WORST_CASE(Omega(n^1),O(n^2)) proof of /export/starexec/sandbox/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 851 ms] (2) CpxRelTRS (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 14 ms] (12) CpxRNTS (13) SimplificationProof [BOTH BOUNDS(ID, ID), 11 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), 246 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 108 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 417 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 116 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 294 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 167 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 199 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 5 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 190 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 54 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 322 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 64 ms] (52) CpxRNTS (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 9707 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 3783 ms] (58) CpxRNTS (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 1776 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 796 ms] (64) CpxRNTS (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 418 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (70) CpxRNTS (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (72) CpxRNTS (73) IntTrsBoundProof [UPPER BOUND(ID), 117 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 22 ms] (76) CpxRNTS (77) FinalProof [FINISHED, 0 ms] (78) BOUNDS(1, n^2) (79) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (80) TRS for Loop Detection (81) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (82) BEST (83) proven lower bound (84) LowerBoundPropagationProof [FINISHED, 0 ms] (85) BOUNDS(n^1, INF) (86) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: HALF(0) -> c HALF(s(0)) -> c1 HALF(s(s(z0))) -> c2(HALF(z0)) LE(0, z0) -> c3 LE(s(z0), 0) -> c4 LE(s(z0), s(z1)) -> c5(LE(z0, z1)) INC(0) -> c6 INC(s(z0)) -> c7(INC(z0)) LOG(z0) -> c8(LOG2(z0, 0)) LOG2(z0, z1) -> c9(IF(le(z0, s(0)), z0, inc(z1)), LE(z0, s(0))) LOG2(z0, z1) -> c10(IF(le(z0, s(0)), z0, inc(z1)), INC(z1)) IF(true, z0, s(z1)) -> c11 IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0)) The (relative) TRS S consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) inc(0) -> 0 inc(s(z0)) -> s(inc(z0)) log(z0) -> log2(z0, 0) log2(z0, z1) -> if(le(z0, s(0)), z0, inc(z1)) if(true, z0, s(z1)) -> z1 if(false, z0, z1) -> log2(half(z0), z1) Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: HALF(0) -> c HALF(s(0)) -> c1 HALF(s(s(z0))) -> c2(HALF(z0)) LE(0, z0) -> c3 LE(s(z0), 0) -> c4 LE(s(z0), s(z1)) -> c5(LE(z0, z1)) INC(0) -> c6 INC(s(z0)) -> c7(INC(z0)) LOG(z0) -> c8(LOG2(z0, 0)) LOG2(z0, z1) -> c9(IF(le(z0, s(0)), z0, inc(z1)), LE(z0, s(0))) LOG2(z0, z1) -> c10(IF(le(z0, s(0)), z0, inc(z1)), INC(z1)) IF(true, z0, s(z1)) -> c11 IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0)) The (relative) TRS S consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) inc(0) -> 0 inc(s(z0)) -> s(inc(z0)) log(z0) -> log2(z0, 0) log2(z0, z1) -> if(le(z0, s(0)), z0, inc(z1)) if(true, z0, s(z1)) -> z1 if(false, z0, z1) -> log2(half(z0), z1) Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: HALF(0) -> c [1] HALF(s(0)) -> c1 [1] HALF(s(s(z0))) -> c2(HALF(z0)) [1] LE(0, z0) -> c3 [1] LE(s(z0), 0) -> c4 [1] LE(s(z0), s(z1)) -> c5(LE(z0, z1)) [1] INC(0) -> c6 [1] INC(s(z0)) -> c7(INC(z0)) [1] LOG(z0) -> c8(LOG2(z0, 0)) [1] LOG2(z0, z1) -> c9(IF(le(z0, s(0)), z0, inc(z1)), LE(z0, s(0))) [1] LOG2(z0, z1) -> c10(IF(le(z0, s(0)), z0, inc(z1)), INC(z1)) [1] IF(true, z0, s(z1)) -> c11 [1] IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0)) [1] half(0) -> 0 [0] half(s(0)) -> 0 [0] half(s(s(z0))) -> s(half(z0)) [0] le(0, z0) -> true [0] le(s(z0), 0) -> false [0] le(s(z0), s(z1)) -> le(z0, z1) [0] inc(0) -> 0 [0] inc(s(z0)) -> s(inc(z0)) [0] log(z0) -> log2(z0, 0) [0] log2(z0, z1) -> if(le(z0, s(0)), z0, inc(z1)) [0] if(true, z0, s(z1)) -> z1 [0] if(false, z0, z1) -> log2(half(z0), z1) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: HALF(0) -> c [1] HALF(s(0)) -> c1 [1] HALF(s(s(z0))) -> c2(HALF(z0)) [1] LE(0, z0) -> c3 [1] LE(s(z0), 0) -> c4 [1] LE(s(z0), s(z1)) -> c5(LE(z0, z1)) [1] INC(0) -> c6 [1] INC(s(z0)) -> c7(INC(z0)) [1] LOG(z0) -> c8(LOG2(z0, 0)) [1] LOG2(z0, z1) -> c9(IF(le(z0, s(0)), z0, inc(z1)), LE(z0, s(0))) [1] LOG2(z0, z1) -> c10(IF(le(z0, s(0)), z0, inc(z1)), INC(z1)) [1] IF(true, z0, s(z1)) -> c11 [1] IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0)) [1] half(0) -> 0 [0] half(s(0)) -> 0 [0] half(s(s(z0))) -> s(half(z0)) [0] le(0, z0) -> true [0] le(s(z0), 0) -> false [0] le(s(z0), s(z1)) -> le(z0, z1) [0] inc(0) -> 0 [0] inc(s(z0)) -> s(inc(z0)) [0] log(z0) -> log2(z0, 0) [0] log2(z0, z1) -> if(le(z0, s(0)), z0, inc(z1)) [0] if(true, z0, s(z1)) -> z1 [0] if(false, z0, z1) -> log2(half(z0), z1) [0] The TRS has the following type information: HALF :: 0:s -> c:c1:c2 0 :: 0:s c :: c:c1:c2 s :: 0:s -> 0:s c1 :: c:c1:c2 c2 :: c:c1:c2 -> c:c1:c2 LE :: 0:s -> 0:s -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c3:c4:c5 c5 :: c3:c4:c5 -> c3:c4:c5 INC :: 0:s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c6:c7 LOG :: 0:s -> c8 c8 :: c9:c10 -> c8 LOG2 :: 0:s -> 0:s -> c9:c10 c9 :: c11:c12 -> c3:c4:c5 -> c9:c10 IF :: true:false -> 0:s -> 0:s -> c11:c12 le :: 0:s -> 0:s -> true:false inc :: 0:s -> 0:s c10 :: c11:c12 -> c6:c7 -> c9:c10 true :: true:false c11 :: c11:c12 false :: true:false c12 :: c9:c10 -> c:c1:c2 -> c11:c12 half :: 0:s -> 0:s log :: 0:s -> 0:s log2 :: 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (7) 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: HALF_1 LE_2 INC_1 LOG_1 LOG2_2 IF_3 (c) The following functions are completely defined: half_1 le_2 inc_1 log_1 log2_2 if_3 Due to the following rules being added: half(v0) -> 0 [0] le(v0, v1) -> null_le [0] inc(v0) -> 0 [0] log(v0) -> 0 [0] log2(v0, v1) -> 0 [0] if(v0, v1, v2) -> 0 [0] And the following fresh constants: null_le, const, const1 ---------------------------------------- (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: HALF(0) -> c [1] HALF(s(0)) -> c1 [1] HALF(s(s(z0))) -> c2(HALF(z0)) [1] LE(0, z0) -> c3 [1] LE(s(z0), 0) -> c4 [1] LE(s(z0), s(z1)) -> c5(LE(z0, z1)) [1] INC(0) -> c6 [1] INC(s(z0)) -> c7(INC(z0)) [1] LOG(z0) -> c8(LOG2(z0, 0)) [1] LOG2(z0, z1) -> c9(IF(le(z0, s(0)), z0, inc(z1)), LE(z0, s(0))) [1] LOG2(z0, z1) -> c10(IF(le(z0, s(0)), z0, inc(z1)), INC(z1)) [1] IF(true, z0, s(z1)) -> c11 [1] IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0)) [1] half(0) -> 0 [0] half(s(0)) -> 0 [0] half(s(s(z0))) -> s(half(z0)) [0] le(0, z0) -> true [0] le(s(z0), 0) -> false [0] le(s(z0), s(z1)) -> le(z0, z1) [0] inc(0) -> 0 [0] inc(s(z0)) -> s(inc(z0)) [0] log(z0) -> log2(z0, 0) [0] log2(z0, z1) -> if(le(z0, s(0)), z0, inc(z1)) [0] if(true, z0, s(z1)) -> z1 [0] if(false, z0, z1) -> log2(half(z0), z1) [0] half(v0) -> 0 [0] le(v0, v1) -> null_le [0] inc(v0) -> 0 [0] log(v0) -> 0 [0] log2(v0, v1) -> 0 [0] if(v0, v1, v2) -> 0 [0] The TRS has the following type information: HALF :: 0:s -> c:c1:c2 0 :: 0:s c :: c:c1:c2 s :: 0:s -> 0:s c1 :: c:c1:c2 c2 :: c:c1:c2 -> c:c1:c2 LE :: 0:s -> 0:s -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c3:c4:c5 c5 :: c3:c4:c5 -> c3:c4:c5 INC :: 0:s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c6:c7 LOG :: 0:s -> c8 c8 :: c9:c10 -> c8 LOG2 :: 0:s -> 0:s -> c9:c10 c9 :: c11:c12 -> c3:c4:c5 -> c9:c10 IF :: true:false:null_le -> 0:s -> 0:s -> c11:c12 le :: 0:s -> 0:s -> true:false:null_le inc :: 0:s -> 0:s c10 :: c11:c12 -> c6:c7 -> c9:c10 true :: true:false:null_le c11 :: c11:c12 false :: true:false:null_le c12 :: c9:c10 -> c:c1:c2 -> c11:c12 half :: 0:s -> 0:s log :: 0:s -> 0:s log2 :: 0:s -> 0:s -> 0:s if :: true:false:null_le -> 0:s -> 0:s -> 0:s null_le :: true:false:null_le const :: c8 const1 :: c9:c10 Rewrite Strategy: INNERMOST ---------------------------------------- (9) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (10) 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: HALF(0) -> c [1] HALF(s(0)) -> c1 [1] HALF(s(s(z0))) -> c2(HALF(z0)) [1] LE(0, z0) -> c3 [1] LE(s(z0), 0) -> c4 [1] LE(s(z0), s(z1)) -> c5(LE(z0, z1)) [1] INC(0) -> c6 [1] INC(s(z0)) -> c7(INC(z0)) [1] LOG(z0) -> c8(LOG2(z0, 0)) [1] LOG2(0, 0) -> c9(IF(true, 0, 0), LE(0, s(0))) [1] LOG2(0, s(z0'')) -> c9(IF(true, 0, s(inc(z0''))), LE(0, s(0))) [1] LOG2(0, z1) -> c9(IF(true, 0, 0), LE(0, s(0))) [1] LOG2(s(z0'), 0) -> c9(IF(le(z0', 0), s(z0'), 0), LE(s(z0'), s(0))) [1] LOG2(s(z0'), s(z01)) -> c9(IF(le(z0', 0), s(z0'), s(inc(z01))), LE(s(z0'), s(0))) [1] LOG2(s(z0'), z1) -> c9(IF(le(z0', 0), s(z0'), 0), LE(s(z0'), s(0))) [1] LOG2(z0, 0) -> c9(IF(null_le, z0, 0), LE(z0, s(0))) [1] LOG2(z0, s(z02)) -> c9(IF(null_le, z0, s(inc(z02))), LE(z0, s(0))) [1] LOG2(z0, z1) -> c9(IF(null_le, z0, 0), LE(z0, s(0))) [1] LOG2(0, 0) -> c10(IF(true, 0, 0), INC(0)) [1] LOG2(0, s(z04)) -> c10(IF(true, 0, s(inc(z04))), INC(s(z04))) [1] LOG2(0, z1) -> c10(IF(true, 0, 0), INC(z1)) [1] LOG2(s(z03), 0) -> c10(IF(le(z03, 0), s(z03), 0), INC(0)) [1] LOG2(s(z03), s(z05)) -> c10(IF(le(z03, 0), s(z03), s(inc(z05))), INC(s(z05))) [1] LOG2(s(z03), z1) -> c10(IF(le(z03, 0), s(z03), 0), INC(z1)) [1] LOG2(z0, 0) -> c10(IF(null_le, z0, 0), INC(0)) [1] LOG2(z0, s(z06)) -> c10(IF(null_le, z0, s(inc(z06))), INC(s(z06))) [1] LOG2(z0, z1) -> c10(IF(null_le, z0, 0), INC(z1)) [1] IF(true, z0, s(z1)) -> c11 [1] IF(false, 0, z1) -> c12(LOG2(0, z1), HALF(0)) [1] IF(false, s(0), z1) -> c12(LOG2(0, z1), HALF(s(0))) [1] IF(false, s(s(z07)), z1) -> c12(LOG2(s(half(z07)), z1), HALF(s(s(z07)))) [1] IF(false, z0, z1) -> c12(LOG2(0, z1), HALF(z0)) [1] half(0) -> 0 [0] half(s(0)) -> 0 [0] half(s(s(z0))) -> s(half(z0)) [0] le(0, z0) -> true [0] le(s(z0), 0) -> false [0] le(s(z0), s(z1)) -> le(z0, z1) [0] inc(0) -> 0 [0] inc(s(z0)) -> s(inc(z0)) [0] log(z0) -> log2(z0, 0) [0] log2(0, 0) -> if(true, 0, 0) [0] log2(0, s(z09)) -> if(true, 0, s(inc(z09))) [0] log2(0, z1) -> if(true, 0, 0) [0] log2(s(z08), 0) -> if(le(z08, 0), s(z08), 0) [0] log2(s(z08), s(z010)) -> if(le(z08, 0), s(z08), s(inc(z010))) [0] log2(s(z08), z1) -> if(le(z08, 0), s(z08), 0) [0] log2(z0, 0) -> if(null_le, z0, 0) [0] log2(z0, s(z011)) -> if(null_le, z0, s(inc(z011))) [0] log2(z0, z1) -> if(null_le, z0, 0) [0] if(true, z0, s(z1)) -> z1 [0] if(false, 0, z1) -> log2(0, z1) [0] if(false, s(0), z1) -> log2(0, z1) [0] if(false, s(s(z012)), z1) -> log2(s(half(z012)), z1) [0] if(false, z0, z1) -> log2(0, z1) [0] half(v0) -> 0 [0] le(v0, v1) -> null_le [0] inc(v0) -> 0 [0] log(v0) -> 0 [0] log2(v0, v1) -> 0 [0] if(v0, v1, v2) -> 0 [0] The TRS has the following type information: HALF :: 0:s -> c:c1:c2 0 :: 0:s c :: c:c1:c2 s :: 0:s -> 0:s c1 :: c:c1:c2 c2 :: c:c1:c2 -> c:c1:c2 LE :: 0:s -> 0:s -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c3:c4:c5 c5 :: c3:c4:c5 -> c3:c4:c5 INC :: 0:s -> c6:c7 c6 :: c6:c7 c7 :: c6:c7 -> c6:c7 LOG :: 0:s -> c8 c8 :: c9:c10 -> c8 LOG2 :: 0:s -> 0:s -> c9:c10 c9 :: c11:c12 -> c3:c4:c5 -> c9:c10 IF :: true:false:null_le -> 0:s -> 0:s -> c11:c12 le :: 0:s -> 0:s -> true:false:null_le inc :: 0:s -> 0:s c10 :: c11:c12 -> c6:c7 -> c9:c10 true :: true:false:null_le c11 :: c11:c12 false :: true:false:null_le c12 :: c9:c10 -> c:c1:c2 -> c11:c12 half :: 0:s -> 0:s log :: 0:s -> 0:s log2 :: 0:s -> 0:s -> 0:s if :: true:false:null_le -> 0:s -> 0:s -> 0:s null_le :: true:false:null_le const :: c8 const1 :: c9:c10 Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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 c => 0 c1 => 1 c3 => 0 c4 => 1 c6 => 0 true => 2 c11 => 0 false => 1 null_le => 0 const => 0 const1 => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 }-> 1 + HALF(z0) :|: z0 >= 0, z = 1 + (1 + z0) IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z1 >= 0, z0 >= 0, z'' = 1 + z1, z' = z0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z1) + HALF(z0) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z1) + HALF(0) :|: z1 >= 0, z = 1, z' = 0, z'' = z1 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z1) + HALF(1 + 0) :|: z1 >= 0, z = 1, z' = 1 + 0, z'' = z1 IF(z, z', z'') -{ 1 }-> 1 + LOG2(1 + half(z07), z1) + HALF(1 + (1 + z07)) :|: z' = 1 + (1 + z07), z1 >= 0, z07 >= 0, z = 1, z'' = z1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z0) :|: z = 1 + z0, z0 >= 0 LE(z, z') -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z0 >= 0, z = 0, z' = z0 LE(z, z') -{ 1 }-> 1 + LE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 LOG(z) -{ 1 }-> 1 + LOG2(z0, 0) :|: z = z0, z0 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z0', 0), 1 + z0', 0) + LE(1 + z0', 1 + 0) :|: z0' >= 0, z = 1 + z0', z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z0', 0), 1 + z0', 0) + LE(1 + z0', 1 + 0) :|: z1 >= 0, z0' >= 0, z' = z1, z = 1 + z0' LOG2(z, z') -{ 1 }-> 1 + IF(le(z0', 0), 1 + z0', 1 + inc(z01)) + LE(1 + z0', 1 + 0) :|: z01 >= 0, z0' >= 0, z' = 1 + z01, z = 1 + z0' LOG2(z, z') -{ 1 }-> 1 + IF(le(z03, 0), 1 + z03, 0) + INC(z1) :|: z1 >= 0, z' = z1, z = 1 + z03, z03 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z03, 0), 1 + z03, 0) + INC(0) :|: z = 1 + z03, z03 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z03, 0), 1 + z03, 1 + inc(z05)) + INC(1 + z05) :|: z = 1 + z03, z03 >= 0, z05 >= 0, z' = 1 + z05 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z1 >= 0, z' = z1, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z1) :|: z1 >= 0, z' = z1, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z0'')) + LE(0, 1 + 0) :|: z' = 1 + z0'', z0'' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z04)) + INC(1 + z04) :|: z04 >= 0, z = 0, z' = 1 + z04 LOG2(z, z') -{ 1 }-> 1 + IF(0, z0, 0) + LE(z0, 1 + 0) :|: z = z0, z0 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z0, 0) + LE(z0, 1 + 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z0, 0) + INC(z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z0, 0) + INC(0) :|: z = z0, z0 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z0, 1 + inc(z02)) + LE(z0, 1 + 0) :|: z = z0, z02 >= 0, z' = 1 + z02, z0 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z0, 1 + inc(z06)) + INC(1 + z06) :|: z = z0, z06 >= 0, z' = 1 + z06, z0 >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 half(z) -{ 0 }-> 1 + half(z0) :|: z0 >= 0, z = 1 + (1 + z0) if(z, z', z'') -{ 0 }-> z1 :|: z = 2, z1 >= 0, z0 >= 0, z'' = 1 + z1, z' = z0 if(z, z', z'') -{ 0 }-> log2(0, z1) :|: z1 >= 0, z = 1, z' = 0, z'' = z1 if(z, z', z'') -{ 0 }-> log2(0, z1) :|: z1 >= 0, z = 1, z' = 1 + 0, z'' = z1 if(z, z', z'') -{ 0 }-> log2(0, z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 if(z, z', z'') -{ 0 }-> log2(1 + half(z012), z1) :|: z' = 1 + (1 + z012), z1 >= 0, z = 1, z012 >= 0, z'' = z1 if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 inc(z) -{ 0 }-> 1 + inc(z0) :|: z = 1 + z0, z0 >= 0 le(z, z') -{ 0 }-> le(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 le(z, z') -{ 0 }-> 2 :|: z0 >= 0, z = 0, z' = z0 le(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 log(z) -{ 0 }-> log2(z0, 0) :|: z = z0, z0 >= 0 log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 log2(z, z') -{ 0 }-> if(le(z08, 0), 1 + z08, 0) :|: z08 >= 0, z = 1 + z08, z' = 0 log2(z, z') -{ 0 }-> if(le(z08, 0), 1 + z08, 0) :|: z08 >= 0, z1 >= 0, z = 1 + z08, z' = z1 log2(z, z') -{ 0 }-> if(le(z08, 0), 1 + z08, 1 + inc(z010)) :|: z08 >= 0, z = 1 + z08, z' = 1 + z010, z010 >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z1 >= 0, z' = z1, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z09)) :|: z' = 1 + z09, z09 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z0, 0) :|: z = z0, z0 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z0, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 log2(z, z') -{ 0 }-> if(0, z0, 1 + inc(z011)) :|: z = z0, z011 >= 0, z' = 1 + z011, z0 >= 0 log2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ---------------------------------------- (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: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 }-> 1 + HALF(z - 2) :|: z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(z') :|: z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(0) :|: z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(1 + 0) :|: z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(1 + half(z' - 2), z'') + HALF(1 + (1 + (z' - 2))) :|: z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + LE(1 + (z - 1), 1 + 0) :|: z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + LE(0, 1 + 0) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + LE(z, 1 + 0) :|: z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { HALF } { le } { LE } { inc } { INC } { half } { IF, LOG2 } { log2, if } { LOG } { log } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 }-> 1 + HALF(z - 2) :|: z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(z') :|: z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(0) :|: z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(1 + 0) :|: z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(1 + half(z' - 2), z'') + HALF(1 + (1 + (z' - 2))) :|: z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + LE(1 + (z - 1), 1 + 0) :|: z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + LE(0, 1 + 0) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + LE(z, 1 + 0) :|: z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {HALF}, {le}, {LE}, {inc}, {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} ---------------------------------------- (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: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 }-> 1 + HALF(z - 2) :|: z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(z') :|: z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(0) :|: z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(1 + 0) :|: z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(1 + half(z' - 2), z'') + HALF(1 + (1 + (z' - 2))) :|: z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + LE(1 + (z - 1), 1 + 0) :|: z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + LE(0, 1 + 0) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + LE(z, 1 + 0) :|: z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {HALF}, {le}, {LE}, {inc}, {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: HALF after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 }-> 1 + HALF(z - 2) :|: z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(z') :|: z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(0) :|: z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(1 + 0) :|: z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(1 + half(z' - 2), z'') + HALF(1 + (1 + (z' - 2))) :|: z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + LE(1 + (z - 1), 1 + 0) :|: z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + LE(0, 1 + 0) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + LE(z, 1 + 0) :|: z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {HALF}, {le}, {LE}, {inc}, {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: ?, size: O(n^1) [z] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: HALF after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 }-> 1 + HALF(z - 2) :|: z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(z') :|: z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(0) :|: z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(0, z'') + HALF(1 + 0) :|: z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 1 }-> 1 + LOG2(1 + half(z' - 2), z'') + HALF(1 + (1 + (z' - 2))) :|: z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + LE(1 + (z - 1), 1 + 0) :|: z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + LE(0, 1 + 0) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + LE(z, 1 + 0) :|: z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {le}, {LE}, {inc}, {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [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: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + LE(1 + (z - 1), 1 + 0) :|: z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + LE(0, 1 + 0) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + LE(z, 1 + 0) :|: z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {le}, {LE}, {inc}, {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: le after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + LE(1 + (z - 1), 1 + 0) :|: z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + LE(0, 1 + 0) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + LE(z, 1 + 0) :|: z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {le}, {LE}, {inc}, {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: ?, size: O(1) [2] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: le after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + LE(1 + (z - 1), 1 + 0) :|: z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + LE(0, 1 + 0) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + LE(z, 1 + 0) :|: z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {LE}, {inc}, {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (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: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s3, 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(s4, 1 + (z - 1), 1 + inc(z' - 1)) + LE(1 + (z - 1), 1 + 0) :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s6, 1 + (z - 1), 0) + INC(0) :|: s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(s7, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s8, 1 + (z - 1), 0) + INC(z') :|: s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + LE(0, 1 + 0) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + LE(z, 1 + 0) :|: z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + inc(z' - 1)) :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {LE}, {inc}, {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: LE after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s3, 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(s4, 1 + (z - 1), 1 + inc(z' - 1)) + LE(1 + (z - 1), 1 + 0) :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s6, 1 + (z - 1), 0) + INC(0) :|: s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(s7, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s8, 1 + (z - 1), 0) + INC(z') :|: s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + LE(0, 1 + 0) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + LE(z, 1 + 0) :|: z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + inc(z' - 1)) :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {LE}, {inc}, {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: ?, size: O(n^1) [1 + z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: LE after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s3, 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(s4, 1 + (z - 1), 1 + inc(z' - 1)) + LE(1 + (z - 1), 1 + 0) :|: s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 0) + LE(1 + (z - 1), 1 + 0) :|: s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s6, 1 + (z - 1), 0) + INC(0) :|: s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(s7, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s8, 1 + (z - 1), 0) + INC(z') :|: s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + LE(0, 1 + 0) :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + LE(0, 1 + 0) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + LE(z, 1 + 0) :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + LE(z, 1 + 0) :|: z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + inc(z' - 1)) :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {inc}, {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [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: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s3, 1 + (z - 1), 0) + s17 :|: s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(s4, 1 + (z - 1), 1 + inc(z' - 1)) + s18 :|: s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s5, 1 + (z - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s6, 1 + (z - 1), 0) + INC(0) :|: s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(s7, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s8, 1 + (z - 1), 0) + INC(z') :|: s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s14 :|: s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s16 :|: s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + s15 :|: s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s20 :|: s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s22 :|: s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + s21 :|: s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + inc(z' - 1)) :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {inc}, {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: inc after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s3, 1 + (z - 1), 0) + s17 :|: s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(s4, 1 + (z - 1), 1 + inc(z' - 1)) + s18 :|: s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s5, 1 + (z - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s6, 1 + (z - 1), 0) + INC(0) :|: s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(s7, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s8, 1 + (z - 1), 0) + INC(z') :|: s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s14 :|: s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s16 :|: s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + s15 :|: s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s20 :|: s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s22 :|: s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + s21 :|: s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + inc(z' - 1)) :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {inc}, {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: ?, size: O(n^1) [z] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: inc after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s3, 1 + (z - 1), 0) + s17 :|: s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(s4, 1 + (z - 1), 1 + inc(z' - 1)) + s18 :|: s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s5, 1 + (z - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s6, 1 + (z - 1), 0) + INC(0) :|: s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(s7, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s8, 1 + (z - 1), 0) + INC(z') :|: s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s14 :|: s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s16 :|: s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + s15 :|: s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s20 :|: s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s22 :|: s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + s21 :|: s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + inc(z' - 1)) :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [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: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s3, 1 + (z - 1), 0) + s17 :|: s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(s4, 1 + (z - 1), 1 + s24) + s18 :|: s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s5, 1 + (z - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s6, 1 + (z - 1), 0) + INC(0) :|: s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(s7, 1 + (z - 1), 1 + s27) + INC(1 + (z' - 1)) :|: s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s8, 1 + (z - 1), 0) + INC(z') :|: s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s14 :|: s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s16 :|: s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 1 + s23) + s15 :|: s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + s26) + INC(1 + (z' - 1)) :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s20 :|: s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s22 :|: s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 1 + s25) + s21 :|: s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + s28) + INC(1 + (z' - 1)) :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + s31) :|: s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + s30) :|: s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + s32) :|: s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: INC 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: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s3, 1 + (z - 1), 0) + s17 :|: s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(s4, 1 + (z - 1), 1 + s24) + s18 :|: s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s5, 1 + (z - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s6, 1 + (z - 1), 0) + INC(0) :|: s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(s7, 1 + (z - 1), 1 + s27) + INC(1 + (z' - 1)) :|: s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s8, 1 + (z - 1), 0) + INC(z') :|: s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s14 :|: s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s16 :|: s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 1 + s23) + s15 :|: s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + s26) + INC(1 + (z' - 1)) :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s20 :|: s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s22 :|: s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 1 + s25) + s21 :|: s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + s28) + INC(1 + (z' - 1)) :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + s31) :|: s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + s30) :|: s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + s32) :|: s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {INC}, {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: ?, size: O(n^1) [z] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: INC after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s3, 1 + (z - 1), 0) + s17 :|: s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(s4, 1 + (z - 1), 1 + s24) + s18 :|: s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s5, 1 + (z - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s6, 1 + (z - 1), 0) + INC(0) :|: s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 1 }-> 1 + IF(s7, 1 + (z - 1), 1 + s27) + INC(1 + (z' - 1)) :|: s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(s8, 1 + (z - 1), 0) + INC(z') :|: s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s14 :|: s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s16 :|: s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 1 + s23) + s15 :|: s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + s26) + INC(1 + (z' - 1)) :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s20 :|: s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s22 :|: s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 1 + s25) + s21 :|: s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 1 }-> 1 + IF(0, z, 1 + s28) + INC(1 + (z' - 1)) :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + s31) :|: s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + s30) :|: s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + s32) :|: s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], 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: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s3, 1 + (z - 1), 0) + s17 :|: s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(s4, 1 + (z - 1), 1 + s24) + s18 :|: s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s5, 1 + (z - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 2 }-> 1 + IF(s6, 1 + (z - 1), 0) + s37 :|: s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(s7, 1 + (z - 1), 1 + s27) + s38 :|: s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(s8, 1 + (z - 1), 0) + s39 :|: s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s14 :|: s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s16 :|: s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s34 :|: s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s36 :|: s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 1 + s23) + s15 :|: s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s26) + s35 :|: s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s20 :|: s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s22 :|: s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s40 :|: s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s42 :|: s42 >= 0, s42 <= z', z' >= 0, z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 1 + s25) + s21 :|: s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s28) + s41 :|: s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + s31) :|: s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + s30) :|: s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + s32) :|: s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: half after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s3, 1 + (z - 1), 0) + s17 :|: s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(s4, 1 + (z - 1), 1 + s24) + s18 :|: s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s5, 1 + (z - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 2 }-> 1 + IF(s6, 1 + (z - 1), 0) + s37 :|: s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(s7, 1 + (z - 1), 1 + s27) + s38 :|: s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(s8, 1 + (z - 1), 0) + s39 :|: s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s14 :|: s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s16 :|: s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s34 :|: s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s36 :|: s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 1 + s23) + s15 :|: s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s26) + s35 :|: s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s20 :|: s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s22 :|: s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s40 :|: s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s42 :|: s42 >= 0, s42 <= z', z' >= 0, z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 1 + s25) + s21 :|: s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s28) + s41 :|: s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + s31) :|: s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + s30) :|: s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + s32) :|: s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {half}, {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: ?, size: O(n^1) [z] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: half after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + half(z' - 2), z'') + s1 :|: s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s3, 1 + (z - 1), 0) + s17 :|: s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(s4, 1 + (z - 1), 1 + s24) + s18 :|: s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s5, 1 + (z - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 2 }-> 1 + IF(s6, 1 + (z - 1), 0) + s37 :|: s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(s7, 1 + (z - 1), 1 + s27) + s38 :|: s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(s8, 1 + (z - 1), 0) + s39 :|: s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s14 :|: s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s16 :|: s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s34 :|: s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s36 :|: s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 1 + s23) + s15 :|: s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s26) + s35 :|: s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s20 :|: s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s22 :|: s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s40 :|: s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s42 :|: s42 >= 0, s42 <= z', z' >= 0, z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 1 + s25) + s21 :|: s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s28) + s41 :|: s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + half(z - 2) :|: z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + s31) :|: s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + s30) :|: s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + s32) :|: s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [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: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + s43, z'') + s1 :|: s43 >= 0, s43 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s3, 1 + (z - 1), 0) + s17 :|: s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(s4, 1 + (z - 1), 1 + s24) + s18 :|: s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s5, 1 + (z - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 2 }-> 1 + IF(s6, 1 + (z - 1), 0) + s37 :|: s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(s7, 1 + (z - 1), 1 + s27) + s38 :|: s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(s8, 1 + (z - 1), 0) + s39 :|: s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s14 :|: s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s16 :|: s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s34 :|: s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s36 :|: s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 1 + s23) + s15 :|: s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s26) + s35 :|: s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s20 :|: s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s22 :|: s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s40 :|: s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s42 :|: s42 >= 0, s42 <= z', z' >= 0, z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 1 + s25) + s21 :|: s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s28) + s41 :|: s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + s45, z'') :|: s45 >= 0, s45 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + s31) :|: s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + s30) :|: s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + s32) :|: s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: IF after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 18 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' Computed SIZE bound using KoAT for: LOG2 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 384 + 80*z + 8*z*z' + 12*z^2 + 30*z' ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + s43, z'') + s1 :|: s43 >= 0, s43 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s3, 1 + (z - 1), 0) + s17 :|: s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(s4, 1 + (z - 1), 1 + s24) + s18 :|: s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s5, 1 + (z - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 2 }-> 1 + IF(s6, 1 + (z - 1), 0) + s37 :|: s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(s7, 1 + (z - 1), 1 + s27) + s38 :|: s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(s8, 1 + (z - 1), 0) + s39 :|: s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s14 :|: s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s16 :|: s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s34 :|: s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s36 :|: s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 1 + s23) + s15 :|: s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s26) + s35 :|: s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s20 :|: s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s22 :|: s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s40 :|: s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s42 :|: s42 >= 0, s42 <= z', z' >= 0, z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 1 + s25) + s21 :|: s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s28) + s41 :|: s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + s45, z'') :|: s45 >= 0, s45 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + s31) :|: s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + s30) :|: s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + s32) :|: s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {IF,LOG2}, {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] IF: runtime: ?, size: O(n^2) [18 + 6*z' + 2*z'*z'' + z'^2 + 4*z''] LOG2: runtime: ?, size: O(n^2) [384 + 80*z + 8*z*z' + 12*z^2 + 30*z'] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: IF after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 79 + 314*z' + 32*z'*z'' + 7*z'^2 + 8*z'' Computed RUNTIME bound using KoAT for: LOG2 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1476 + 3768*z + 128*z*z' + 84*z^2 + 54*z' ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 3 }-> 1 + LOG2(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + LOG2(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + LOG2(1 + s43, z'') + s1 :|: s43 >= 0, s43 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1 }-> 1 + LOG2(z, 0) :|: z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s3, 1 + (z - 1), 0) + s17 :|: s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(s4, 1 + (z - 1), 1 + s24) + s18 :|: s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(s5, 1 + (z - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 2 }-> 1 + IF(s6, 1 + (z - 1), 0) + s37 :|: s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(s7, 1 + (z - 1), 1 + s27) + s38 :|: s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(s8, 1 + (z - 1), 0) + s39 :|: s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s14 :|: s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 0) + s16 :|: s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s34 :|: s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s36 :|: s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(2, 0, 1 + s23) + s15 :|: s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s26) + s35 :|: s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s20 :|: s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 0) + s22 :|: s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s40 :|: s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s42 :|: s42 >= 0, s42 <= z', z' >= 0, z >= 0 LOG2(z, z') -{ 4 }-> 1 + IF(0, z, 1 + s25) + s21 :|: s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s28) + s41 :|: s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + s45, z'') :|: s45 >= 0, s45 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + s31) :|: s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + s30) :|: s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + s32) :|: s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] IF: runtime: O(n^2) [79 + 314*z' + 32*z'*z'' + 7*z'^2 + 8*z''], size: O(n^2) [18 + 6*z' + 2*z'*z'' + z'^2 + 4*z''] LOG2: runtime: O(n^2) [1476 + 3768*z + 128*z*z' + 84*z^2 + 54*z'], size: O(n^2) [384 + 80*z + 8*z*z' + 12*z^2 + 30*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: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1479 + 54*z'' }-> 1 + s65 + s' :|: s65 >= 0, s65 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1480 + 54*z'' }-> 1 + s66 + s'' :|: s66 >= 0, s66 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 5331 + 3936*s43 + 128*s43*z'' + 84*s43^2 + z' + 182*z'' }-> 1 + s67 + s1 :|: s67 >= 0, s67 <= 30 * z'' + 80 * (1 + s43) + 12 * ((1 + s43) * (1 + s43)) + 8 * (z'' * (1 + s43)) + 384, s43 >= 0, s43 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 1479 + z' + 54*z'' }-> 1 + s68 + s2 :|: s68 >= 0, s68 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1477 + 3768*z + 84*z^2 }-> 1 + s46 :|: s46 >= 0, s46 <= 30 * 0 + 80 * z + 12 * (z * z) + 8 * (0 * z) + 384, z >= 0 LOG2(z, z') -{ 83 }-> 1 + s47 + s14 :|: s47 >= 0, s47 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 91 + 8*s23 }-> 1 + s48 + s15 :|: s48 >= 0, s48 <= 6 * 0 + 4 * (1 + s23) + 2 * ((1 + s23) * 0) + 0 * 0 + 18, s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 83 }-> 1 + s49 + s16 :|: s49 >= 0, s49 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s50 + s17 :|: s50 >= 0, s50 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s24 + 32*s24*z + 346*z + 7*z^2 }-> 1 + s51 + s18 :|: s51 >= 0, s51 <= 6 * (1 + (z - 1)) + 4 * (1 + s24) + 2 * ((1 + s24) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s52 + s19 :|: s52 >= 0, s52 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s53 + s20 :|: s53 >= 0, s53 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s25 + 32*s25*z + 346*z + 7*z^2 }-> 1 + s54 + s21 :|: s54 >= 0, s54 <= 6 * z + 4 * (1 + s25) + 2 * ((1 + s25) * z) + z * z + 18, s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s55 + s22 :|: s55 >= 0, s55 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 81 }-> 1 + s56 + s34 :|: s56 >= 0, s56 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 89 + 8*s26 + z' }-> 1 + s57 + s35 :|: s57 >= 0, s57 <= 6 * 0 + 4 * (1 + s26) + 2 * ((1 + s26) * 0) + 0 * 0 + 18, s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 81 + z' }-> 1 + s58 + s36 :|: s58 >= 0, s58 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s59 + s37 :|: s59 >= 0, s59 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s27 + 32*s27*z + 346*z + 7*z^2 + z' }-> 1 + s60 + s38 :|: s60 >= 0, s60 <= 6 * (1 + (z - 1)) + 4 * (1 + s27) + 2 * ((1 + s27) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s61 + s39 :|: s61 >= 0, s61 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s62 + s40 :|: s62 >= 0, s62 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s28 + 32*s28*z + 346*z + 7*z^2 + z' }-> 1 + s63 + s41 :|: s63 >= 0, s63 <= 6 * z + 4 * (1 + s28) + 2 * ((1 + s28) * z) + z * z + 18, s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s64 + s42 :|: s64 >= 0, s64 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s42 >= 0, s42 <= z', z' >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + s45, z'') :|: s45 >= 0, s45 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + s31) :|: s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + s30) :|: s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + s32) :|: s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] IF: runtime: O(n^2) [79 + 314*z' + 32*z'*z'' + 7*z'^2 + 8*z''], size: O(n^2) [18 + 6*z' + 2*z'*z'' + z'^2 + 4*z''] LOG2: runtime: O(n^2) [1476 + 3768*z + 128*z*z' + 84*z^2 + 54*z'], size: O(n^2) [384 + 80*z + 8*z*z' + 12*z^2 + 30*z'] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: log2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' Computed SIZE bound using KoAT for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1479 + 54*z'' }-> 1 + s65 + s' :|: s65 >= 0, s65 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1480 + 54*z'' }-> 1 + s66 + s'' :|: s66 >= 0, s66 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 5331 + 3936*s43 + 128*s43*z'' + 84*s43^2 + z' + 182*z'' }-> 1 + s67 + s1 :|: s67 >= 0, s67 <= 30 * z'' + 80 * (1 + s43) + 12 * ((1 + s43) * (1 + s43)) + 8 * (z'' * (1 + s43)) + 384, s43 >= 0, s43 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 1479 + z' + 54*z'' }-> 1 + s68 + s2 :|: s68 >= 0, s68 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1477 + 3768*z + 84*z^2 }-> 1 + s46 :|: s46 >= 0, s46 <= 30 * 0 + 80 * z + 12 * (z * z) + 8 * (0 * z) + 384, z >= 0 LOG2(z, z') -{ 83 }-> 1 + s47 + s14 :|: s47 >= 0, s47 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 91 + 8*s23 }-> 1 + s48 + s15 :|: s48 >= 0, s48 <= 6 * 0 + 4 * (1 + s23) + 2 * ((1 + s23) * 0) + 0 * 0 + 18, s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 83 }-> 1 + s49 + s16 :|: s49 >= 0, s49 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s50 + s17 :|: s50 >= 0, s50 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s24 + 32*s24*z + 346*z + 7*z^2 }-> 1 + s51 + s18 :|: s51 >= 0, s51 <= 6 * (1 + (z - 1)) + 4 * (1 + s24) + 2 * ((1 + s24) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s52 + s19 :|: s52 >= 0, s52 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s53 + s20 :|: s53 >= 0, s53 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s25 + 32*s25*z + 346*z + 7*z^2 }-> 1 + s54 + s21 :|: s54 >= 0, s54 <= 6 * z + 4 * (1 + s25) + 2 * ((1 + s25) * z) + z * z + 18, s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s55 + s22 :|: s55 >= 0, s55 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 81 }-> 1 + s56 + s34 :|: s56 >= 0, s56 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 89 + 8*s26 + z' }-> 1 + s57 + s35 :|: s57 >= 0, s57 <= 6 * 0 + 4 * (1 + s26) + 2 * ((1 + s26) * 0) + 0 * 0 + 18, s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 81 + z' }-> 1 + s58 + s36 :|: s58 >= 0, s58 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s59 + s37 :|: s59 >= 0, s59 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s27 + 32*s27*z + 346*z + 7*z^2 + z' }-> 1 + s60 + s38 :|: s60 >= 0, s60 <= 6 * (1 + (z - 1)) + 4 * (1 + s27) + 2 * ((1 + s27) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s61 + s39 :|: s61 >= 0, s61 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s62 + s40 :|: s62 >= 0, s62 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s28 + 32*s28*z + 346*z + 7*z^2 + z' }-> 1 + s63 + s41 :|: s63 >= 0, s63 <= 6 * z + 4 * (1 + s28) + 2 * ((1 + s28) * z) + z * z + 18, s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s64 + s42 :|: s64 >= 0, s64 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s42 >= 0, s42 <= z', z' >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + s45, z'') :|: s45 >= 0, s45 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + s31) :|: s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + s30) :|: s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + s32) :|: s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {log2,if}, {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] IF: runtime: O(n^2) [79 + 314*z' + 32*z'*z'' + 7*z'^2 + 8*z''], size: O(n^2) [18 + 6*z' + 2*z'*z'' + z'^2 + 4*z''] LOG2: runtime: O(n^2) [1476 + 3768*z + 128*z*z' + 84*z^2 + 54*z'], size: O(n^2) [384 + 80*z + 8*z*z' + 12*z^2 + 30*z'] log2: runtime: ?, size: O(n^1) [z'] if: runtime: ?, size: O(n^1) [z''] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: log2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1479 + 54*z'' }-> 1 + s65 + s' :|: s65 >= 0, s65 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1480 + 54*z'' }-> 1 + s66 + s'' :|: s66 >= 0, s66 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 5331 + 3936*s43 + 128*s43*z'' + 84*s43^2 + z' + 182*z'' }-> 1 + s67 + s1 :|: s67 >= 0, s67 <= 30 * z'' + 80 * (1 + s43) + 12 * ((1 + s43) * (1 + s43)) + 8 * (z'' * (1 + s43)) + 384, s43 >= 0, s43 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 1479 + z' + 54*z'' }-> 1 + s68 + s2 :|: s68 >= 0, s68 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1477 + 3768*z + 84*z^2 }-> 1 + s46 :|: s46 >= 0, s46 <= 30 * 0 + 80 * z + 12 * (z * z) + 8 * (0 * z) + 384, z >= 0 LOG2(z, z') -{ 83 }-> 1 + s47 + s14 :|: s47 >= 0, s47 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 91 + 8*s23 }-> 1 + s48 + s15 :|: s48 >= 0, s48 <= 6 * 0 + 4 * (1 + s23) + 2 * ((1 + s23) * 0) + 0 * 0 + 18, s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 83 }-> 1 + s49 + s16 :|: s49 >= 0, s49 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s50 + s17 :|: s50 >= 0, s50 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s24 + 32*s24*z + 346*z + 7*z^2 }-> 1 + s51 + s18 :|: s51 >= 0, s51 <= 6 * (1 + (z - 1)) + 4 * (1 + s24) + 2 * ((1 + s24) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s52 + s19 :|: s52 >= 0, s52 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s53 + s20 :|: s53 >= 0, s53 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s25 + 32*s25*z + 346*z + 7*z^2 }-> 1 + s54 + s21 :|: s54 >= 0, s54 <= 6 * z + 4 * (1 + s25) + 2 * ((1 + s25) * z) + z * z + 18, s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s55 + s22 :|: s55 >= 0, s55 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 81 }-> 1 + s56 + s34 :|: s56 >= 0, s56 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 89 + 8*s26 + z' }-> 1 + s57 + s35 :|: s57 >= 0, s57 <= 6 * 0 + 4 * (1 + s26) + 2 * ((1 + s26) * 0) + 0 * 0 + 18, s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 81 + z' }-> 1 + s58 + s36 :|: s58 >= 0, s58 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s59 + s37 :|: s59 >= 0, s59 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s27 + 32*s27*z + 346*z + 7*z^2 + z' }-> 1 + s60 + s38 :|: s60 >= 0, s60 <= 6 * (1 + (z - 1)) + 4 * (1 + s27) + 2 * ((1 + s27) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s61 + s39 :|: s61 >= 0, s61 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s62 + s40 :|: s62 >= 0, s62 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s28 + 32*s28*z + 346*z + 7*z^2 + z' }-> 1 + s63 + s41 :|: s63 >= 0, s63 <= 6 * z + 4 * (1 + s28) + 2 * ((1 + s28) * z) + z * z + 18, s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s64 + s42 :|: s64 >= 0, s64 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s42 >= 0, s42 <= z', z' >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> log2(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> log2(1 + s45, z'') :|: s45 >= 0, s45 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> log2(z, 0) :|: z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> if(s10, 1 + (z - 1), 0) :|: s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> if(s11, 1 + (z - 1), 1 + s31) :|: s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> if(s12, 1 + (z - 1), 0) :|: s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 log2(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 log2(z, z') -{ 0 }-> if(2, 0, 1 + s30) :|: s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 log2(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 log2(z, z') -{ 0 }-> if(0, z, 1 + s32) :|: s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] IF: runtime: O(n^2) [79 + 314*z' + 32*z'*z'' + 7*z'^2 + 8*z''], size: O(n^2) [18 + 6*z' + 2*z'*z'' + z'^2 + 4*z''] LOG2: runtime: O(n^2) [1476 + 3768*z + 128*z*z' + 84*z^2 + 54*z'], size: O(n^2) [384 + 80*z + 8*z*z' + 12*z^2 + 30*z'] log2: runtime: O(1) [0], size: O(n^1) [z'] if: runtime: O(1) [0], size: O(n^1) [z''] ---------------------------------------- (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: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1479 + 54*z'' }-> 1 + s65 + s' :|: s65 >= 0, s65 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1480 + 54*z'' }-> 1 + s66 + s'' :|: s66 >= 0, s66 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 5331 + 3936*s43 + 128*s43*z'' + 84*s43^2 + z' + 182*z'' }-> 1 + s67 + s1 :|: s67 >= 0, s67 <= 30 * z'' + 80 * (1 + s43) + 12 * ((1 + s43) * (1 + s43)) + 8 * (z'' * (1 + s43)) + 384, s43 >= 0, s43 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 1479 + z' + 54*z'' }-> 1 + s68 + s2 :|: s68 >= 0, s68 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1477 + 3768*z + 84*z^2 }-> 1 + s46 :|: s46 >= 0, s46 <= 30 * 0 + 80 * z + 12 * (z * z) + 8 * (0 * z) + 384, z >= 0 LOG2(z, z') -{ 83 }-> 1 + s47 + s14 :|: s47 >= 0, s47 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 91 + 8*s23 }-> 1 + s48 + s15 :|: s48 >= 0, s48 <= 6 * 0 + 4 * (1 + s23) + 2 * ((1 + s23) * 0) + 0 * 0 + 18, s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 83 }-> 1 + s49 + s16 :|: s49 >= 0, s49 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s50 + s17 :|: s50 >= 0, s50 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s24 + 32*s24*z + 346*z + 7*z^2 }-> 1 + s51 + s18 :|: s51 >= 0, s51 <= 6 * (1 + (z - 1)) + 4 * (1 + s24) + 2 * ((1 + s24) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s52 + s19 :|: s52 >= 0, s52 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s53 + s20 :|: s53 >= 0, s53 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s25 + 32*s25*z + 346*z + 7*z^2 }-> 1 + s54 + s21 :|: s54 >= 0, s54 <= 6 * z + 4 * (1 + s25) + 2 * ((1 + s25) * z) + z * z + 18, s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s55 + s22 :|: s55 >= 0, s55 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 81 }-> 1 + s56 + s34 :|: s56 >= 0, s56 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 89 + 8*s26 + z' }-> 1 + s57 + s35 :|: s57 >= 0, s57 <= 6 * 0 + 4 * (1 + s26) + 2 * ((1 + s26) * 0) + 0 * 0 + 18, s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 81 + z' }-> 1 + s58 + s36 :|: s58 >= 0, s58 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s59 + s37 :|: s59 >= 0, s59 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s27 + 32*s27*z + 346*z + 7*z^2 + z' }-> 1 + s60 + s38 :|: s60 >= 0, s60 <= 6 * (1 + (z - 1)) + 4 * (1 + s27) + 2 * ((1 + s27) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s61 + s39 :|: s61 >= 0, s61 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s62 + s40 :|: s62 >= 0, s62 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s28 + 32*s28*z + 346*z + 7*z^2 + z' }-> 1 + s63 + s41 :|: s63 >= 0, s63 <= 6 * z + 4 * (1 + s28) + 2 * ((1 + s28) * z) + z * z + 18, s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s64 + s42 :|: s64 >= 0, s64 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s42 >= 0, s42 <= z', z' >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> s79 :|: s79 >= 0, s79 <= z'', z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> s80 :|: s80 >= 0, s80 <= z'', z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> s81 :|: s81 >= 0, s81 <= z'', s45 >= 0, s45 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> s82 :|: s82 >= 0, s82 <= z'', z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 0, z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 0, z = 0, z' = 0 log2(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + s30, s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> s72 :|: s72 >= 0, s72 <= 0, z' >= 0, z = 0 log2(z, z') -{ 0 }-> s73 :|: s73 >= 0, s73 <= 0, s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> s74 :|: s74 >= 0, s74 <= 1 + s31, s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> s75 :|: s75 >= 0, s75 <= 0, s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> s76 :|: s76 >= 0, s76 <= 0, z >= 0, z' = 0 log2(z, z') -{ 0 }-> s77 :|: s77 >= 0, s77 <= 1 + s32, s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> s78 :|: s78 >= 0, s78 <= 0, z' >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] IF: runtime: O(n^2) [79 + 314*z' + 32*z'*z'' + 7*z'^2 + 8*z''], size: O(n^2) [18 + 6*z' + 2*z'*z'' + z'^2 + 4*z''] LOG2: runtime: O(n^2) [1476 + 3768*z + 128*z*z' + 84*z^2 + 54*z'], size: O(n^2) [384 + 80*z + 8*z*z' + 12*z^2 + 30*z'] log2: runtime: O(1) [0], size: O(n^1) [z'] if: runtime: O(1) [0], size: O(n^1) [z''] ---------------------------------------- (67) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: LOG after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 385 + 80*z + 12*z^2 ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1479 + 54*z'' }-> 1 + s65 + s' :|: s65 >= 0, s65 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1480 + 54*z'' }-> 1 + s66 + s'' :|: s66 >= 0, s66 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 5331 + 3936*s43 + 128*s43*z'' + 84*s43^2 + z' + 182*z'' }-> 1 + s67 + s1 :|: s67 >= 0, s67 <= 30 * z'' + 80 * (1 + s43) + 12 * ((1 + s43) * (1 + s43)) + 8 * (z'' * (1 + s43)) + 384, s43 >= 0, s43 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 1479 + z' + 54*z'' }-> 1 + s68 + s2 :|: s68 >= 0, s68 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1477 + 3768*z + 84*z^2 }-> 1 + s46 :|: s46 >= 0, s46 <= 30 * 0 + 80 * z + 12 * (z * z) + 8 * (0 * z) + 384, z >= 0 LOG2(z, z') -{ 83 }-> 1 + s47 + s14 :|: s47 >= 0, s47 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 91 + 8*s23 }-> 1 + s48 + s15 :|: s48 >= 0, s48 <= 6 * 0 + 4 * (1 + s23) + 2 * ((1 + s23) * 0) + 0 * 0 + 18, s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 83 }-> 1 + s49 + s16 :|: s49 >= 0, s49 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s50 + s17 :|: s50 >= 0, s50 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s24 + 32*s24*z + 346*z + 7*z^2 }-> 1 + s51 + s18 :|: s51 >= 0, s51 <= 6 * (1 + (z - 1)) + 4 * (1 + s24) + 2 * ((1 + s24) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s52 + s19 :|: s52 >= 0, s52 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s53 + s20 :|: s53 >= 0, s53 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s25 + 32*s25*z + 346*z + 7*z^2 }-> 1 + s54 + s21 :|: s54 >= 0, s54 <= 6 * z + 4 * (1 + s25) + 2 * ((1 + s25) * z) + z * z + 18, s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s55 + s22 :|: s55 >= 0, s55 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 81 }-> 1 + s56 + s34 :|: s56 >= 0, s56 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 89 + 8*s26 + z' }-> 1 + s57 + s35 :|: s57 >= 0, s57 <= 6 * 0 + 4 * (1 + s26) + 2 * ((1 + s26) * 0) + 0 * 0 + 18, s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 81 + z' }-> 1 + s58 + s36 :|: s58 >= 0, s58 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s59 + s37 :|: s59 >= 0, s59 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s27 + 32*s27*z + 346*z + 7*z^2 + z' }-> 1 + s60 + s38 :|: s60 >= 0, s60 <= 6 * (1 + (z - 1)) + 4 * (1 + s27) + 2 * ((1 + s27) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s61 + s39 :|: s61 >= 0, s61 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s62 + s40 :|: s62 >= 0, s62 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s28 + 32*s28*z + 346*z + 7*z^2 + z' }-> 1 + s63 + s41 :|: s63 >= 0, s63 <= 6 * z + 4 * (1 + s28) + 2 * ((1 + s28) * z) + z * z + 18, s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s64 + s42 :|: s64 >= 0, s64 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s42 >= 0, s42 <= z', z' >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> s79 :|: s79 >= 0, s79 <= z'', z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> s80 :|: s80 >= 0, s80 <= z'', z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> s81 :|: s81 >= 0, s81 <= z'', s45 >= 0, s45 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> s82 :|: s82 >= 0, s82 <= z'', z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 0, z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 0, z = 0, z' = 0 log2(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + s30, s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> s72 :|: s72 >= 0, s72 <= 0, z' >= 0, z = 0 log2(z, z') -{ 0 }-> s73 :|: s73 >= 0, s73 <= 0, s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> s74 :|: s74 >= 0, s74 <= 1 + s31, s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> s75 :|: s75 >= 0, s75 <= 0, s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> s76 :|: s76 >= 0, s76 <= 0, z >= 0, z' = 0 log2(z, z') -{ 0 }-> s77 :|: s77 >= 0, s77 <= 1 + s32, s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> s78 :|: s78 >= 0, s78 <= 0, z' >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {LOG}, {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] IF: runtime: O(n^2) [79 + 314*z' + 32*z'*z'' + 7*z'^2 + 8*z''], size: O(n^2) [18 + 6*z' + 2*z'*z'' + z'^2 + 4*z''] LOG2: runtime: O(n^2) [1476 + 3768*z + 128*z*z' + 84*z^2 + 54*z'], size: O(n^2) [384 + 80*z + 8*z*z' + 12*z^2 + 30*z'] log2: runtime: O(1) [0], size: O(n^1) [z'] if: runtime: O(1) [0], size: O(n^1) [z''] LOG: runtime: ?, size: O(n^2) [385 + 80*z + 12*z^2] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: LOG after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1477 + 3768*z + 84*z^2 ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1479 + 54*z'' }-> 1 + s65 + s' :|: s65 >= 0, s65 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1480 + 54*z'' }-> 1 + s66 + s'' :|: s66 >= 0, s66 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 5331 + 3936*s43 + 128*s43*z'' + 84*s43^2 + z' + 182*z'' }-> 1 + s67 + s1 :|: s67 >= 0, s67 <= 30 * z'' + 80 * (1 + s43) + 12 * ((1 + s43) * (1 + s43)) + 8 * (z'' * (1 + s43)) + 384, s43 >= 0, s43 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 1479 + z' + 54*z'' }-> 1 + s68 + s2 :|: s68 >= 0, s68 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1477 + 3768*z + 84*z^2 }-> 1 + s46 :|: s46 >= 0, s46 <= 30 * 0 + 80 * z + 12 * (z * z) + 8 * (0 * z) + 384, z >= 0 LOG2(z, z') -{ 83 }-> 1 + s47 + s14 :|: s47 >= 0, s47 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 91 + 8*s23 }-> 1 + s48 + s15 :|: s48 >= 0, s48 <= 6 * 0 + 4 * (1 + s23) + 2 * ((1 + s23) * 0) + 0 * 0 + 18, s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 83 }-> 1 + s49 + s16 :|: s49 >= 0, s49 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s50 + s17 :|: s50 >= 0, s50 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s24 + 32*s24*z + 346*z + 7*z^2 }-> 1 + s51 + s18 :|: s51 >= 0, s51 <= 6 * (1 + (z - 1)) + 4 * (1 + s24) + 2 * ((1 + s24) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s52 + s19 :|: s52 >= 0, s52 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s53 + s20 :|: s53 >= 0, s53 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s25 + 32*s25*z + 346*z + 7*z^2 }-> 1 + s54 + s21 :|: s54 >= 0, s54 <= 6 * z + 4 * (1 + s25) + 2 * ((1 + s25) * z) + z * z + 18, s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s55 + s22 :|: s55 >= 0, s55 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 81 }-> 1 + s56 + s34 :|: s56 >= 0, s56 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 89 + 8*s26 + z' }-> 1 + s57 + s35 :|: s57 >= 0, s57 <= 6 * 0 + 4 * (1 + s26) + 2 * ((1 + s26) * 0) + 0 * 0 + 18, s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 81 + z' }-> 1 + s58 + s36 :|: s58 >= 0, s58 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s59 + s37 :|: s59 >= 0, s59 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s27 + 32*s27*z + 346*z + 7*z^2 + z' }-> 1 + s60 + s38 :|: s60 >= 0, s60 <= 6 * (1 + (z - 1)) + 4 * (1 + s27) + 2 * ((1 + s27) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s61 + s39 :|: s61 >= 0, s61 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s62 + s40 :|: s62 >= 0, s62 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s28 + 32*s28*z + 346*z + 7*z^2 + z' }-> 1 + s63 + s41 :|: s63 >= 0, s63 <= 6 * z + 4 * (1 + s28) + 2 * ((1 + s28) * z) + z * z + 18, s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s64 + s42 :|: s64 >= 0, s64 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s42 >= 0, s42 <= z', z' >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> s79 :|: s79 >= 0, s79 <= z'', z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> s80 :|: s80 >= 0, s80 <= z'', z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> s81 :|: s81 >= 0, s81 <= z'', s45 >= 0, s45 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> s82 :|: s82 >= 0, s82 <= z'', z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 0, z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 0, z = 0, z' = 0 log2(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + s30, s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> s72 :|: s72 >= 0, s72 <= 0, z' >= 0, z = 0 log2(z, z') -{ 0 }-> s73 :|: s73 >= 0, s73 <= 0, s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> s74 :|: s74 >= 0, s74 <= 1 + s31, s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> s75 :|: s75 >= 0, s75 <= 0, s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> s76 :|: s76 >= 0, s76 <= 0, z >= 0, z' = 0 log2(z, z') -{ 0 }-> s77 :|: s77 >= 0, s77 <= 1 + s32, s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> s78 :|: s78 >= 0, s78 <= 0, z' >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] IF: runtime: O(n^2) [79 + 314*z' + 32*z'*z'' + 7*z'^2 + 8*z''], size: O(n^2) [18 + 6*z' + 2*z'*z'' + z'^2 + 4*z''] LOG2: runtime: O(n^2) [1476 + 3768*z + 128*z*z' + 84*z^2 + 54*z'], size: O(n^2) [384 + 80*z + 8*z*z' + 12*z^2 + 30*z'] log2: runtime: O(1) [0], size: O(n^1) [z'] if: runtime: O(1) [0], size: O(n^1) [z''] LOG: runtime: O(n^2) [1477 + 3768*z + 84*z^2], size: O(n^2) [385 + 80*z + 12*z^2] ---------------------------------------- (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: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1479 + 54*z'' }-> 1 + s65 + s' :|: s65 >= 0, s65 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1480 + 54*z'' }-> 1 + s66 + s'' :|: s66 >= 0, s66 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 5331 + 3936*s43 + 128*s43*z'' + 84*s43^2 + z' + 182*z'' }-> 1 + s67 + s1 :|: s67 >= 0, s67 <= 30 * z'' + 80 * (1 + s43) + 12 * ((1 + s43) * (1 + s43)) + 8 * (z'' * (1 + s43)) + 384, s43 >= 0, s43 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 1479 + z' + 54*z'' }-> 1 + s68 + s2 :|: s68 >= 0, s68 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1477 + 3768*z + 84*z^2 }-> 1 + s46 :|: s46 >= 0, s46 <= 30 * 0 + 80 * z + 12 * (z * z) + 8 * (0 * z) + 384, z >= 0 LOG2(z, z') -{ 83 }-> 1 + s47 + s14 :|: s47 >= 0, s47 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 91 + 8*s23 }-> 1 + s48 + s15 :|: s48 >= 0, s48 <= 6 * 0 + 4 * (1 + s23) + 2 * ((1 + s23) * 0) + 0 * 0 + 18, s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 83 }-> 1 + s49 + s16 :|: s49 >= 0, s49 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s50 + s17 :|: s50 >= 0, s50 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s24 + 32*s24*z + 346*z + 7*z^2 }-> 1 + s51 + s18 :|: s51 >= 0, s51 <= 6 * (1 + (z - 1)) + 4 * (1 + s24) + 2 * ((1 + s24) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s52 + s19 :|: s52 >= 0, s52 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s53 + s20 :|: s53 >= 0, s53 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s25 + 32*s25*z + 346*z + 7*z^2 }-> 1 + s54 + s21 :|: s54 >= 0, s54 <= 6 * z + 4 * (1 + s25) + 2 * ((1 + s25) * z) + z * z + 18, s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s55 + s22 :|: s55 >= 0, s55 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 81 }-> 1 + s56 + s34 :|: s56 >= 0, s56 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 89 + 8*s26 + z' }-> 1 + s57 + s35 :|: s57 >= 0, s57 <= 6 * 0 + 4 * (1 + s26) + 2 * ((1 + s26) * 0) + 0 * 0 + 18, s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 81 + z' }-> 1 + s58 + s36 :|: s58 >= 0, s58 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s59 + s37 :|: s59 >= 0, s59 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s27 + 32*s27*z + 346*z + 7*z^2 + z' }-> 1 + s60 + s38 :|: s60 >= 0, s60 <= 6 * (1 + (z - 1)) + 4 * (1 + s27) + 2 * ((1 + s27) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s61 + s39 :|: s61 >= 0, s61 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s62 + s40 :|: s62 >= 0, s62 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s28 + 32*s28*z + 346*z + 7*z^2 + z' }-> 1 + s63 + s41 :|: s63 >= 0, s63 <= 6 * z + 4 * (1 + s28) + 2 * ((1 + s28) * z) + z * z + 18, s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s64 + s42 :|: s64 >= 0, s64 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s42 >= 0, s42 <= z', z' >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> s79 :|: s79 >= 0, s79 <= z'', z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> s80 :|: s80 >= 0, s80 <= z'', z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> s81 :|: s81 >= 0, s81 <= z'', s45 >= 0, s45 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> s82 :|: s82 >= 0, s82 <= z'', z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 0, z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 0, z = 0, z' = 0 log2(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + s30, s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> s72 :|: s72 >= 0, s72 <= 0, z' >= 0, z = 0 log2(z, z') -{ 0 }-> s73 :|: s73 >= 0, s73 <= 0, s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> s74 :|: s74 >= 0, s74 <= 1 + s31, s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> s75 :|: s75 >= 0, s75 <= 0, s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> s76 :|: s76 >= 0, s76 <= 0, z >= 0, z' = 0 log2(z, z') -{ 0 }-> s77 :|: s77 >= 0, s77 <= 1 + s32, s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> s78 :|: s78 >= 0, s78 <= 0, z' >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] IF: runtime: O(n^2) [79 + 314*z' + 32*z'*z'' + 7*z'^2 + 8*z''], size: O(n^2) [18 + 6*z' + 2*z'*z'' + z'^2 + 4*z''] LOG2: runtime: O(n^2) [1476 + 3768*z + 128*z*z' + 84*z^2 + 54*z'], size: O(n^2) [384 + 80*z + 8*z*z' + 12*z^2 + 30*z'] log2: runtime: O(1) [0], size: O(n^1) [z'] if: runtime: O(1) [0], size: O(n^1) [z''] LOG: runtime: O(n^2) [1477 + 3768*z + 84*z^2], size: O(n^2) [385 + 80*z + 12*z^2] ---------------------------------------- (73) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: log after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1479 + 54*z'' }-> 1 + s65 + s' :|: s65 >= 0, s65 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1480 + 54*z'' }-> 1 + s66 + s'' :|: s66 >= 0, s66 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 5331 + 3936*s43 + 128*s43*z'' + 84*s43^2 + z' + 182*z'' }-> 1 + s67 + s1 :|: s67 >= 0, s67 <= 30 * z'' + 80 * (1 + s43) + 12 * ((1 + s43) * (1 + s43)) + 8 * (z'' * (1 + s43)) + 384, s43 >= 0, s43 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 1479 + z' + 54*z'' }-> 1 + s68 + s2 :|: s68 >= 0, s68 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1477 + 3768*z + 84*z^2 }-> 1 + s46 :|: s46 >= 0, s46 <= 30 * 0 + 80 * z + 12 * (z * z) + 8 * (0 * z) + 384, z >= 0 LOG2(z, z') -{ 83 }-> 1 + s47 + s14 :|: s47 >= 0, s47 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 91 + 8*s23 }-> 1 + s48 + s15 :|: s48 >= 0, s48 <= 6 * 0 + 4 * (1 + s23) + 2 * ((1 + s23) * 0) + 0 * 0 + 18, s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 83 }-> 1 + s49 + s16 :|: s49 >= 0, s49 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s50 + s17 :|: s50 >= 0, s50 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s24 + 32*s24*z + 346*z + 7*z^2 }-> 1 + s51 + s18 :|: s51 >= 0, s51 <= 6 * (1 + (z - 1)) + 4 * (1 + s24) + 2 * ((1 + s24) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s52 + s19 :|: s52 >= 0, s52 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s53 + s20 :|: s53 >= 0, s53 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s25 + 32*s25*z + 346*z + 7*z^2 }-> 1 + s54 + s21 :|: s54 >= 0, s54 <= 6 * z + 4 * (1 + s25) + 2 * ((1 + s25) * z) + z * z + 18, s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s55 + s22 :|: s55 >= 0, s55 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 81 }-> 1 + s56 + s34 :|: s56 >= 0, s56 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 89 + 8*s26 + z' }-> 1 + s57 + s35 :|: s57 >= 0, s57 <= 6 * 0 + 4 * (1 + s26) + 2 * ((1 + s26) * 0) + 0 * 0 + 18, s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 81 + z' }-> 1 + s58 + s36 :|: s58 >= 0, s58 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s59 + s37 :|: s59 >= 0, s59 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s27 + 32*s27*z + 346*z + 7*z^2 + z' }-> 1 + s60 + s38 :|: s60 >= 0, s60 <= 6 * (1 + (z - 1)) + 4 * (1 + s27) + 2 * ((1 + s27) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s61 + s39 :|: s61 >= 0, s61 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s62 + s40 :|: s62 >= 0, s62 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s28 + 32*s28*z + 346*z + 7*z^2 + z' }-> 1 + s63 + s41 :|: s63 >= 0, s63 <= 6 * z + 4 * (1 + s28) + 2 * ((1 + s28) * z) + z * z + 18, s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s64 + s42 :|: s64 >= 0, s64 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s42 >= 0, s42 <= z', z' >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> s79 :|: s79 >= 0, s79 <= z'', z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> s80 :|: s80 >= 0, s80 <= z'', z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> s81 :|: s81 >= 0, s81 <= z'', s45 >= 0, s45 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> s82 :|: s82 >= 0, s82 <= z'', z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 0, z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 0, z = 0, z' = 0 log2(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + s30, s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> s72 :|: s72 >= 0, s72 <= 0, z' >= 0, z = 0 log2(z, z') -{ 0 }-> s73 :|: s73 >= 0, s73 <= 0, s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> s74 :|: s74 >= 0, s74 <= 1 + s31, s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> s75 :|: s75 >= 0, s75 <= 0, s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> s76 :|: s76 >= 0, s76 <= 0, z >= 0, z' = 0 log2(z, z') -{ 0 }-> s77 :|: s77 >= 0, s77 <= 1 + s32, s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> s78 :|: s78 >= 0, s78 <= 0, z' >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {log} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] IF: runtime: O(n^2) [79 + 314*z' + 32*z'*z'' + 7*z'^2 + 8*z''], size: O(n^2) [18 + 6*z' + 2*z'*z'' + z'^2 + 4*z''] LOG2: runtime: O(n^2) [1476 + 3768*z + 128*z*z' + 84*z^2 + 54*z'], size: O(n^2) [384 + 80*z + 8*z*z' + 12*z^2 + 30*z'] log2: runtime: O(1) [0], size: O(n^1) [z'] if: runtime: O(1) [0], size: O(n^1) [z''] LOG: runtime: O(n^2) [1477 + 3768*z + 84*z^2], size: O(n^2) [385 + 80*z + 12*z^2] log: runtime: ?, size: O(1) [0] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: log after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: HALF(z) -{ 1 }-> 1 :|: z = 1 + 0 HALF(z) -{ 1 }-> 0 :|: z = 0 HALF(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 IF(z, z', z'') -{ 1 }-> 0 :|: z = 2, z'' - 1 >= 0, z' >= 0 IF(z, z', z'') -{ 1479 + 54*z'' }-> 1 + s65 + s' :|: s65 >= 0, s65 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1480 + 54*z'' }-> 1 + s66 + s'' :|: s66 >= 0, s66 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 5331 + 3936*s43 + 128*s43*z'' + 84*s43^2 + z' + 182*z'' }-> 1 + s67 + s1 :|: s67 >= 0, s67 <= 30 * z'' + 80 * (1 + s43) + 12 * ((1 + s43) * (1 + s43)) + 8 * (z'' * (1 + s43)) + 384, s43 >= 0, s43 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 1479 + z' + 54*z'' }-> 1 + s68 + s2 :|: s68 >= 0, s68 <= 30 * z'' + 80 * 0 + 12 * (0 * 0) + 8 * (z'' * 0) + 384, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s33 :|: s33 >= 0, s33 <= z - 1, z - 1 >= 0 LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 LE(z, z') -{ 2 + z' }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0 LOG(z) -{ 1477 + 3768*z + 84*z^2 }-> 1 + s46 :|: s46 >= 0, s46 <= 30 * 0 + 80 * z + 12 * (z * z) + 8 * (0 * z) + 384, z >= 0 LOG2(z, z') -{ 83 }-> 1 + s47 + s14 :|: s47 >= 0, s47 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s14 >= 0, s14 <= 1 + 0 + 1, z = 0, z' = 0 LOG2(z, z') -{ 91 + 8*s23 }-> 1 + s48 + s15 :|: s48 >= 0, s48 <= 6 * 0 + 4 * (1 + s23) + 2 * ((1 + s23) * 0) + 0 * 0 + 18, s23 >= 0, s23 <= z' - 1, s15 >= 0, s15 <= 1 + 0 + 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 83 }-> 1 + s49 + s16 :|: s49 >= 0, s49 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s16 >= 0, s16 <= 1 + 0 + 1, z' >= 0, z = 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s50 + s17 :|: s50 >= 0, s50 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s17 >= 0, s17 <= 1 + 0 + 1, s3 >= 0, s3 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s24 + 32*s24*z + 346*z + 7*z^2 }-> 1 + s51 + s18 :|: s51 >= 0, s51 <= 6 * (1 + (z - 1)) + 4 * (1 + s24) + 2 * ((1 + s24) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s24 >= 0, s24 <= z' - 1, s18 >= 0, s18 <= 1 + 0 + 1, s4 >= 0, s4 <= 2, z' - 1 >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s52 + s19 :|: s52 >= 0, s52 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s19 >= 0, s19 <= 1 + 0 + 1, s5 >= 0, s5 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s53 + s20 :|: s53 >= 0, s53 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s20 >= 0, s20 <= 1 + 0 + 1, z >= 0, z' = 0 LOG2(z, z') -{ 91 + 8*s25 + 32*s25*z + 346*z + 7*z^2 }-> 1 + s54 + s21 :|: s54 >= 0, s54 <= 6 * z + 4 * (1 + s25) + 2 * ((1 + s25) * z) + z * z + 18, s25 >= 0, s25 <= z' - 1, s21 >= 0, s21 <= 1 + 0 + 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 83 + 314*z + 7*z^2 }-> 1 + s55 + s22 :|: s55 >= 0, s55 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s22 >= 0, s22 <= 1 + 0 + 1, z' >= 0, z >= 0 LOG2(z, z') -{ 81 }-> 1 + s56 + s34 :|: s56 >= 0, s56 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s34 >= 0, s34 <= 0, z = 0, z' = 0 LOG2(z, z') -{ 89 + 8*s26 + z' }-> 1 + s57 + s35 :|: s57 >= 0, s57 <= 6 * 0 + 4 * (1 + s26) + 2 * ((1 + s26) * 0) + 0 * 0 + 18, s35 >= 0, s35 <= 1 + (z' - 1), s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z = 0 LOG2(z, z') -{ 81 + z' }-> 1 + s58 + s36 :|: s58 >= 0, s58 <= 6 * 0 + 4 * 0 + 2 * (0 * 0) + 0 * 0 + 18, s36 >= 0, s36 <= z', z' >= 0, z = 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s59 + s37 :|: s59 >= 0, s59 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s37 >= 0, s37 <= 0, s6 >= 0, s6 <= 2, z - 1 >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s27 + 32*s27*z + 346*z + 7*z^2 + z' }-> 1 + s60 + s38 :|: s60 >= 0, s60 <= 6 * (1 + (z - 1)) + 4 * (1 + s27) + 2 * ((1 + s27) * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s38 >= 0, s38 <= 1 + (z' - 1), s27 >= 0, s27 <= z' - 1, s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s61 + s39 :|: s61 >= 0, s61 <= 6 * (1 + (z - 1)) + 4 * 0 + 2 * (0 * (1 + (z - 1))) + (1 + (z - 1)) * (1 + (z - 1)) + 18, s39 >= 0, s39 <= z', s8 >= 0, s8 <= 2, z' >= 0, z - 1 >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 }-> 1 + s62 + s40 :|: s62 >= 0, s62 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s40 >= 0, s40 <= 0, z >= 0, z' = 0 LOG2(z, z') -{ 89 + 8*s28 + 32*s28*z + 346*z + 7*z^2 + z' }-> 1 + s63 + s41 :|: s63 >= 0, s63 <= 6 * z + 4 * (1 + s28) + 2 * ((1 + s28) * z) + z * z + 18, s41 >= 0, s41 <= 1 + (z' - 1), s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z >= 0 LOG2(z, z') -{ 81 + 314*z + 7*z^2 + z' }-> 1 + s64 + s42 :|: s64 >= 0, s64 <= 6 * z + 4 * 0 + 2 * (0 * z) + z * z + 18, s42 >= 0, s42 <= z', z' >= 0, z >= 0 half(z) -{ 0 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 0 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> s79 :|: s79 >= 0, s79 <= z'', z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> s80 :|: s80 >= 0, s80 <= z'', z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> s81 :|: s81 >= 0, s81 <= z'', s45 >= 0, s45 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> s82 :|: s82 >= 0, s82 <= z'', z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' - 1 >= 0, z' >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s29 :|: s29 >= 0, s29 <= z - 1, z - 1 >= 0 le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 log(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 0, z >= 0 log(z) -{ 0 }-> 0 :|: z >= 0 log2(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 0, z = 0, z' = 0 log2(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + s30, s30 >= 0, s30 <= z' - 1, z' - 1 >= 0, z = 0 log2(z, z') -{ 0 }-> s72 :|: s72 >= 0, s72 <= 0, z' >= 0, z = 0 log2(z, z') -{ 0 }-> s73 :|: s73 >= 0, s73 <= 0, s10 >= 0, s10 <= 2, z - 1 >= 0, z' = 0 log2(z, z') -{ 0 }-> s74 :|: s74 >= 0, s74 <= 1 + s31, s31 >= 0, s31 <= z' - 1, s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 log2(z, z') -{ 0 }-> s75 :|: s75 >= 0, s75 <= 0, s12 >= 0, s12 <= 2, z - 1 >= 0, z' >= 0 log2(z, z') -{ 0 }-> s76 :|: s76 >= 0, s76 <= 0, z >= 0, z' = 0 log2(z, z') -{ 0 }-> s77 :|: s77 >= 0, s77 <= 1 + s32, s32 >= 0, s32 <= z' - 1, z' - 1 >= 0, z >= 0 log2(z, z') -{ 0 }-> s78 :|: s78 >= 0, s78 <= 0, z' >= 0, z >= 0 log2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] le: runtime: O(1) [0], size: O(1) [2] LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z'] inc: runtime: O(1) [0], size: O(n^1) [z] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] IF: runtime: O(n^2) [79 + 314*z' + 32*z'*z'' + 7*z'^2 + 8*z''], size: O(n^2) [18 + 6*z' + 2*z'*z'' + z'^2 + 4*z''] LOG2: runtime: O(n^2) [1476 + 3768*z + 128*z*z' + 84*z^2 + 54*z'], size: O(n^2) [384 + 80*z + 8*z*z' + 12*z^2 + 30*z'] log2: runtime: O(1) [0], size: O(n^1) [z'] if: runtime: O(1) [0], size: O(n^1) [z''] LOG: runtime: O(n^2) [1477 + 3768*z + 84*z^2], size: O(n^2) [385 + 80*z + 12*z^2] log: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (77) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (78) BOUNDS(1, n^2) ---------------------------------------- (79) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (80) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: HALF(0) -> c HALF(s(0)) -> c1 HALF(s(s(z0))) -> c2(HALF(z0)) LE(0, z0) -> c3 LE(s(z0), 0) -> c4 LE(s(z0), s(z1)) -> c5(LE(z0, z1)) INC(0) -> c6 INC(s(z0)) -> c7(INC(z0)) LOG(z0) -> c8(LOG2(z0, 0)) LOG2(z0, z1) -> c9(IF(le(z0, s(0)), z0, inc(z1)), LE(z0, s(0))) LOG2(z0, z1) -> c10(IF(le(z0, s(0)), z0, inc(z1)), INC(z1)) IF(true, z0, s(z1)) -> c11 IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0)) The (relative) TRS S consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) inc(0) -> 0 inc(s(z0)) -> s(inc(z0)) log(z0) -> log2(z0, 0) log2(z0, z1) -> if(le(z0, s(0)), z0, inc(z1)) if(true, z0, s(z1)) -> z1 if(false, z0, z1) -> log2(half(z0), z1) Rewrite Strategy: INNERMOST ---------------------------------------- (81) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence INC(s(z0)) ->^+ c7(INC(z0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [z0 / s(z0)]. The result substitution is [ ]. ---------------------------------------- (82) Complex Obligation (BEST) ---------------------------------------- (83) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: HALF(0) -> c HALF(s(0)) -> c1 HALF(s(s(z0))) -> c2(HALF(z0)) LE(0, z0) -> c3 LE(s(z0), 0) -> c4 LE(s(z0), s(z1)) -> c5(LE(z0, z1)) INC(0) -> c6 INC(s(z0)) -> c7(INC(z0)) LOG(z0) -> c8(LOG2(z0, 0)) LOG2(z0, z1) -> c9(IF(le(z0, s(0)), z0, inc(z1)), LE(z0, s(0))) LOG2(z0, z1) -> c10(IF(le(z0, s(0)), z0, inc(z1)), INC(z1)) IF(true, z0, s(z1)) -> c11 IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0)) The (relative) TRS S consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) inc(0) -> 0 inc(s(z0)) -> s(inc(z0)) log(z0) -> log2(z0, 0) log2(z0, z1) -> if(le(z0, s(0)), z0, inc(z1)) if(true, z0, s(z1)) -> z1 if(false, z0, z1) -> log2(half(z0), z1) Rewrite Strategy: INNERMOST ---------------------------------------- (84) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (85) BOUNDS(n^1, INF) ---------------------------------------- (86) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: HALF(0) -> c HALF(s(0)) -> c1 HALF(s(s(z0))) -> c2(HALF(z0)) LE(0, z0) -> c3 LE(s(z0), 0) -> c4 LE(s(z0), s(z1)) -> c5(LE(z0, z1)) INC(0) -> c6 INC(s(z0)) -> c7(INC(z0)) LOG(z0) -> c8(LOG2(z0, 0)) LOG2(z0, z1) -> c9(IF(le(z0, s(0)), z0, inc(z1)), LE(z0, s(0))) LOG2(z0, z1) -> c10(IF(le(z0, s(0)), z0, inc(z1)), INC(z1)) IF(true, z0, s(z1)) -> c11 IF(false, z0, z1) -> c12(LOG2(half(z0), z1), HALF(z0)) The (relative) TRS S consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) inc(0) -> 0 inc(s(z0)) -> s(inc(z0)) log(z0) -> log2(z0, 0) log2(z0, z1) -> if(le(z0, s(0)), z0, inc(z1)) if(true, z0, s(z1)) -> z1 if(false, z0, z1) -> log2(half(z0), z1) Rewrite Strategy: INNERMOST