WORST_CASE(?,O(n^1)) proof of input_hsm0wWezvu.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). (0) CpxTRS (1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxWeightedTrs (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTypedWeightedTrs (5) CompletionProof [UPPER BOUND(ID), 0 ms] (6) CpxTypedWeightedCompleteTrs (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (10) CpxRNTS (11) InliningProof [UPPER BOUND(ID), 828 ms] (12) CpxRNTS (13) SimplificationProof [BOTH BOUNDS(ID, ID), 10 ms] (14) CpxRNTS (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 227 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 46 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 141 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 54 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 129 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 43 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 7 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 13 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 138 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 42 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 202 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 54 ms] (52) CpxRNTS (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 406 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 85 ms] (58) CpxRNTS (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 1665 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 414 ms] (64) CpxRNTS (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 1038 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 8 ms] (70) CpxRNTS (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (72) CpxRNTS (73) IntTrsBoundProof [UPPER BOUND(ID), 1112 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 31 ms] (76) CpxRNTS (77) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (78) CpxRNTS (79) IntTrsBoundProof [UPPER BOUND(ID), 628 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 62 ms] (82) CpxRNTS (83) FinalProof [FINISHED, 0 ms] (84) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-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 S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative 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)) 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)) 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_1 head_1 tail_1 sel_2 take_2 (c) The following functions are completely defined: splitAt_2 activate_1 afterNth_2 natsFrom_1 snd_1 u_4 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)) 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)) 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)) Inlined the following terminating rules on right-hand sides where appropriate: natsFrom(z) -{ 1 }-> 1 + X :|: X >= 0, z = X natsFrom(z) -{ 1 }-> 1 + N + (1 + (1 + N)) :|: z = N, N >= 0 fst(z) -{ 1 }-> XS :|: z = 1 + XS + YS, YS >= 0, XS >= 0 snd(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 snd(z) -{ 1 }-> YS :|: z = 1 + XS + YS, YS >= 0, XS >= 0 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)) 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)) 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) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (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} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: natsFrom after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z ---------------------------------------- (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: {natsFrom}, {snd}, {fst}, {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel} Previous analysis results are: natsFrom: runtime: ?, size: O(n^1) [3 + 2*z] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: natsFrom after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (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') -{ 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(n^1) [3 + 2*z] ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (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(n^1) [3 + 2*z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: snd after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (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: {snd}, {fst}, {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel} Previous analysis results are: natsFrom: runtime: O(1) [1], size: O(n^1) [3 + 2*z] snd: runtime: ?, size: O(n^1) [z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: snd after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: fst after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (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: {fst}, {activate}, {head}, {tail}, {u}, {splitAt}, {take}, {afterNth}, {sel} Previous analysis results are: natsFrom: runtime: O(1) [1], size: O(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: ?, size: O(n^1) [z] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: fst after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (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') -{ 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: activate after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 2*z ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: ?, size: O(n^1) [1 + 2*z] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: activate after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 5 ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: head after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: ?, size: O(n^1) [z] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: head after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (47) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: tail after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: tail after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] ---------------------------------------- (53) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: u after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z + 2*z'' ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: ?, size: O(n^1) [2 + z + 2*z''] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: u after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z''] ---------------------------------------- (59) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z''] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: splitAt after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z''] splitAt: runtime: ?, size: EXP ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: splitAt after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 66*z ---------------------------------------- (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') -{ 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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z''] splitAt: runtime: O(n^1) [1 + 66*z], size: EXP ---------------------------------------- (65) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (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 <= s42 + 2 * X2 + 2, s44 >= 0, s44 <= 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 <= s45 + 2 * X2 + 2, s47 >= 0, s47 <= 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 <= s48 + 2 * X2 + 2, s50 >= 0, s50 <= 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 <= s51 + 2 * X2 + 2, s53 >= 0, s53 <= 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 <= s27, s29 >= 0, s29 <= 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 <= 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 <= s11 + 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') -{ -57 + 66*z }-> s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= s13 + 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 }-> s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= s15 + 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 }-> s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= s17 + 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') -{ -58 + 66*z }-> s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= s19 + 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 }-> s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= s21 + 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 }-> s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= s23 + 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 }-> s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 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') -{ -55 + 66*z }-> s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= s30 + 2 * X1 + 2, s32 >= 0, s32 <= 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 <= s33 + 2 * X1 + 2, s35 >= 0, s35 <= 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 <= s36 + 2 * X1 + 2, s38 >= 0, s38 <= 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 <= s39 + 2 * X1 + 2, s41 >= 0, s41 <= 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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z''] splitAt: runtime: O(n^1) [1 + 66*z], size: EXP ---------------------------------------- (67) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: take after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (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 <= s42 + 2 * X2 + 2, s44 >= 0, s44 <= 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 <= s45 + 2 * X2 + 2, s47 >= 0, s47 <= 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 <= s48 + 2 * X2 + 2, s50 >= 0, s50 <= 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 <= s51 + 2 * X2 + 2, s53 >= 0, s53 <= 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 <= s27, s29 >= 0, s29 <= 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 <= 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 <= s11 + 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') -{ -57 + 66*z }-> s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= s13 + 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 }-> s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= s15 + 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 }-> s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= s17 + 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') -{ -58 + 66*z }-> s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= s19 + 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 }-> s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= s21 + 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 }-> s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= s23 + 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 }-> s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 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') -{ -55 + 66*z }-> s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= s30 + 2 * X1 + 2, s32 >= 0, s32 <= 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 <= s33 + 2 * X1 + 2, s35 >= 0, s35 <= 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 <= s36 + 2 * X1 + 2, s38 >= 0, s38 <= 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 <= s39 + 2 * X1 + 2, s41 >= 0, s41 <= 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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z''] splitAt: runtime: O(n^1) [1 + 66*z], size: EXP take: runtime: ?, size: INF ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: take after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 5 + 264*z ---------------------------------------- (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 <= s42 + 2 * X2 + 2, s44 >= 0, s44 <= 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 <= s45 + 2 * X2 + 2, s47 >= 0, s47 <= 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 <= s48 + 2 * X2 + 2, s50 >= 0, s50 <= 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 <= s51 + 2 * X2 + 2, s53 >= 0, s53 <= 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 <= s27, s29 >= 0, s29 <= 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 <= 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 <= s11 + 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') -{ -57 + 66*z }-> s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= s13 + 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 }-> s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= s15 + 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 }-> s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= s17 + 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') -{ -58 + 66*z }-> s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= s19 + 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 }-> s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= s21 + 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 }-> s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= s23 + 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 }-> s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 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') -{ -55 + 66*z }-> s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= s30 + 2 * X1 + 2, s32 >= 0, s32 <= 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 <= s33 + 2 * X1 + 2, s35 >= 0, s35 <= 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 <= s36 + 2 * X1 + 2, s38 >= 0, s38 <= 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 <= s39 + 2 * X1 + 2, s41 >= 0, s41 <= 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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z''] splitAt: runtime: O(n^1) [1 + 66*z], size: EXP take: runtime: O(n^1) [5 + 264*z], size: INF ---------------------------------------- (71) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (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 <= s42 + 2 * X2 + 2, s44 >= 0, s44 <= 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 <= s45 + 2 * X2 + 2, s47 >= 0, s47 <= 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 <= s48 + 2 * X2 + 2, s50 >= 0, s50 <= 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 <= s51 + 2 * X2 + 2, s53 >= 0, s53 <= 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 <= s27, s29 >= 0, s29 <= 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 <= 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 <= s11 + 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') -{ -57 + 66*z }-> s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= s13 + 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 }-> s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= s15 + 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 }-> s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= s17 + 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') -{ -58 + 66*z }-> s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= s19 + 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 }-> s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= s21 + 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 }-> s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= s23 + 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 }-> s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 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') -{ -55 + 66*z }-> s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= s30 + 2 * X1 + 2, s32 >= 0, s32 <= 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 <= s33 + 2 * X1 + 2, s35 >= 0, s35 <= 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 <= s36 + 2 * X1 + 2, s38 >= 0, s38 <= 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 <= s39 + 2 * X1 + 2, s41 >= 0, s41 <= 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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z''] splitAt: runtime: O(n^1) [1 + 66*z], size: EXP take: runtime: O(n^1) [5 + 264*z], size: INF ---------------------------------------- (73) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: afterNth after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (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 <= s42 + 2 * X2 + 2, s44 >= 0, s44 <= 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 <= s45 + 2 * X2 + 2, s47 >= 0, s47 <= 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 <= s48 + 2 * X2 + 2, s50 >= 0, s50 <= 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 <= s51 + 2 * X2 + 2, s53 >= 0, s53 <= 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 <= s27, s29 >= 0, s29 <= 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 <= 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 <= s11 + 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') -{ -57 + 66*z }-> s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= s13 + 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 }-> s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= s15 + 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 }-> s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= s17 + 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') -{ -58 + 66*z }-> s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= s19 + 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 }-> s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= s21 + 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 }-> s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= s23 + 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 }-> s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 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') -{ -55 + 66*z }-> s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= s30 + 2 * X1 + 2, s32 >= 0, s32 <= 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 <= s33 + 2 * X1 + 2, s35 >= 0, s35 <= 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 <= s36 + 2 * X1 + 2, s38 >= 0, s38 <= 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 <= s39 + 2 * X1 + 2, s41 >= 0, s41 <= 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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z''] splitAt: runtime: O(n^1) [1 + 66*z], size: EXP take: runtime: O(n^1) [5 + 264*z], size: INF afterNth: runtime: ?, size: INF ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: afterNth after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 6 + 264*z ---------------------------------------- (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 <= s42 + 2 * X2 + 2, s44 >= 0, s44 <= 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 <= s45 + 2 * X2 + 2, s47 >= 0, s47 <= 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 <= s48 + 2 * X2 + 2, s50 >= 0, s50 <= 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 <= s51 + 2 * X2 + 2, s53 >= 0, s53 <= 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 <= s27, s29 >= 0, s29 <= 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 <= 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 <= s11 + 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') -{ -57 + 66*z }-> s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= s13 + 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 }-> s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= s15 + 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 }-> s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= s17 + 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') -{ -58 + 66*z }-> s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= s19 + 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 }-> s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= s21 + 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 }-> s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= s23 + 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 }-> s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 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') -{ -55 + 66*z }-> s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= s30 + 2 * X1 + 2, s32 >= 0, s32 <= 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 <= s33 + 2 * X1 + 2, s35 >= 0, s35 <= 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 <= s36 + 2 * X1 + 2, s38 >= 0, s38 <= 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 <= s39 + 2 * X1 + 2, s41 >= 0, s41 <= 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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z''] splitAt: runtime: O(n^1) [1 + 66*z], size: EXP take: runtime: O(n^1) [5 + 264*z], size: INF afterNth: runtime: O(n^1) [6 + 264*z], size: INF ---------------------------------------- (77) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (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 <= s42 + 2 * X2 + 2, s44 >= 0, s44 <= 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 <= s45 + 2 * X2 + 2, s47 >= 0, s47 <= 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 <= s48 + 2 * X2 + 2, s50 >= 0, s50 <= 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 <= s51 + 2 * X2 + 2, s53 >= 0, s53 <= 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 <= s27, s29 >= 0, s29 <= 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 <= 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 <= s11 + 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') -{ -57 + 66*z }-> s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= s13 + 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 }-> s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= s15 + 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 }-> s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= s17 + 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') -{ -58 + 66*z }-> s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= s19 + 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 }-> s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= s21 + 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 }-> s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= s23 + 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 }-> s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 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') -{ -55 + 66*z }-> s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= s30 + 2 * X1 + 2, s32 >= 0, s32 <= 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 <= s33 + 2 * X1 + 2, s35 >= 0, s35 <= 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 <= s36 + 2 * X1 + 2, s38 >= 0, s38 <= 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 <= s39 + 2 * X1 + 2, s41 >= 0, s41 <= 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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z''] splitAt: runtime: O(n^1) [1 + 66*z], size: EXP take: runtime: O(n^1) [5 + 264*z], size: INF afterNth: runtime: O(n^1) [6 + 264*z], size: INF ---------------------------------------- (79) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: sel after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (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 <= s42 + 2 * X2 + 2, s44 >= 0, s44 <= 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 <= s45 + 2 * X2 + 2, s47 >= 0, s47 <= 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 <= s48 + 2 * X2 + 2, s50 >= 0, s50 <= 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 <= s51 + 2 * X2 + 2, s53 >= 0, s53 <= 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 <= s27, s29 >= 0, s29 <= 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 <= 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 <= s11 + 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') -{ -57 + 66*z }-> s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= s13 + 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 }-> s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= s15 + 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 }-> s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= s17 + 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') -{ -58 + 66*z }-> s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= s19 + 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 }-> s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= s21 + 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 }-> s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= s23 + 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 }-> s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 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') -{ -55 + 66*z }-> s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= s30 + 2 * X1 + 2, s32 >= 0, s32 <= 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 <= s33 + 2 * X1 + 2, s35 >= 0, s35 <= 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 <= s36 + 2 * X1 + 2, s38 >= 0, s38 <= 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 <= s39 + 2 * X1 + 2, s41 >= 0, s41 <= 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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z''] splitAt: runtime: O(n^1) [1 + 66*z], size: EXP take: runtime: O(n^1) [5 + 264*z], size: INF afterNth: runtime: O(n^1) [6 + 264*z], size: INF sel: runtime: ?, size: INF ---------------------------------------- (81) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: sel after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 5 + 66*z ---------------------------------------- (82) 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 <= s42 + 2 * X2 + 2, s44 >= 0, s44 <= 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 <= s45 + 2 * X2 + 2, s47 >= 0, s47 <= 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 <= s48 + 2 * X2 + 2, s50 >= 0, s50 <= 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 <= s51 + 2 * X2 + 2, s53 >= 0, s53 <= 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 <= s27, s29 >= 0, s29 <= 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 <= 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 <= s11 + 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') -{ -57 + 66*z }-> s14 :|: s13 >= 0, s13 <= inf'', s14 >= 0, s14 <= s13 + 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 }-> s16 :|: s15 >= 0, s15 <= inf1, s16 >= 0, s16 <= s15 + 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 }-> s18 :|: s17 >= 0, s17 <= inf2, s18 >= 0, s18 <= s17 + 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') -{ -58 + 66*z }-> s20 :|: s19 >= 0, s19 <= inf3, s20 >= 0, s20 <= s19 + 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 }-> s22 :|: s21 >= 0, s21 <= inf4, s22 >= 0, s22 <= s21 + 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 }-> s24 :|: s23 >= 0, s23 <= inf5, s24 >= 0, s24 <= s23 + 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 }-> s26 :|: s25 >= 0, s25 <= inf6, s26 >= 0, s26 <= s25 + 2 * X + 2, z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 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') -{ -55 + 66*z }-> s32 :|: s30 >= 0, s30 <= inf8, s31 >= 0, s31 <= s30 + 2 * X1 + 2, s32 >= 0, s32 <= 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 <= s33 + 2 * X1 + 2, s35 >= 0, s35 <= 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 <= s36 + 2 * X1 + 2, s38 >= 0, s38 <= 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 <= s39 + 2 * X1 + 2, s41 >= 0, s41 <= 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 <= 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(n^1) [3 + 2*z] snd: runtime: O(1) [1], size: O(n^1) [z] fst: runtime: O(1) [1], size: O(n^1) [z] activate: runtime: O(1) [5], size: O(n^1) [1 + 2*z] head: runtime: O(1) [1], size: O(n^1) [z] tail: runtime: O(1) [3], size: O(n^1) [2*z] u: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z''] splitAt: runtime: O(n^1) [1 + 66*z], size: EXP take: runtime: O(n^1) [5 + 264*z], size: INF afterNth: runtime: O(n^1) [6 + 264*z], size: INF sel: runtime: O(n^1) [5 + 66*z], size: INF ---------------------------------------- (83) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (84) BOUNDS(1, n^1)