{ 3 }→ XS' :|: z = 0, z' >= 0, 1 + 1 + z' = 1 + XS' + YS, YS >= 0, XS' >= 0
take(z, z') -{ -55 + 66·z }→ s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= 1 * s30 + 2 * X1 + 2, s32 >= 0, s32 <= 1 * s31, s'' >= 0, s'' <= 2 * X + 3, s1 >= 0, s1 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, XS' = 1 + X', X' >= 0
take(z, z') -{ -56 + 66·z }→ s35 :|: s33 >= 0, s33 <= inf9, s34 >= 0, s34 <= 1 * s33 + 2 * X1 + 2, s35 >= 0, s35 <= 1 * s34, s2 >= 0, s2 <= 2 * X + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, XS' = 1 + X, X >= 0, X' >= 0, XS' = X'
take(z, z') -{ -56 + 66·z }→ s38 :|: s36 >= 0, s36 <= inf10, s37 >= 0, s37 <= 1 * s36 + 2 * X1 + 2, s38 >= 0, s38 <= 1 * s37, s3 >= 0, s3 <= 2 * X' + 3, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, XS' = 1 + X', X' >= 0
take(z, z') -{ -57 + 66·z }→ s41 :|: s39 >= 0, s39 <= inf11, s40 >= 0, s40 <= 1 * s39 + 2 * X1 + 2, s41 >= 0, s41 <= 1 * s40, X1 >= 0, XS' >= 0, z' = 1 + X1 + XS', z - 1 >= 0, X >= 0, XS' = X, X' >= 0, XS' = X'
take(z, z') -{ 2 }→ s8 :|: s8 >= 0, s8 <= 1 * 0, z' >= 0, z >= 0
u(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
u(z, z', z'', z1) -{ 3 }→ 1 + (1 + s + YS) + ZS :|: s >= 0, s <= 2 * (z'' - 1) + 3, z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0, z'' - 1 >= 0
u(z, z', z'', z1) -{ 2 }→ 1 + (1 + z'' + YS) + ZS :|: z = 1 + YS + ZS, YS >= 0, z'' >= 0, ZS >= 0, z1 >= 0, z' >= 0

Function symbols to be analyzed:
Previous analysis results are:

natsFrom: runtime: O(1) [1], size: O(n1) [3 + 2·z] snd: runtime: O(1) [1], size: O(n1) [z] fst: runtime: O(1) [1], size: O(n1) [z] activate: runtime: O(1) [5], size: O(n1) [1 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] tail: runtime: O(1) [3], size: O(n1) [2·z] u: runtime: O(1) [3], size: O(n1) [2 + z + 2·z''] splitAt: runtime: O(n1) [1 + 66·z], size: EXP take: runtime: O(n1) [5 + 264·z], size: INF afterNth: runtime: O(n1) [6 + 264·z], size: INF sel: runtime: O(n1) [5 + 66·z], size: INF

(81) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(82) BOUNDS(1, n^1)