WORST_CASE(Omega(n^1),O(n^2)) proof of /export/starexec/sandbox2/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).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), 743 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), 4 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) InliningProof [UPPER BOUND(ID), 580 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 2 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 296 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 98 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 139 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 89 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 14 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 201 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 54 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 139 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 12 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 128 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 230 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 63 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 75 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (66) CpxRNTS (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 8332 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 3660 ms] (72) CpxRNTS (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 4845 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 915 ms] (78) CpxRNTS (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 376 ms] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (84) CpxRNTS (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (86) CpxRNTS (87) IntTrsBoundProof [UPPER BOUND(ID), 56 ms] (88) CpxRNTS (89) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (90) CpxRNTS (91) FinalProof [FINISHED, 0 ms] (92) BOUNDS(1, n^2) (93) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CpxRelTRS (95) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (96) typed CpxTrs (97) OrderProof [LOWER BOUND(ID), 0 ms] (98) typed CpxTrs (99) RewriteLemmaProof [LOWER BOUND(ID), 277 ms] (100) BEST (101) proven lower bound (102) LowerBoundPropagationProof [FINISHED, 0 ms] (103) BOUNDS(n^1, INF) (104) typed CpxTrs (105) RewriteLemmaProof [LOWER BOUND(ID), 9 ms] (106) typed CpxTrs (107) RewriteLemmaProof [LOWER BOUND(ID), 13 ms] (108) typed CpxTrs (109) RewriteLemmaProof [LOWER BOUND(ID), 93 ms] (110) typed CpxTrs ---------------------------------------- (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)) INC(0) -> c3 INC(s(z0)) -> c4(INC(z0)) ZERO(0) -> c5 ZERO(s(z0)) -> c6 P(0) -> c7 P(s(z0)) -> c8 BITS(z0) -> c9(BITITER(z0, 0)) BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0)) BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1)) IF(true, z0, z1) -> c12(P(z1)) IF(false, z0, z1) -> c13(BITITER(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)) inc(0) -> 0 inc(s(z0)) -> s(inc(z0)) zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(z0)) -> z0 bits(z0) -> bitIter(z0, 0) bitIter(z0, z1) -> if(zero(z0), z0, inc(z1)) if(true, z0, z1) -> p(z1) if(false, z0, z1) -> bitIter(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)) INC(0) -> c3 INC(s(z0)) -> c4(INC(z0)) ZERO(0) -> c5 ZERO(s(z0)) -> c6 P(0) -> c7 P(s(z0)) -> c8 BITS(z0) -> c9(BITITER(z0, 0)) BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0)) BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1)) IF(true, z0, z1) -> c12(P(z1)) IF(false, z0, z1) -> c13(BITITER(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)) inc(0) -> 0 inc(s(z0)) -> s(inc(z0)) zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(z0)) -> z0 bits(z0) -> bitIter(z0, 0) bitIter(z0, z1) -> if(zero(z0), z0, inc(z1)) if(true, z0, z1) -> p(z1) if(false, z0, z1) -> bitIter(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] INC(0) -> c3 [1] INC(s(z0)) -> c4(INC(z0)) [1] ZERO(0) -> c5 [1] ZERO(s(z0)) -> c6 [1] P(0) -> c7 [1] P(s(z0)) -> c8 [1] BITS(z0) -> c9(BITITER(z0, 0)) [1] BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0)) [1] BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1)) [1] IF(true, z0, z1) -> c12(P(z1)) [1] IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0)) [1] half(0) -> 0 [0] half(s(0)) -> 0 [0] half(s(s(z0))) -> s(half(z0)) [0] inc(0) -> 0 [0] inc(s(z0)) -> s(inc(z0)) [0] zero(0) -> true [0] zero(s(z0)) -> false [0] p(0) -> 0 [0] p(s(z0)) -> z0 [0] bits(z0) -> bitIter(z0, 0) [0] bitIter(z0, z1) -> if(zero(z0), z0, inc(z1)) [0] if(true, z0, z1) -> p(z1) [0] if(false, z0, z1) -> bitIter(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] INC(0) -> c3 [1] INC(s(z0)) -> c4(INC(z0)) [1] ZERO(0) -> c5 [1] ZERO(s(z0)) -> c6 [1] P(0) -> c7 [1] P(s(z0)) -> c8 [1] BITS(z0) -> c9(BITITER(z0, 0)) [1] BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0)) [1] BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1)) [1] IF(true, z0, z1) -> c12(P(z1)) [1] IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0)) [1] half(0) -> 0 [0] half(s(0)) -> 0 [0] half(s(s(z0))) -> s(half(z0)) [0] inc(0) -> 0 [0] inc(s(z0)) -> s(inc(z0)) [0] zero(0) -> true [0] zero(s(z0)) -> false [0] p(0) -> 0 [0] p(s(z0)) -> z0 [0] bits(z0) -> bitIter(z0, 0) [0] bitIter(z0, z1) -> if(zero(z0), z0, inc(z1)) [0] if(true, z0, z1) -> p(z1) [0] if(false, z0, z1) -> bitIter(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 INC :: 0:s -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> c3:c4 ZERO :: 0:s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 P :: 0:s -> c7:c8 c7 :: c7:c8 c8 :: c7:c8 BITS :: 0:s -> c9 c9 :: c10:c11 -> c9 BITITER :: 0:s -> 0:s -> c10:c11 c10 :: c12:c13 -> c5:c6 -> c10:c11 IF :: true:false -> 0:s -> 0:s -> c12:c13 zero :: 0:s -> true:false inc :: 0:s -> 0:s c11 :: c12:c13 -> c3:c4 -> c10:c11 true :: true:false c12 :: c7:c8 -> c12:c13 false :: true:false c13 :: c10:c11 -> c:c1:c2 -> c12:c13 half :: 0:s -> 0:s p :: 0:s -> 0:s bits :: 0:s -> 0:s bitIter :: 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 INC_1 ZERO_1 P_1 BITS_1 BITITER_2 IF_3 (c) The following functions are completely defined: half_1 inc_1 zero_1 p_1 bits_1 bitIter_2 if_3 Due to the following rules being added: half(v0) -> 0 [0] inc(v0) -> 0 [0] zero(v0) -> null_zero [0] p(v0) -> 0 [0] bits(v0) -> 0 [0] bitIter(v0, v1) -> 0 [0] if(v0, v1, v2) -> 0 [0] And the following fresh constants: null_zero, const, const1, const2 ---------------------------------------- (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] INC(0) -> c3 [1] INC(s(z0)) -> c4(INC(z0)) [1] ZERO(0) -> c5 [1] ZERO(s(z0)) -> c6 [1] P(0) -> c7 [1] P(s(z0)) -> c8 [1] BITS(z0) -> c9(BITITER(z0, 0)) [1] BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0)) [1] BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1)) [1] IF(true, z0, z1) -> c12(P(z1)) [1] IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0)) [1] half(0) -> 0 [0] half(s(0)) -> 0 [0] half(s(s(z0))) -> s(half(z0)) [0] inc(0) -> 0 [0] inc(s(z0)) -> s(inc(z0)) [0] zero(0) -> true [0] zero(s(z0)) -> false [0] p(0) -> 0 [0] p(s(z0)) -> z0 [0] bits(z0) -> bitIter(z0, 0) [0] bitIter(z0, z1) -> if(zero(z0), z0, inc(z1)) [0] if(true, z0, z1) -> p(z1) [0] if(false, z0, z1) -> bitIter(half(z0), z1) [0] half(v0) -> 0 [0] inc(v0) -> 0 [0] zero(v0) -> null_zero [0] p(v0) -> 0 [0] bits(v0) -> 0 [0] bitIter(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 INC :: 0:s -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> c3:c4 ZERO :: 0:s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 P :: 0:s -> c7:c8 c7 :: c7:c8 c8 :: c7:c8 BITS :: 0:s -> c9 c9 :: c10:c11 -> c9 BITITER :: 0:s -> 0:s -> c10:c11 c10 :: c12:c13 -> c5:c6 -> c10:c11 IF :: true:false:null_zero -> 0:s -> 0:s -> c12:c13 zero :: 0:s -> true:false:null_zero inc :: 0:s -> 0:s c11 :: c12:c13 -> c3:c4 -> c10:c11 true :: true:false:null_zero c12 :: c7:c8 -> c12:c13 false :: true:false:null_zero c13 :: c10:c11 -> c:c1:c2 -> c12:c13 half :: 0:s -> 0:s p :: 0:s -> 0:s bits :: 0:s -> 0:s bitIter :: 0:s -> 0:s -> 0:s if :: true:false:null_zero -> 0:s -> 0:s -> 0:s null_zero :: true:false:null_zero const :: c9 const1 :: c10:c11 const2 :: c12:c13 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] INC(0) -> c3 [1] INC(s(z0)) -> c4(INC(z0)) [1] ZERO(0) -> c5 [1] ZERO(s(z0)) -> c6 [1] P(0) -> c7 [1] P(s(z0)) -> c8 [1] BITS(z0) -> c9(BITITER(z0, 0)) [1] BITITER(0, 0) -> c10(IF(true, 0, 0), ZERO(0)) [1] BITITER(0, s(z0'')) -> c10(IF(true, 0, s(inc(z0''))), ZERO(0)) [1] BITITER(0, z1) -> c10(IF(true, 0, 0), ZERO(0)) [1] BITITER(s(z0'), 0) -> c10(IF(false, s(z0'), 0), ZERO(s(z0'))) [1] BITITER(s(z0'), s(z01)) -> c10(IF(false, s(z0'), s(inc(z01))), ZERO(s(z0'))) [1] BITITER(s(z0'), z1) -> c10(IF(false, s(z0'), 0), ZERO(s(z0'))) [1] BITITER(z0, 0) -> c10(IF(null_zero, z0, 0), ZERO(z0)) [1] BITITER(z0, s(z02)) -> c10(IF(null_zero, z0, s(inc(z02))), ZERO(z0)) [1] BITITER(z0, z1) -> c10(IF(null_zero, z0, 0), ZERO(z0)) [1] BITITER(0, 0) -> c11(IF(true, 0, 0), INC(0)) [1] BITITER(0, s(z04)) -> c11(IF(true, 0, s(inc(z04))), INC(s(z04))) [1] BITITER(0, z1) -> c11(IF(true, 0, 0), INC(z1)) [1] BITITER(s(z03), 0) -> c11(IF(false, s(z03), 0), INC(0)) [1] BITITER(s(z03), s(z05)) -> c11(IF(false, s(z03), s(inc(z05))), INC(s(z05))) [1] BITITER(s(z03), z1) -> c11(IF(false, s(z03), 0), INC(z1)) [1] BITITER(z0, 0) -> c11(IF(null_zero, z0, 0), INC(0)) [1] BITITER(z0, s(z06)) -> c11(IF(null_zero, z0, s(inc(z06))), INC(s(z06))) [1] BITITER(z0, z1) -> c11(IF(null_zero, z0, 0), INC(z1)) [1] IF(true, z0, z1) -> c12(P(z1)) [1] IF(false, 0, z1) -> c13(BITITER(0, z1), HALF(0)) [1] IF(false, s(0), z1) -> c13(BITITER(0, z1), HALF(s(0))) [1] IF(false, s(s(z07)), z1) -> c13(BITITER(s(half(z07)), z1), HALF(s(s(z07)))) [1] IF(false, z0, z1) -> c13(BITITER(0, z1), HALF(z0)) [1] half(0) -> 0 [0] half(s(0)) -> 0 [0] half(s(s(z0))) -> s(half(z0)) [0] inc(0) -> 0 [0] inc(s(z0)) -> s(inc(z0)) [0] zero(0) -> true [0] zero(s(z0)) -> false [0] p(0) -> 0 [0] p(s(z0)) -> z0 [0] bits(z0) -> bitIter(z0, 0) [0] bitIter(0, 0) -> if(true, 0, 0) [0] bitIter(0, s(z09)) -> if(true, 0, s(inc(z09))) [0] bitIter(0, z1) -> if(true, 0, 0) [0] bitIter(s(z08), 0) -> if(false, s(z08), 0) [0] bitIter(s(z08), s(z010)) -> if(false, s(z08), s(inc(z010))) [0] bitIter(s(z08), z1) -> if(false, s(z08), 0) [0] bitIter(z0, 0) -> if(null_zero, z0, 0) [0] bitIter(z0, s(z011)) -> if(null_zero, z0, s(inc(z011))) [0] bitIter(z0, z1) -> if(null_zero, z0, 0) [0] if(true, z0, z1) -> p(z1) [0] if(false, 0, z1) -> bitIter(0, z1) [0] if(false, s(0), z1) -> bitIter(0, z1) [0] if(false, s(s(z012)), z1) -> bitIter(s(half(z012)), z1) [0] if(false, z0, z1) -> bitIter(0, z1) [0] half(v0) -> 0 [0] inc(v0) -> 0 [0] zero(v0) -> null_zero [0] p(v0) -> 0 [0] bits(v0) -> 0 [0] bitIter(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 INC :: 0:s -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> c3:c4 ZERO :: 0:s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 P :: 0:s -> c7:c8 c7 :: c7:c8 c8 :: c7:c8 BITS :: 0:s -> c9 c9 :: c10:c11 -> c9 BITITER :: 0:s -> 0:s -> c10:c11 c10 :: c12:c13 -> c5:c6 -> c10:c11 IF :: true:false:null_zero -> 0:s -> 0:s -> c12:c13 zero :: 0:s -> true:false:null_zero inc :: 0:s -> 0:s c11 :: c12:c13 -> c3:c4 -> c10:c11 true :: true:false:null_zero c12 :: c7:c8 -> c12:c13 false :: true:false:null_zero c13 :: c10:c11 -> c:c1:c2 -> c12:c13 half :: 0:s -> 0:s p :: 0:s -> 0:s bits :: 0:s -> 0:s bitIter :: 0:s -> 0:s -> 0:s if :: true:false:null_zero -> 0:s -> 0:s -> 0:s null_zero :: true:false:null_zero const :: c9 const1 :: c10:c11 const2 :: c12:c13 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 c5 => 0 c6 => 1 c7 => 0 c8 => 1 true => 2 false => 1 null_zero => 0 const => 0 const1 => 0 const2 => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + ZERO(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + ZERO(0) :|: z1 >= 0, z' = z1, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z1) :|: z1 >= 0, z' = z1, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z0'')) + ZERO(0) :|: z' = 1 + z0'', z0'' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z04)) + INC(1 + z04) :|: z04 >= 0, z = 0, z' = 1 + z04 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + z0', 0) + ZERO(1 + z0') :|: z0' >= 0, z = 1 + z0', z' = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + z0', 0) + ZERO(1 + z0') :|: z1 >= 0, z0' >= 0, z' = z1, z = 1 + z0' BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + z0', 1 + inc(z01)) + ZERO(1 + z0') :|: z01 >= 0, z0' >= 0, z' = 1 + z01, z = 1 + z0' BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + z03, 0) + INC(z1) :|: z1 >= 0, z' = z1, z = 1 + z03, z03 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + z03, 0) + INC(0) :|: z = 1 + z03, z03 >= 0, z' = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + z03, 1 + inc(z05)) + INC(1 + z05) :|: z = 1 + z03, z03 >= 0, z05 >= 0, z' = 1 + z05 BITITER(z, z') -{ 1 }-> 1 + IF(0, z0, 0) + ZERO(z0) :|: z = z0, z0 >= 0, z' = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z0, 0) + ZERO(z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z0, 0) + INC(z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z0, 0) + INC(0) :|: z = z0, z0 >= 0, z' = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z0, 1 + inc(z02)) + ZERO(z0) :|: z = z0, z02 >= 0, z' = 1 + z02, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z0, 1 + inc(z06)) + INC(1 + z06) :|: z = z0, z06 >= 0, z' = 1 + z06, z0 >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z0, 0) :|: z = z0, z0 >= 0 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 }-> 1 + P(z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z1) + HALF(z0) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z1) + HALF(0) :|: z1 >= 0, z = 1, z' = 0, z'' = z1 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z1) + HALF(1 + 0) :|: z1 >= 0, z = 1, z' = 1 + 0, z'' = z1 IF(z, z', z'') -{ 1 }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z1 >= 0, z' = z1, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z09)) :|: z' = 1 + z09, z09 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + z08, 0) :|: z08 >= 0, z = 1 + z08, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + z08, 0) :|: z08 >= 0, z1 >= 0, z = 1 + z08, z' = z1 bitIter(z, z') -{ 0 }-> if(1, 1 + z08, 1 + inc(z010)) :|: z08 >= 0, z = 1 + z08, z' = 1 + z010, z010 >= 0 bitIter(z, z') -{ 0 }-> if(0, z0, 0) :|: z = z0, z0 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z0, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 bitIter(z, z') -{ 0 }-> if(0, z0, 1 + inc(z011)) :|: z = z0, z011 >= 0, z' = 1 + z011, z0 >= 0 bitIter(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 bits(z) -{ 0 }-> bitIter(z0, 0) :|: z = z0, z0 >= 0 bits(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 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 }-> p(z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1 if(z, z', z'') -{ 0 }-> bitIter(0, z1) :|: z1 >= 0, z = 1, z' = 0, z'' = z1 if(z, z', z'') -{ 0 }-> bitIter(0, z1) :|: z1 >= 0, z = 1, z' = 1 + 0, z'' = z1 if(z, z', z'') -{ 0 }-> bitIter(0, z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 if(z, z', z'') -{ 0 }-> bitIter(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 p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0 zero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (13) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: ZERO(z) -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 P(z) -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z1) :|: z1 >= 0, z' = z1, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z1 >= 0, z' = z1, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z0'')) + 0 :|: z' = 1 + z0'', z0'' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z04)) + INC(1 + z04) :|: z04 >= 0, z = 0, z' = 1 + z04 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + z0', 0) + 1 :|: z0' >= 0, z = 1 + z0', z' = 0, 1 + z0' = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + z0', 0) + 1 :|: z1 >= 0, z0' >= 0, z' = z1, z = 1 + z0', 1 + z0' = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + z0', 1 + inc(z01)) + 1 :|: z01 >= 0, z0' >= 0, z' = 1 + z01, z = 1 + z0', 1 + z0' = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + z03, 0) + INC(z1) :|: z1 >= 0, z' = z1, z = 1 + z03, z03 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + z03, 0) + INC(0) :|: z = 1 + z03, z03 >= 0, z' = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + z03, 1 + inc(z05)) + INC(1 + z05) :|: z = 1 + z03, z03 >= 0, z05 >= 0, z' = 1 + z05 BITITER(z, z') -{ 1 }-> 1 + IF(0, z0, 0) + INC(z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z0, 0) + INC(0) :|: z = z0, z0 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z0, 0) + 1 :|: z = z0, z0 >= 0, z' = 0, z0 = 1 + z0', z0' >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z0, 0) + 1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, z0 = 1 + z0', z0' >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z0, 0) + 0 :|: z = z0, z0 >= 0, z' = 0, z0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z0, 0) + 0 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, z0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z0, 1 + inc(z02)) + 1 :|: z = z0, z02 >= 0, z' = 1 + z02, z0 >= 0, z0 = 1 + z0', z0' >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z0, 1 + inc(z02)) + 0 :|: z = z0, z02 >= 0, z' = 1 + z02, z0 >= 0, z0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z0, 1 + inc(z06)) + INC(1 + z06) :|: z = z0, z06 >= 0, z' = 1 + z06, z0 >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z0, 0) :|: z = z0, z0 >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1, z1 = 1 + z0', z0' >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1, z1 = 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z1) + HALF(z0) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z1) + HALF(0) :|: z1 >= 0, z = 1, z' = 0, z'' = z1 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z1) + HALF(1 + 0) :|: z1 >= 0, z = 1, z' = 1 + 0, z'' = z1 IF(z, z', z'') -{ 1 }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z1 >= 0, z' = z1, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z09)) :|: z' = 1 + z09, z09 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + z08, 0) :|: z08 >= 0, z = 1 + z08, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + z08, 0) :|: z08 >= 0, z1 >= 0, z = 1 + z08, z' = z1 bitIter(z, z') -{ 0 }-> if(1, 1 + z08, 1 + inc(z010)) :|: z08 >= 0, z = 1 + z08, z' = 1 + z010, z010 >= 0 bitIter(z, z') -{ 0 }-> if(0, z0, 0) :|: z = z0, z0 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z0, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 bitIter(z, z') -{ 0 }-> if(0, z0, 1 + inc(z011)) :|: z = z0, z011 >= 0, z' = 1 + z011, z0 >= 0 bitIter(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 bits(z) -{ 0 }-> bitIter(z0, 0) :|: z = z0, z0 >= 0 bits(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 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 }-> z0' :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1, z1 = 1 + z0', z0' >= 0 if(z, z', z'') -{ 0 }-> bitIter(0, z1) :|: z1 >= 0, z = 1, z' = 0, z'' = z1 if(z, z', z'') -{ 0 }-> bitIter(0, z1) :|: z1 >= 0, z = 1, z' = 1 + 0, z'' = z1 if(z, z', z'') -{ 0 }-> bitIter(0, z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 if(z, z', z'') -{ 0 }-> bitIter(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 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1, z1 = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1, v0 >= 0, z1 = v0 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 p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0 zero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(z') :|: z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(0) :|: z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(1 + 0) :|: z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { HALF } { P } { ZERO } { inc } { zero } { INC } { half } { p } { BITITER, IF } { if, bitIter } { BITS } { bits } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(z') :|: z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(0) :|: z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(1 + 0) :|: z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {HALF}, {P}, {ZERO}, {inc}, {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(z') :|: z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(0) :|: z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(1 + 0) :|: z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {HALF}, {P}, {ZERO}, {inc}, {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} ---------------------------------------- (21) 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 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(z') :|: z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(0) :|: z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(1 + 0) :|: z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {HALF}, {P}, {ZERO}, {inc}, {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: ?, size: O(n^1) [z] ---------------------------------------- (23) 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 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(z') :|: z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(0) :|: z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(0, z'') + HALF(1 + 0) :|: z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 1 }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {P}, {ZERO}, {inc}, {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {P}, {ZERO}, {inc}, {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: P after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {P}, {ZERO}, {inc}, {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: ?, size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: P after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {ZERO}, {inc}, {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {ZERO}, {inc}, {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: ZERO after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {ZERO}, {inc}, {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: ?, size: O(1) [1] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: ZERO after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {inc}, {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {inc}, {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (39) 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 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {inc}, {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: ?, size: O(n^1) [z] ---------------------------------------- (41) 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 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + INC(1 + (z' - 1)) :|: z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 1 :|: z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + inc(z' - 1)) + 0 :|: z' - 1 >= 0, z >= 0, z = 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + inc(z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + s3) + INC(1 + (z' - 1)) :|: s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + INC(1 + (z' - 1)) :|: s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + s5) + INC(1 + (z' - 1)) :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: zero after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + s3) + INC(1 + (z' - 1)) :|: s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + INC(1 + (z' - 1)) :|: s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + s5) + INC(1 + (z' - 1)) :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {zero}, {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: ?, size: O(1) [2] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: zero after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + s3) + INC(1 + (z' - 1)) :|: s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + INC(1 + (z' - 1)) :|: s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + s5) + INC(1 + (z' - 1)) :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + s3) + INC(1 + (z' - 1)) :|: s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + INC(1 + (z' - 1)) :|: s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + s5) + INC(1 + (z' - 1)) :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (51) 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 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + s3) + INC(1 + (z' - 1)) :|: s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + INC(1 + (z' - 1)) :|: s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + s5) + INC(1 + (z' - 1)) :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {INC}, {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: ?, size: O(n^1) [z] ---------------------------------------- (53) 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 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(z') :|: z' >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 0) + INC(0) :|: z = 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1 }-> 1 + IF(2, 0, 1 + s3) + INC(1 + (z' - 1)) :|: s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(z') :|: z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 0) + INC(0) :|: z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + INC(1 + (z' - 1)) :|: s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(z') :|: z' >= 0, z >= 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 0) + INC(0) :|: z >= 0, z' = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1 }-> 1 + IF(0, z, 1 + s5) + INC(1 + (z' - 1)) :|: s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s15 :|: s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s17 :|: s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s3) + s16 :|: s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + s18 :|: s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 0) + s20 :|: s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s21 :|: s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s23 :|: s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s5) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (57) 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 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s15 :|: s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s17 :|: s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s3) + s16 :|: s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + s18 :|: s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 0) + s20 :|: s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s21 :|: s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s23 :|: s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s5) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {half}, {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: ?, size: O(n^1) [z] ---------------------------------------- (59) 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 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s15 :|: s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s17 :|: s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s3) + s16 :|: s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + s18 :|: s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 0) + s20 :|: s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s21 :|: s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s23 :|: s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s5) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(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 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (61) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s15 :|: s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s17 :|: s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s3) + s16 :|: s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + s18 :|: s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 0) + s20 :|: s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s21 :|: s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s23 :|: s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s5) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(1 + s24, z'') + s1 :|: s24 >= 0, s24 <= 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 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(1 + s26, z'') :|: s26 >= 0, s26 <= 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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s15 :|: s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s17 :|: s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s3) + s16 :|: s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + s18 :|: s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 0) + s20 :|: s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s21 :|: s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s23 :|: s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s5) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(1 + s24, z'') + s1 :|: s24 >= 0, s24 <= 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 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(1 + s26, z'') :|: s26 >= 0, s26 <= 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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {p}, {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: p after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s15 :|: s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s17 :|: s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s3) + s16 :|: s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + s18 :|: s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 0) + s20 :|: s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s21 :|: s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s23 :|: s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s5) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(1 + s24, z'') + s1 :|: s24 >= 0, s24 <= 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 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(1 + s26, z'') :|: s26 >= 0, s26 <= 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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (67) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s15 :|: s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s17 :|: s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s3) + s16 :|: s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + s18 :|: s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 0) + s20 :|: s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s21 :|: s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s23 :|: s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s5) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(1 + s24, z'') + s1 :|: s24 >= 0, s24 <= 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 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(1 + s26, z'') :|: s26 >= 0, s26 <= 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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: BITITER after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 47 + 12*z + 4*z*z' + 2*z^2 + 12*z' Computed SIZE bound using KoAT for: IF after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 210 + 18*z' + 4*z'*z'' + 2*z'^2 + 52*z'' ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s15 :|: s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s17 :|: s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s3) + s16 :|: s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + s18 :|: s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 0) + s20 :|: s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s21 :|: s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s23 :|: s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s5) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(1 + s24, z'') + s1 :|: s24 >= 0, s24 <= 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 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(1 + s26, z'') :|: s26 >= 0, s26 <= 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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {BITITER,IF}, {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: O(1) [0], size: O(n^1) [z] BITITER: runtime: ?, size: O(n^2) [47 + 12*z + 4*z*z' + 2*z^2 + 12*z'] IF: runtime: ?, size: O(n^2) [210 + 18*z' + 4*z'*z'' + 2*z'^2 + 52*z''] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: BITITER after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 425 + 230*z + 32*z*z' + 7*z^2 + 62*z' Computed RUNTIME bound using KoAT for: IF after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1494 + 218*z' + 32*z'*z'' + 7*z'^2 + 216*z'' ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + s15 :|: s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 0) + s17 :|: s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 0) + 0 :|: z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 }-> 1 + IF(2, 0, 1 + s10) + 0 :|: s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(2, 0, 1 + s3) + s16 :|: s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + s18 :|: s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 0) + s20 :|: s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 0) + 1 :|: z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(1, 1 + (z - 1), 1 + s11) + 1 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(1, 1 + (z - 1), 1 + s4) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + s21 :|: s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 0) + s23 :|: s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 1 :|: z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 0) + 0 :|: z' >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s12) + 1 :|: s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 2 }-> 1 + IF(0, z, 1 + s13) + 0 :|: s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 2 + z' }-> 1 + IF(0, z, 1 + s5) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITS(z) -{ 1 }-> 1 + BITITER(z, 0) :|: z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 3 }-> 1 + BITITER(0, z'') + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 4 }-> 1 + BITITER(0, z'') + s'' :|: s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(0, z'') + s2 :|: s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 IF(z, z', z'') -{ 3 + z' }-> 1 + BITITER(1 + s24, z'') + s1 :|: s24 >= 0, s24 <= 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 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(1 + s26, z'') :|: s26 >= 0, s26 <= 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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: O(1) [0], size: O(n^1) [z] BITITER: runtime: O(n^2) [425 + 230*z + 32*z*z' + 7*z^2 + 62*z'], size: O(n^2) [47 + 12*z + 4*z*z' + 2*z^2 + 12*z'] IF: runtime: O(n^2) [1494 + 218*z' + 32*z'*z'' + 7*z'^2 + 216*z''], size: O(n^2) [210 + 18*z' + 4*z'*z'' + 2*z'^2 + 52*z''] ---------------------------------------- (73) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1496 }-> 1 + s28 + s15 :|: s28 >= 0, s28 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s3 + z' }-> 1 + s29 + s16 :|: s29 >= 0, s29 <= 18 * 0 + 52 * (1 + s3) + 4 * ((1 + s3) * 0) + 2 * (0 * 0) + 210, s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1496 + z' }-> 1 + s30 + s17 :|: s30 >= 0, s30 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s31 + s18 :|: s31 >= 0, s31 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s4 + 32*s4*z + 250*z + 7*z^2 + z' }-> 1 + s32 + s19 :|: s32 >= 0, s32 <= 18 * (1 + (z - 1)) + 52 * (1 + s4) + 4 * ((1 + s4) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s33 + s20 :|: s33 >= 0, s33 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s34 + s21 :|: s34 >= 0, s34 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s5 + 32*s5*z + 250*z + 7*z^2 + z' }-> 1 + s35 + s22 :|: s35 >= 0, s35 <= 18 * z + 52 * (1 + s5) + 4 * ((1 + s5) * z) + 2 * (z * z) + 210, s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s36 + s23 :|: s36 >= 0, s36 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 1496 }-> 1 + s41 + 0 :|: s41 >= 0, s41 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 1712 + 216*s10 }-> 1 + s42 + 0 :|: s42 >= 0, s42 <= 18 * 0 + 52 * (1 + s10) + 4 * ((1 + s10) * 0) + 2 * (0 * 0) + 210, s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 }-> 1 + s43 + 0 :|: s43 >= 0, s43 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s44 + 1 :|: s44 >= 0, s44 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1712 + 216*s11 + 32*s11*z + 250*z + 7*z^2 }-> 1 + s45 + 1 :|: s45 >= 0, s45 <= 18 * (1 + (z - 1)) + 52 * (1 + s11) + 4 * ((1 + s11) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s46 + 1 :|: s46 >= 0, s46 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s47 + 1 :|: s47 >= 0, s47 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s48 + 0 :|: s48 >= 0, s48 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 1712 + 216*s12 + 32*s12*z + 250*z + 7*z^2 }-> 1 + s49 + 1 :|: s49 >= 0, s49 <= 18 * z + 52 * (1 + s12) + 4 * ((1 + s12) * z) + 2 * (z * z) + 210, s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1712 + 216*s13 + 32*s13*z + 250*z + 7*z^2 }-> 1 + s50 + 0 :|: s50 >= 0, s50 <= 18 * z + 52 * (1 + s13) + 4 * ((1 + s13) * z) + 2 * (z * z) + 210, s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s51 + 1 :|: s51 >= 0, s51 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z = 0 BITS(z) -{ 426 + 230*z + 7*z^2 }-> 1 + s27 :|: s27 >= 0, s27 <= 12 * z + 12 * 0 + 4 * (0 * z) + 2 * (z * z) + 47, z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 428 + 62*z'' }-> 1 + s37 + s' :|: s37 >= 0, s37 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 429 + 62*z'' }-> 1 + s38 + s'' :|: s38 >= 0, s38 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 665 + 244*s24 + 32*s24*z'' + 7*s24^2 + z' + 94*z'' }-> 1 + s39 + s1 :|: s39 >= 0, s39 <= 12 * (1 + s24) + 12 * z'' + 4 * (z'' * (1 + s24)) + 2 * ((1 + s24) * (1 + s24)) + 47, s24 >= 0, s24 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 428 + z' + 62*z'' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(1 + s26, z'') :|: s26 >= 0, s26 <= 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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: O(1) [0], size: O(n^1) [z] BITITER: runtime: O(n^2) [425 + 230*z + 32*z*z' + 7*z^2 + 62*z'], size: O(n^2) [47 + 12*z + 4*z*z' + 2*z^2 + 12*z'] IF: runtime: O(n^2) [1494 + 218*z' + 32*z'*z'' + 7*z'^2 + 216*z''], size: O(n^2) [210 + 18*z' + 4*z'*z'' + 2*z'^2 + 52*z''] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' Computed SIZE bound using CoFloCo for: bitIter after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1496 }-> 1 + s28 + s15 :|: s28 >= 0, s28 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s3 + z' }-> 1 + s29 + s16 :|: s29 >= 0, s29 <= 18 * 0 + 52 * (1 + s3) + 4 * ((1 + s3) * 0) + 2 * (0 * 0) + 210, s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1496 + z' }-> 1 + s30 + s17 :|: s30 >= 0, s30 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s31 + s18 :|: s31 >= 0, s31 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s4 + 32*s4*z + 250*z + 7*z^2 + z' }-> 1 + s32 + s19 :|: s32 >= 0, s32 <= 18 * (1 + (z - 1)) + 52 * (1 + s4) + 4 * ((1 + s4) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s33 + s20 :|: s33 >= 0, s33 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s34 + s21 :|: s34 >= 0, s34 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s5 + 32*s5*z + 250*z + 7*z^2 + z' }-> 1 + s35 + s22 :|: s35 >= 0, s35 <= 18 * z + 52 * (1 + s5) + 4 * ((1 + s5) * z) + 2 * (z * z) + 210, s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s36 + s23 :|: s36 >= 0, s36 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 1496 }-> 1 + s41 + 0 :|: s41 >= 0, s41 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 1712 + 216*s10 }-> 1 + s42 + 0 :|: s42 >= 0, s42 <= 18 * 0 + 52 * (1 + s10) + 4 * ((1 + s10) * 0) + 2 * (0 * 0) + 210, s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 }-> 1 + s43 + 0 :|: s43 >= 0, s43 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s44 + 1 :|: s44 >= 0, s44 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1712 + 216*s11 + 32*s11*z + 250*z + 7*z^2 }-> 1 + s45 + 1 :|: s45 >= 0, s45 <= 18 * (1 + (z - 1)) + 52 * (1 + s11) + 4 * ((1 + s11) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s46 + 1 :|: s46 >= 0, s46 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s47 + 1 :|: s47 >= 0, s47 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s48 + 0 :|: s48 >= 0, s48 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 1712 + 216*s12 + 32*s12*z + 250*z + 7*z^2 }-> 1 + s49 + 1 :|: s49 >= 0, s49 <= 18 * z + 52 * (1 + s12) + 4 * ((1 + s12) * z) + 2 * (z * z) + 210, s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1712 + 216*s13 + 32*s13*z + 250*z + 7*z^2 }-> 1 + s50 + 0 :|: s50 >= 0, s50 <= 18 * z + 52 * (1 + s13) + 4 * ((1 + s13) * z) + 2 * (z * z) + 210, s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s51 + 1 :|: s51 >= 0, s51 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z = 0 BITS(z) -{ 426 + 230*z + 7*z^2 }-> 1 + s27 :|: s27 >= 0, s27 <= 12 * z + 12 * 0 + 4 * (0 * z) + 2 * (z * z) + 47, z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 428 + 62*z'' }-> 1 + s37 + s' :|: s37 >= 0, s37 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 429 + 62*z'' }-> 1 + s38 + s'' :|: s38 >= 0, s38 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 665 + 244*s24 + 32*s24*z'' + 7*s24^2 + z' + 94*z'' }-> 1 + s39 + s1 :|: s39 >= 0, s39 <= 12 * (1 + s24) + 12 * z'' + 4 * (z'' * (1 + s24)) + 2 * ((1 + s24) * (1 + s24)) + 47, s24 >= 0, s24 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 428 + z' + 62*z'' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(1 + s26, z'') :|: s26 >= 0, s26 <= 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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {if,bitIter}, {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: O(1) [0], size: O(n^1) [z] BITITER: runtime: O(n^2) [425 + 230*z + 32*z*z' + 7*z^2 + 62*z'], size: O(n^2) [47 + 12*z + 4*z*z' + 2*z^2 + 12*z'] IF: runtime: O(n^2) [1494 + 218*z' + 32*z'*z'' + 7*z'^2 + 216*z''], size: O(n^2) [210 + 18*z' + 4*z'*z'' + 2*z'^2 + 52*z''] if: runtime: ?, size: O(n^1) [z''] bitIter: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed RUNTIME bound using CoFloCo for: bitIter after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1496 }-> 1 + s28 + s15 :|: s28 >= 0, s28 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s3 + z' }-> 1 + s29 + s16 :|: s29 >= 0, s29 <= 18 * 0 + 52 * (1 + s3) + 4 * ((1 + s3) * 0) + 2 * (0 * 0) + 210, s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1496 + z' }-> 1 + s30 + s17 :|: s30 >= 0, s30 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s31 + s18 :|: s31 >= 0, s31 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s4 + 32*s4*z + 250*z + 7*z^2 + z' }-> 1 + s32 + s19 :|: s32 >= 0, s32 <= 18 * (1 + (z - 1)) + 52 * (1 + s4) + 4 * ((1 + s4) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s33 + s20 :|: s33 >= 0, s33 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s34 + s21 :|: s34 >= 0, s34 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s5 + 32*s5*z + 250*z + 7*z^2 + z' }-> 1 + s35 + s22 :|: s35 >= 0, s35 <= 18 * z + 52 * (1 + s5) + 4 * ((1 + s5) * z) + 2 * (z * z) + 210, s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s36 + s23 :|: s36 >= 0, s36 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 1496 }-> 1 + s41 + 0 :|: s41 >= 0, s41 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 1712 + 216*s10 }-> 1 + s42 + 0 :|: s42 >= 0, s42 <= 18 * 0 + 52 * (1 + s10) + 4 * ((1 + s10) * 0) + 2 * (0 * 0) + 210, s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 }-> 1 + s43 + 0 :|: s43 >= 0, s43 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s44 + 1 :|: s44 >= 0, s44 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1712 + 216*s11 + 32*s11*z + 250*z + 7*z^2 }-> 1 + s45 + 1 :|: s45 >= 0, s45 <= 18 * (1 + (z - 1)) + 52 * (1 + s11) + 4 * ((1 + s11) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s46 + 1 :|: s46 >= 0, s46 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s47 + 1 :|: s47 >= 0, s47 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s48 + 0 :|: s48 >= 0, s48 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 1712 + 216*s12 + 32*s12*z + 250*z + 7*z^2 }-> 1 + s49 + 1 :|: s49 >= 0, s49 <= 18 * z + 52 * (1 + s12) + 4 * ((1 + s12) * z) + 2 * (z * z) + 210, s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1712 + 216*s13 + 32*s13*z + 250*z + 7*z^2 }-> 1 + s50 + 0 :|: s50 >= 0, s50 <= 18 * z + 52 * (1 + s13) + 4 * ((1 + s13) * z) + 2 * (z * z) + 210, s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s51 + 1 :|: s51 >= 0, s51 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z = 0 BITS(z) -{ 426 + 230*z + 7*z^2 }-> 1 + s27 :|: s27 >= 0, s27 <= 12 * z + 12 * 0 + 4 * (0 * z) + 2 * (z * z) + 47, z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 428 + 62*z'' }-> 1 + s37 + s' :|: s37 >= 0, s37 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 429 + 62*z'' }-> 1 + s38 + s'' :|: s38 >= 0, s38 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 665 + 244*s24 + 32*s24*z'' + 7*s24^2 + z' + 94*z'' }-> 1 + s39 + s1 :|: s39 >= 0, s39 <= 12 * (1 + s24) + 12 * z'' + 4 * (z'' * (1 + s24)) + 2 * ((1 + s24) * (1 + s24)) + 47, s24 >= 0, s24 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 428 + z' + 62*z'' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z = 0, z' = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 0) :|: z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(2, 0, 1 + s7) :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> if(1, 1 + (z - 1), 1 + s8) :|: s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> if(0, z, 0) :|: z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> if(0, z, 1 + s9) :|: s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> bitIter(z, 0) :|: z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> bitIter(0, z'') :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> bitIter(1 + s26, z'') :|: s26 >= 0, s26 <= 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 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: O(1) [0], size: O(n^1) [z] BITITER: runtime: O(n^2) [425 + 230*z + 32*z*z' + 7*z^2 + 62*z'], size: O(n^2) [47 + 12*z + 4*z*z' + 2*z^2 + 12*z'] IF: runtime: O(n^2) [1494 + 218*z' + 32*z'*z'' + 7*z'^2 + 216*z''], size: O(n^2) [210 + 18*z' + 4*z'*z'' + 2*z'^2 + 52*z''] if: runtime: O(1) [0], size: O(n^1) [z''] bitIter: runtime: O(1) [0], size: O(n^1) [z'] ---------------------------------------- (79) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1496 }-> 1 + s28 + s15 :|: s28 >= 0, s28 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s3 + z' }-> 1 + s29 + s16 :|: s29 >= 0, s29 <= 18 * 0 + 52 * (1 + s3) + 4 * ((1 + s3) * 0) + 2 * (0 * 0) + 210, s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1496 + z' }-> 1 + s30 + s17 :|: s30 >= 0, s30 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s31 + s18 :|: s31 >= 0, s31 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s4 + 32*s4*z + 250*z + 7*z^2 + z' }-> 1 + s32 + s19 :|: s32 >= 0, s32 <= 18 * (1 + (z - 1)) + 52 * (1 + s4) + 4 * ((1 + s4) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s33 + s20 :|: s33 >= 0, s33 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s34 + s21 :|: s34 >= 0, s34 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s5 + 32*s5*z + 250*z + 7*z^2 + z' }-> 1 + s35 + s22 :|: s35 >= 0, s35 <= 18 * z + 52 * (1 + s5) + 4 * ((1 + s5) * z) + 2 * (z * z) + 210, s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s36 + s23 :|: s36 >= 0, s36 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 1496 }-> 1 + s41 + 0 :|: s41 >= 0, s41 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 1712 + 216*s10 }-> 1 + s42 + 0 :|: s42 >= 0, s42 <= 18 * 0 + 52 * (1 + s10) + 4 * ((1 + s10) * 0) + 2 * (0 * 0) + 210, s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 }-> 1 + s43 + 0 :|: s43 >= 0, s43 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s44 + 1 :|: s44 >= 0, s44 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1712 + 216*s11 + 32*s11*z + 250*z + 7*z^2 }-> 1 + s45 + 1 :|: s45 >= 0, s45 <= 18 * (1 + (z - 1)) + 52 * (1 + s11) + 4 * ((1 + s11) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s46 + 1 :|: s46 >= 0, s46 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s47 + 1 :|: s47 >= 0, s47 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s48 + 0 :|: s48 >= 0, s48 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 1712 + 216*s12 + 32*s12*z + 250*z + 7*z^2 }-> 1 + s49 + 1 :|: s49 >= 0, s49 <= 18 * z + 52 * (1 + s12) + 4 * ((1 + s12) * z) + 2 * (z * z) + 210, s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1712 + 216*s13 + 32*s13*z + 250*z + 7*z^2 }-> 1 + s50 + 0 :|: s50 >= 0, s50 <= 18 * z + 52 * (1 + s13) + 4 * ((1 + s13) * z) + 2 * (z * z) + 210, s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s51 + 1 :|: s51 >= 0, s51 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z = 0 BITS(z) -{ 426 + 230*z + 7*z^2 }-> 1 + s27 :|: s27 >= 0, s27 <= 12 * z + 12 * 0 + 4 * (0 * z) + 2 * (z * z) + 47, z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 428 + 62*z'' }-> 1 + s37 + s' :|: s37 >= 0, s37 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 429 + 62*z'' }-> 1 + s38 + s'' :|: s38 >= 0, s38 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 665 + 244*s24 + 32*s24*z'' + 7*s24^2 + z' + 94*z'' }-> 1 + s39 + s1 :|: s39 >= 0, s39 <= 12 * (1 + s24) + 12 * z'' + 4 * (z'' * (1 + s24)) + 2 * ((1 + s24) * (1 + s24)) + 47, s24 >= 0, s24 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 428 + z' + 62*z'' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 0, z = 0, z' = 0 bitIter(z, z') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s7, s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0, z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0, z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s8, s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0, z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 0, z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s9, s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0, z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 0, z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z'', z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= z'', z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= z'', s26 >= 0, s26 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> s66 :|: s66 >= 0, s66 <= z'', z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: O(1) [0], size: O(n^1) [z] BITITER: runtime: O(n^2) [425 + 230*z + 32*z*z' + 7*z^2 + 62*z'], size: O(n^2) [47 + 12*z + 4*z*z' + 2*z^2 + 12*z'] IF: runtime: O(n^2) [1494 + 218*z' + 32*z'*z'' + 7*z'^2 + 216*z''], size: O(n^2) [210 + 18*z' + 4*z'*z'' + 2*z'^2 + 52*z''] if: runtime: O(1) [0], size: O(n^1) [z''] bitIter: runtime: O(1) [0], size: O(n^1) [z'] ---------------------------------------- (81) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: BITS after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 48 + 12*z + 2*z^2 ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1496 }-> 1 + s28 + s15 :|: s28 >= 0, s28 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s3 + z' }-> 1 + s29 + s16 :|: s29 >= 0, s29 <= 18 * 0 + 52 * (1 + s3) + 4 * ((1 + s3) * 0) + 2 * (0 * 0) + 210, s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1496 + z' }-> 1 + s30 + s17 :|: s30 >= 0, s30 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s31 + s18 :|: s31 >= 0, s31 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s4 + 32*s4*z + 250*z + 7*z^2 + z' }-> 1 + s32 + s19 :|: s32 >= 0, s32 <= 18 * (1 + (z - 1)) + 52 * (1 + s4) + 4 * ((1 + s4) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s33 + s20 :|: s33 >= 0, s33 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s34 + s21 :|: s34 >= 0, s34 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s5 + 32*s5*z + 250*z + 7*z^2 + z' }-> 1 + s35 + s22 :|: s35 >= 0, s35 <= 18 * z + 52 * (1 + s5) + 4 * ((1 + s5) * z) + 2 * (z * z) + 210, s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s36 + s23 :|: s36 >= 0, s36 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 1496 }-> 1 + s41 + 0 :|: s41 >= 0, s41 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 1712 + 216*s10 }-> 1 + s42 + 0 :|: s42 >= 0, s42 <= 18 * 0 + 52 * (1 + s10) + 4 * ((1 + s10) * 0) + 2 * (0 * 0) + 210, s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 }-> 1 + s43 + 0 :|: s43 >= 0, s43 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s44 + 1 :|: s44 >= 0, s44 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1712 + 216*s11 + 32*s11*z + 250*z + 7*z^2 }-> 1 + s45 + 1 :|: s45 >= 0, s45 <= 18 * (1 + (z - 1)) + 52 * (1 + s11) + 4 * ((1 + s11) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s46 + 1 :|: s46 >= 0, s46 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s47 + 1 :|: s47 >= 0, s47 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s48 + 0 :|: s48 >= 0, s48 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 1712 + 216*s12 + 32*s12*z + 250*z + 7*z^2 }-> 1 + s49 + 1 :|: s49 >= 0, s49 <= 18 * z + 52 * (1 + s12) + 4 * ((1 + s12) * z) + 2 * (z * z) + 210, s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1712 + 216*s13 + 32*s13*z + 250*z + 7*z^2 }-> 1 + s50 + 0 :|: s50 >= 0, s50 <= 18 * z + 52 * (1 + s13) + 4 * ((1 + s13) * z) + 2 * (z * z) + 210, s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s51 + 1 :|: s51 >= 0, s51 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z = 0 BITS(z) -{ 426 + 230*z + 7*z^2 }-> 1 + s27 :|: s27 >= 0, s27 <= 12 * z + 12 * 0 + 4 * (0 * z) + 2 * (z * z) + 47, z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 428 + 62*z'' }-> 1 + s37 + s' :|: s37 >= 0, s37 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 429 + 62*z'' }-> 1 + s38 + s'' :|: s38 >= 0, s38 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 665 + 244*s24 + 32*s24*z'' + 7*s24^2 + z' + 94*z'' }-> 1 + s39 + s1 :|: s39 >= 0, s39 <= 12 * (1 + s24) + 12 * z'' + 4 * (z'' * (1 + s24)) + 2 * ((1 + s24) * (1 + s24)) + 47, s24 >= 0, s24 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 428 + z' + 62*z'' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 0, z = 0, z' = 0 bitIter(z, z') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s7, s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0, z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0, z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s8, s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0, z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 0, z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s9, s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0, z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 0, z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z'', z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= z'', z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= z'', s26 >= 0, s26 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> s66 :|: s66 >= 0, s66 <= z'', z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {BITS}, {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: O(1) [0], size: O(n^1) [z] BITITER: runtime: O(n^2) [425 + 230*z + 32*z*z' + 7*z^2 + 62*z'], size: O(n^2) [47 + 12*z + 4*z*z' + 2*z^2 + 12*z'] IF: runtime: O(n^2) [1494 + 218*z' + 32*z'*z'' + 7*z'^2 + 216*z''], size: O(n^2) [210 + 18*z' + 4*z'*z'' + 2*z'^2 + 52*z''] if: runtime: O(1) [0], size: O(n^1) [z''] bitIter: runtime: O(1) [0], size: O(n^1) [z'] BITS: runtime: ?, size: O(n^2) [48 + 12*z + 2*z^2] ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: BITS after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 426 + 230*z + 7*z^2 ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1496 }-> 1 + s28 + s15 :|: s28 >= 0, s28 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s3 + z' }-> 1 + s29 + s16 :|: s29 >= 0, s29 <= 18 * 0 + 52 * (1 + s3) + 4 * ((1 + s3) * 0) + 2 * (0 * 0) + 210, s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1496 + z' }-> 1 + s30 + s17 :|: s30 >= 0, s30 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s31 + s18 :|: s31 >= 0, s31 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s4 + 32*s4*z + 250*z + 7*z^2 + z' }-> 1 + s32 + s19 :|: s32 >= 0, s32 <= 18 * (1 + (z - 1)) + 52 * (1 + s4) + 4 * ((1 + s4) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s33 + s20 :|: s33 >= 0, s33 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s34 + s21 :|: s34 >= 0, s34 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s5 + 32*s5*z + 250*z + 7*z^2 + z' }-> 1 + s35 + s22 :|: s35 >= 0, s35 <= 18 * z + 52 * (1 + s5) + 4 * ((1 + s5) * z) + 2 * (z * z) + 210, s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s36 + s23 :|: s36 >= 0, s36 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 1496 }-> 1 + s41 + 0 :|: s41 >= 0, s41 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 1712 + 216*s10 }-> 1 + s42 + 0 :|: s42 >= 0, s42 <= 18 * 0 + 52 * (1 + s10) + 4 * ((1 + s10) * 0) + 2 * (0 * 0) + 210, s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 }-> 1 + s43 + 0 :|: s43 >= 0, s43 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s44 + 1 :|: s44 >= 0, s44 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1712 + 216*s11 + 32*s11*z + 250*z + 7*z^2 }-> 1 + s45 + 1 :|: s45 >= 0, s45 <= 18 * (1 + (z - 1)) + 52 * (1 + s11) + 4 * ((1 + s11) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s46 + 1 :|: s46 >= 0, s46 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s47 + 1 :|: s47 >= 0, s47 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s48 + 0 :|: s48 >= 0, s48 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 1712 + 216*s12 + 32*s12*z + 250*z + 7*z^2 }-> 1 + s49 + 1 :|: s49 >= 0, s49 <= 18 * z + 52 * (1 + s12) + 4 * ((1 + s12) * z) + 2 * (z * z) + 210, s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1712 + 216*s13 + 32*s13*z + 250*z + 7*z^2 }-> 1 + s50 + 0 :|: s50 >= 0, s50 <= 18 * z + 52 * (1 + s13) + 4 * ((1 + s13) * z) + 2 * (z * z) + 210, s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s51 + 1 :|: s51 >= 0, s51 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z = 0 BITS(z) -{ 426 + 230*z + 7*z^2 }-> 1 + s27 :|: s27 >= 0, s27 <= 12 * z + 12 * 0 + 4 * (0 * z) + 2 * (z * z) + 47, z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 428 + 62*z'' }-> 1 + s37 + s' :|: s37 >= 0, s37 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 429 + 62*z'' }-> 1 + s38 + s'' :|: s38 >= 0, s38 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 665 + 244*s24 + 32*s24*z'' + 7*s24^2 + z' + 94*z'' }-> 1 + s39 + s1 :|: s39 >= 0, s39 <= 12 * (1 + s24) + 12 * z'' + 4 * (z'' * (1 + s24)) + 2 * ((1 + s24) * (1 + s24)) + 47, s24 >= 0, s24 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 428 + z' + 62*z'' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 0, z = 0, z' = 0 bitIter(z, z') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s7, s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0, z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0, z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s8, s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0, z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 0, z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s9, s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0, z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 0, z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z'', z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= z'', z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= z'', s26 >= 0, s26 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> s66 :|: s66 >= 0, s66 <= z'', z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: O(1) [0], size: O(n^1) [z] BITITER: runtime: O(n^2) [425 + 230*z + 32*z*z' + 7*z^2 + 62*z'], size: O(n^2) [47 + 12*z + 4*z*z' + 2*z^2 + 12*z'] IF: runtime: O(n^2) [1494 + 218*z' + 32*z'*z'' + 7*z'^2 + 216*z''], size: O(n^2) [210 + 18*z' + 4*z'*z'' + 2*z'^2 + 52*z''] if: runtime: O(1) [0], size: O(n^1) [z''] bitIter: runtime: O(1) [0], size: O(n^1) [z'] BITS: runtime: O(n^2) [426 + 230*z + 7*z^2], size: O(n^2) [48 + 12*z + 2*z^2] ---------------------------------------- (85) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (86) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1496 }-> 1 + s28 + s15 :|: s28 >= 0, s28 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s3 + z' }-> 1 + s29 + s16 :|: s29 >= 0, s29 <= 18 * 0 + 52 * (1 + s3) + 4 * ((1 + s3) * 0) + 2 * (0 * 0) + 210, s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1496 + z' }-> 1 + s30 + s17 :|: s30 >= 0, s30 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s31 + s18 :|: s31 >= 0, s31 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s4 + 32*s4*z + 250*z + 7*z^2 + z' }-> 1 + s32 + s19 :|: s32 >= 0, s32 <= 18 * (1 + (z - 1)) + 52 * (1 + s4) + 4 * ((1 + s4) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s33 + s20 :|: s33 >= 0, s33 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s34 + s21 :|: s34 >= 0, s34 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s5 + 32*s5*z + 250*z + 7*z^2 + z' }-> 1 + s35 + s22 :|: s35 >= 0, s35 <= 18 * z + 52 * (1 + s5) + 4 * ((1 + s5) * z) + 2 * (z * z) + 210, s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s36 + s23 :|: s36 >= 0, s36 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 1496 }-> 1 + s41 + 0 :|: s41 >= 0, s41 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 1712 + 216*s10 }-> 1 + s42 + 0 :|: s42 >= 0, s42 <= 18 * 0 + 52 * (1 + s10) + 4 * ((1 + s10) * 0) + 2 * (0 * 0) + 210, s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 }-> 1 + s43 + 0 :|: s43 >= 0, s43 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s44 + 1 :|: s44 >= 0, s44 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1712 + 216*s11 + 32*s11*z + 250*z + 7*z^2 }-> 1 + s45 + 1 :|: s45 >= 0, s45 <= 18 * (1 + (z - 1)) + 52 * (1 + s11) + 4 * ((1 + s11) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s46 + 1 :|: s46 >= 0, s46 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s47 + 1 :|: s47 >= 0, s47 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s48 + 0 :|: s48 >= 0, s48 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 1712 + 216*s12 + 32*s12*z + 250*z + 7*z^2 }-> 1 + s49 + 1 :|: s49 >= 0, s49 <= 18 * z + 52 * (1 + s12) + 4 * ((1 + s12) * z) + 2 * (z * z) + 210, s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1712 + 216*s13 + 32*s13*z + 250*z + 7*z^2 }-> 1 + s50 + 0 :|: s50 >= 0, s50 <= 18 * z + 52 * (1 + s13) + 4 * ((1 + s13) * z) + 2 * (z * z) + 210, s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s51 + 1 :|: s51 >= 0, s51 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z = 0 BITS(z) -{ 426 + 230*z + 7*z^2 }-> 1 + s27 :|: s27 >= 0, s27 <= 12 * z + 12 * 0 + 4 * (0 * z) + 2 * (z * z) + 47, z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 428 + 62*z'' }-> 1 + s37 + s' :|: s37 >= 0, s37 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 429 + 62*z'' }-> 1 + s38 + s'' :|: s38 >= 0, s38 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 665 + 244*s24 + 32*s24*z'' + 7*s24^2 + z' + 94*z'' }-> 1 + s39 + s1 :|: s39 >= 0, s39 <= 12 * (1 + s24) + 12 * z'' + 4 * (z'' * (1 + s24)) + 2 * ((1 + s24) * (1 + s24)) + 47, s24 >= 0, s24 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 428 + z' + 62*z'' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 0, z = 0, z' = 0 bitIter(z, z') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s7, s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0, z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0, z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s8, s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0, z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 0, z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s9, s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0, z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 0, z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z'', z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= z'', z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= z'', s26 >= 0, s26 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> s66 :|: s66 >= 0, s66 <= z'', z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: O(1) [0], size: O(n^1) [z] BITITER: runtime: O(n^2) [425 + 230*z + 32*z*z' + 7*z^2 + 62*z'], size: O(n^2) [47 + 12*z + 4*z*z' + 2*z^2 + 12*z'] IF: runtime: O(n^2) [1494 + 218*z' + 32*z'*z'' + 7*z'^2 + 216*z''], size: O(n^2) [210 + 18*z' + 4*z'*z'' + 2*z'^2 + 52*z''] if: runtime: O(1) [0], size: O(n^1) [z''] bitIter: runtime: O(1) [0], size: O(n^1) [z'] BITS: runtime: O(n^2) [426 + 230*z + 7*z^2], size: O(n^2) [48 + 12*z + 2*z^2] ---------------------------------------- (87) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: bits after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1496 }-> 1 + s28 + s15 :|: s28 >= 0, s28 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s3 + z' }-> 1 + s29 + s16 :|: s29 >= 0, s29 <= 18 * 0 + 52 * (1 + s3) + 4 * ((1 + s3) * 0) + 2 * (0 * 0) + 210, s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1496 + z' }-> 1 + s30 + s17 :|: s30 >= 0, s30 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s31 + s18 :|: s31 >= 0, s31 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s4 + 32*s4*z + 250*z + 7*z^2 + z' }-> 1 + s32 + s19 :|: s32 >= 0, s32 <= 18 * (1 + (z - 1)) + 52 * (1 + s4) + 4 * ((1 + s4) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s33 + s20 :|: s33 >= 0, s33 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s34 + s21 :|: s34 >= 0, s34 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s5 + 32*s5*z + 250*z + 7*z^2 + z' }-> 1 + s35 + s22 :|: s35 >= 0, s35 <= 18 * z + 52 * (1 + s5) + 4 * ((1 + s5) * z) + 2 * (z * z) + 210, s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s36 + s23 :|: s36 >= 0, s36 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 1496 }-> 1 + s41 + 0 :|: s41 >= 0, s41 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 1712 + 216*s10 }-> 1 + s42 + 0 :|: s42 >= 0, s42 <= 18 * 0 + 52 * (1 + s10) + 4 * ((1 + s10) * 0) + 2 * (0 * 0) + 210, s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 }-> 1 + s43 + 0 :|: s43 >= 0, s43 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s44 + 1 :|: s44 >= 0, s44 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1712 + 216*s11 + 32*s11*z + 250*z + 7*z^2 }-> 1 + s45 + 1 :|: s45 >= 0, s45 <= 18 * (1 + (z - 1)) + 52 * (1 + s11) + 4 * ((1 + s11) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s46 + 1 :|: s46 >= 0, s46 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s47 + 1 :|: s47 >= 0, s47 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s48 + 0 :|: s48 >= 0, s48 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 1712 + 216*s12 + 32*s12*z + 250*z + 7*z^2 }-> 1 + s49 + 1 :|: s49 >= 0, s49 <= 18 * z + 52 * (1 + s12) + 4 * ((1 + s12) * z) + 2 * (z * z) + 210, s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1712 + 216*s13 + 32*s13*z + 250*z + 7*z^2 }-> 1 + s50 + 0 :|: s50 >= 0, s50 <= 18 * z + 52 * (1 + s13) + 4 * ((1 + s13) * z) + 2 * (z * z) + 210, s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s51 + 1 :|: s51 >= 0, s51 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z = 0 BITS(z) -{ 426 + 230*z + 7*z^2 }-> 1 + s27 :|: s27 >= 0, s27 <= 12 * z + 12 * 0 + 4 * (0 * z) + 2 * (z * z) + 47, z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 428 + 62*z'' }-> 1 + s37 + s' :|: s37 >= 0, s37 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 429 + 62*z'' }-> 1 + s38 + s'' :|: s38 >= 0, s38 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 665 + 244*s24 + 32*s24*z'' + 7*s24^2 + z' + 94*z'' }-> 1 + s39 + s1 :|: s39 >= 0, s39 <= 12 * (1 + s24) + 12 * z'' + 4 * (z'' * (1 + s24)) + 2 * ((1 + s24) * (1 + s24)) + 47, s24 >= 0, s24 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 428 + z' + 62*z'' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 0, z = 0, z' = 0 bitIter(z, z') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s7, s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0, z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0, z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s8, s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0, z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 0, z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s9, s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0, z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 0, z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z'', z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= z'', z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= z'', s26 >= 0, s26 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> s66 :|: s66 >= 0, s66 <= z'', z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {bits} Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: O(1) [0], size: O(n^1) [z] BITITER: runtime: O(n^2) [425 + 230*z + 32*z*z' + 7*z^2 + 62*z'], size: O(n^2) [47 + 12*z + 4*z*z' + 2*z^2 + 12*z'] IF: runtime: O(n^2) [1494 + 218*z' + 32*z'*z'' + 7*z'^2 + 216*z''], size: O(n^2) [210 + 18*z' + 4*z'*z'' + 2*z'^2 + 52*z''] if: runtime: O(1) [0], size: O(n^1) [z''] bitIter: runtime: O(1) [0], size: O(n^1) [z'] BITS: runtime: O(n^2) [426 + 230*z + 7*z^2], size: O(n^2) [48 + 12*z + 2*z^2] bits: runtime: ?, size: O(1) [0] ---------------------------------------- (89) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: bits after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: BITITER(z, z') -{ 1496 }-> 1 + s28 + s15 :|: s28 >= 0, s28 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s15 >= 0, s15 <= 0, z = 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s3 + z' }-> 1 + s29 + s16 :|: s29 >= 0, s29 <= 18 * 0 + 52 * (1 + s3) + 4 * ((1 + s3) * 0) + 2 * (0 * 0) + 210, s16 >= 0, s16 <= 1 + (z' - 1), s3 >= 0, s3 <= z' - 1, z' - 1 >= 0, z = 0 BITITER(z, z') -{ 1496 + z' }-> 1 + s30 + s17 :|: s30 >= 0, s30 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, s17 >= 0, s17 <= z', z' >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s31 + s18 :|: s31 >= 0, s31 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s18 >= 0, s18 <= 0, z - 1 >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s4 + 32*s4*z + 250*z + 7*z^2 + z' }-> 1 + s32 + s19 :|: s32 >= 0, s32 <= 18 * (1 + (z - 1)) + 52 * (1 + s4) + 4 * ((1 + s4) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s19 >= 0, s19 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s33 + s20 :|: s33 >= 0, s33 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s20 >= 0, s20 <= z', z' >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s34 + s21 :|: s34 >= 0, s34 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s21 >= 0, s21 <= 0, z >= 0, z' = 0 BITITER(z, z') -{ 1712 + 216*s5 + 32*s5*z + 250*z + 7*z^2 + z' }-> 1 + s35 + s22 :|: s35 >= 0, s35 <= 18 * z + 52 * (1 + s5) + 4 * ((1 + s5) * z) + 2 * (z * z) + 210, s22 >= 0, s22 <= 1 + (z' - 1), s5 >= 0, s5 <= z' - 1, z' - 1 >= 0, z >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 + z' }-> 1 + s36 + s23 :|: s36 >= 0, s36 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, s23 >= 0, s23 <= z', z' >= 0, z >= 0 BITITER(z, z') -{ 1496 }-> 1 + s41 + 0 :|: s41 >= 0, s41 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z = 0, z' = 0, 0 = 0 BITITER(z, z') -{ 1712 + 216*s10 }-> 1 + s42 + 0 :|: s42 >= 0, s42 <= 18 * 0 + 52 * (1 + s10) + 4 * ((1 + s10) * 0) + 2 * (0 * 0) + 210, s10 >= 0, s10 <= z' - 1, z' - 1 >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 }-> 1 + s43 + 0 :|: s43 >= 0, s43 <= 18 * 0 + 52 * 0 + 4 * (0 * 0) + 2 * (0 * 0) + 210, z' >= 0, z = 0, 0 = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s44 + 1 :|: s44 >= 0, s44 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z - 1 >= 0, z' = 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1712 + 216*s11 + 32*s11*z + 250*z + 7*z^2 }-> 1 + s45 + 1 :|: s45 >= 0, s45 <= 18 * (1 + (z - 1)) + 52 * (1 + s11) + 4 * ((1 + s11) * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s46 + 1 :|: s46 >= 0, s46 <= 18 * (1 + (z - 1)) + 52 * 0 + 4 * (0 * (1 + (z - 1))) + 2 * ((1 + (z - 1)) * (1 + (z - 1))) + 210, z' >= 0, z - 1 >= 0, 1 + (z - 1) = 1 + z0, z0 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s47 + 1 :|: s47 >= 0, s47 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s48 + 0 :|: s48 >= 0, s48 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z >= 0, z' = 0, z = 0 BITITER(z, z') -{ 1712 + 216*s12 + 32*s12*z + 250*z + 7*z^2 }-> 1 + s49 + 1 :|: s49 >= 0, s49 <= 18 * z + 52 * (1 + s12) + 4 * ((1 + s12) * z) + 2 * (z * z) + 210, s12 >= 0, s12 <= z' - 1, z' - 1 >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1712 + 216*s13 + 32*s13*z + 250*z + 7*z^2 }-> 1 + s50 + 0 :|: s50 >= 0, s50 <= 18 * z + 52 * (1 + s13) + 4 * ((1 + s13) * z) + 2 * (z * z) + 210, s13 >= 0, s13 <= z' - 1, z' - 1 >= 0, z >= 0, z = 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s51 + 1 :|: s51 >= 0, s51 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z - 1 >= 0 BITITER(z, z') -{ 1496 + 218*z + 7*z^2 }-> 1 + s52 + 0 :|: s52 >= 0, s52 <= 18 * z + 52 * 0 + 4 * (0 * z) + 2 * (z * z) + 210, z' >= 0, z >= 0, z = 0 BITS(z) -{ 426 + 230*z + 7*z^2 }-> 1 + s27 :|: s27 >= 0, s27 <= 12 * z + 12 * 0 + 4 * (0 * z) + 2 * (z * z) + 47, z >= 0 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'') -{ 2 }-> 1 + 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 IF(z, z', z'') -{ 2 }-> 1 + 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 IF(z, z', z'') -{ 428 + 62*z'' }-> 1 + s37 + s' :|: s37 >= 0, s37 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s' >= 0, s' <= 0, z'' >= 0, z = 1, z' = 0 IF(z, z', z'') -{ 429 + 62*z'' }-> 1 + s38 + s'' :|: s38 >= 0, s38 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s'' >= 0, s'' <= 1 + 0, z'' >= 0, z = 1, z' = 1 + 0 IF(z, z', z'') -{ 665 + 244*s24 + 32*s24*z'' + 7*s24^2 + z' + 94*z'' }-> 1 + s39 + s1 :|: s39 >= 0, s39 <= 12 * (1 + s24) + 12 * z'' + 4 * (z'' * (1 + s24)) + 2 * ((1 + s24) * (1 + s24)) + 47, s24 >= 0, s24 <= z' - 2, s1 >= 0, s1 <= 1 + (1 + (z' - 2)), z'' >= 0, z' - 2 >= 0, z = 1 IF(z, z', z'') -{ 428 + z' + 62*z'' }-> 1 + s40 + s2 :|: s40 >= 0, s40 <= 12 * 0 + 12 * z'' + 4 * (z'' * 0) + 2 * (0 * 0) + 47, s2 >= 0, s2 <= z', z'' >= 0, z = 1, z' >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s14 :|: s14 >= 0, s14 <= z - 1, z - 1 >= 0 P(z) -{ 1 }-> 1 :|: z - 1 >= 0 P(z) -{ 1 }-> 0 :|: z = 0 ZERO(z) -{ 1 }-> 1 :|: z - 1 >= 0 ZERO(z) -{ 1 }-> 0 :|: z = 0 bitIter(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 0, z = 0, z' = 0 bitIter(z, z') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s7, s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z = 0 bitIter(z, z') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0, z' >= 0, z = 0 bitIter(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0, z - 1 >= 0, z' = 0 bitIter(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s8, s8 >= 0, s8 <= z' - 1, z - 1 >= 0, z' - 1 >= 0 bitIter(z, z') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0, z - 1 >= 0, z' >= 0 bitIter(z, z') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 0, z >= 0, z' = 0 bitIter(z, z') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s9, s9 >= 0, s9 <= z' - 1, z' - 1 >= 0, z >= 0 bitIter(z, z') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0, z' >= 0, z >= 0 bitIter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 bits(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 0, z >= 0 bits(z) -{ 0 }-> 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 + s25 :|: s25 >= 0, s25 <= z - 2, z - 2 >= 0 if(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z'', z'' >= 0, z = 1, z' = 0 if(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= z'', z'' >= 0, z = 1, z' = 1 + 0 if(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= z'', s26 >= 0, s26 <= z' - 2, z'' >= 0, z = 1, z' - 2 >= 0 if(z, z', z'') -{ 0 }-> s66 :|: s66 >= 0, s66 <= z'', z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z'' = 0 if(z, z', z'') -{ 0 }-> 0 :|: z = 2, z'' >= 0, z' >= 0 if(z, z', z'') -{ 0 }-> z'' - 1 :|: z = 2, z'' >= 0, z' >= 0, z'' - 1 >= 0 inc(z) -{ 0 }-> 0 :|: z = 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 2 :|: z = 0 zero(z) -{ 0 }-> 1 :|: z - 1 >= 0 zero(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: Previous analysis results are: HALF: runtime: O(n^1) [2 + z], size: O(n^1) [z] P: runtime: O(1) [1], size: O(1) [1] ZERO: runtime: O(1) [1], size: O(1) [1] inc: runtime: O(1) [0], size: O(n^1) [z] zero: runtime: O(1) [0], size: O(1) [2] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] half: runtime: O(1) [0], size: O(n^1) [z] p: runtime: O(1) [0], size: O(n^1) [z] BITITER: runtime: O(n^2) [425 + 230*z + 32*z*z' + 7*z^2 + 62*z'], size: O(n^2) [47 + 12*z + 4*z*z' + 2*z^2 + 12*z'] IF: runtime: O(n^2) [1494 + 218*z' + 32*z'*z'' + 7*z'^2 + 216*z''], size: O(n^2) [210 + 18*z' + 4*z'*z'' + 2*z'^2 + 52*z''] if: runtime: O(1) [0], size: O(n^1) [z''] bitIter: runtime: O(1) [0], size: O(n^1) [z'] BITS: runtime: O(n^2) [426 + 230*z + 7*z^2], size: O(n^2) [48 + 12*z + 2*z^2] bits: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (91) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (92) BOUNDS(1, n^2) ---------------------------------------- (93) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (94) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: HALF(0') -> c HALF(s(0')) -> c1 HALF(s(s(z0))) -> c2(HALF(z0)) INC(0') -> c3 INC(s(z0)) -> c4(INC(z0)) ZERO(0') -> c5 ZERO(s(z0)) -> c6 P(0') -> c7 P(s(z0)) -> c8 BITS(z0) -> c9(BITITER(z0, 0')) BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0)) BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1)) IF(true, z0, z1) -> c12(P(z1)) IF(false, z0, z1) -> c13(BITITER(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)) inc(0') -> 0' inc(s(z0)) -> s(inc(z0)) zero(0') -> true zero(s(z0)) -> false p(0') -> 0' p(s(z0)) -> z0 bits(z0) -> bitIter(z0, 0') bitIter(z0, z1) -> if(zero(z0), z0, inc(z1)) if(true, z0, z1) -> p(z1) if(false, z0, z1) -> bitIter(half(z0), z1) Rewrite Strategy: INNERMOST ---------------------------------------- (95) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (96) Obligation: Innermost TRS: Rules: HALF(0') -> c HALF(s(0')) -> c1 HALF(s(s(z0))) -> c2(HALF(z0)) INC(0') -> c3 INC(s(z0)) -> c4(INC(z0)) ZERO(0') -> c5 ZERO(s(z0)) -> c6 P(0') -> c7 P(s(z0)) -> c8 BITS(z0) -> c9(BITITER(z0, 0')) BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0)) BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1)) IF(true, z0, z1) -> c12(P(z1)) IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0)) half(0') -> 0' half(s(0')) -> 0' half(s(s(z0))) -> s(half(z0)) inc(0') -> 0' inc(s(z0)) -> s(inc(z0)) zero(0') -> true zero(s(z0)) -> false p(0') -> 0' p(s(z0)) -> z0 bits(z0) -> bitIter(z0, 0') bitIter(z0, z1) -> if(zero(z0), z0, inc(z1)) if(true, z0, z1) -> p(z1) if(false, z0, z1) -> bitIter(half(z0), z1) Types: 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 INC :: 0':s -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> c3:c4 ZERO :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 P :: 0':s -> c7:c8 c7 :: c7:c8 c8 :: c7:c8 BITS :: 0':s -> c9 c9 :: c10:c11 -> c9 BITITER :: 0':s -> 0':s -> c10:c11 c10 :: c12:c13 -> c5:c6 -> c10:c11 IF :: true:false -> 0':s -> 0':s -> c12:c13 zero :: 0':s -> true:false inc :: 0':s -> 0':s c11 :: c12:c13 -> c3:c4 -> c10:c11 true :: true:false c12 :: c7:c8 -> c12:c13 false :: true:false c13 :: c10:c11 -> c:c1:c2 -> c12:c13 half :: 0':s -> 0':s p :: 0':s -> 0':s bits :: 0':s -> 0':s bitIter :: 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s hole_c:c1:c21_14 :: c:c1:c2 hole_0':s2_14 :: 0':s hole_c3:c43_14 :: c3:c4 hole_c5:c64_14 :: c5:c6 hole_c7:c85_14 :: c7:c8 hole_c96_14 :: c9 hole_c10:c117_14 :: c10:c11 hole_c12:c138_14 :: c12:c13 hole_true:false9_14 :: true:false gen_c:c1:c210_14 :: Nat -> c:c1:c2 gen_0':s11_14 :: Nat -> 0':s gen_c3:c412_14 :: Nat -> c3:c4 ---------------------------------------- (97) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: HALF, INC, BITITER, inc, half, bitIter They will be analysed ascendingly in the following order: HALF < BITITER INC < BITITER inc < BITITER half < BITITER inc < bitIter half < bitIter ---------------------------------------- (98) Obligation: Innermost TRS: Rules: HALF(0') -> c HALF(s(0')) -> c1 HALF(s(s(z0))) -> c2(HALF(z0)) INC(0') -> c3 INC(s(z0)) -> c4(INC(z0)) ZERO(0') -> c5 ZERO(s(z0)) -> c6 P(0') -> c7 P(s(z0)) -> c8 BITS(z0) -> c9(BITITER(z0, 0')) BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0)) BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1)) IF(true, z0, z1) -> c12(P(z1)) IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0)) half(0') -> 0' half(s(0')) -> 0' half(s(s(z0))) -> s(half(z0)) inc(0') -> 0' inc(s(z0)) -> s(inc(z0)) zero(0') -> true zero(s(z0)) -> false p(0') -> 0' p(s(z0)) -> z0 bits(z0) -> bitIter(z0, 0') bitIter(z0, z1) -> if(zero(z0), z0, inc(z1)) if(true, z0, z1) -> p(z1) if(false, z0, z1) -> bitIter(half(z0), z1) Types: 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 INC :: 0':s -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> c3:c4 ZERO :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 P :: 0':s -> c7:c8 c7 :: c7:c8 c8 :: c7:c8 BITS :: 0':s -> c9 c9 :: c10:c11 -> c9 BITITER :: 0':s -> 0':s -> c10:c11 c10 :: c12:c13 -> c5:c6 -> c10:c11 IF :: true:false -> 0':s -> 0':s -> c12:c13 zero :: 0':s -> true:false inc :: 0':s -> 0':s c11 :: c12:c13 -> c3:c4 -> c10:c11 true :: true:false c12 :: c7:c8 -> c12:c13 false :: true:false c13 :: c10:c11 -> c:c1:c2 -> c12:c13 half :: 0':s -> 0':s p :: 0':s -> 0':s bits :: 0':s -> 0':s bitIter :: 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s hole_c:c1:c21_14 :: c:c1:c2 hole_0':s2_14 :: 0':s hole_c3:c43_14 :: c3:c4 hole_c5:c64_14 :: c5:c6 hole_c7:c85_14 :: c7:c8 hole_c96_14 :: c9 hole_c10:c117_14 :: c10:c11 hole_c12:c138_14 :: c12:c13 hole_true:false9_14 :: true:false gen_c:c1:c210_14 :: Nat -> c:c1:c2 gen_0':s11_14 :: Nat -> 0':s gen_c3:c412_14 :: Nat -> c3:c4 Generator Equations: gen_c:c1:c210_14(0) <=> c gen_c:c1:c210_14(+(x, 1)) <=> c2(gen_c:c1:c210_14(x)) gen_0':s11_14(0) <=> 0' gen_0':s11_14(+(x, 1)) <=> s(gen_0':s11_14(x)) gen_c3:c412_14(0) <=> c3 gen_c3:c412_14(+(x, 1)) <=> c4(gen_c3:c412_14(x)) The following defined symbols remain to be analysed: HALF, INC, BITITER, inc, half, bitIter They will be analysed ascendingly in the following order: HALF < BITITER INC < BITITER inc < BITITER half < BITITER inc < bitIter half < bitIter ---------------------------------------- (99) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: HALF(gen_0':s11_14(*(2, n14_14))) -> gen_c:c1:c210_14(n14_14), rt in Omega(1 + n14_14) Induction Base: HALF(gen_0':s11_14(*(2, 0))) ->_R^Omega(1) c Induction Step: HALF(gen_0':s11_14(*(2, +(n14_14, 1)))) ->_R^Omega(1) c2(HALF(gen_0':s11_14(*(2, n14_14)))) ->_IH c2(gen_c:c1:c210_14(c15_14)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (100) Complex Obligation (BEST) ---------------------------------------- (101) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: HALF(0') -> c HALF(s(0')) -> c1 HALF(s(s(z0))) -> c2(HALF(z0)) INC(0') -> c3 INC(s(z0)) -> c4(INC(z0)) ZERO(0') -> c5 ZERO(s(z0)) -> c6 P(0') -> c7 P(s(z0)) -> c8 BITS(z0) -> c9(BITITER(z0, 0')) BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0)) BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1)) IF(true, z0, z1) -> c12(P(z1)) IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0)) half(0') -> 0' half(s(0')) -> 0' half(s(s(z0))) -> s(half(z0)) inc(0') -> 0' inc(s(z0)) -> s(inc(z0)) zero(0') -> true zero(s(z0)) -> false p(0') -> 0' p(s(z0)) -> z0 bits(z0) -> bitIter(z0, 0') bitIter(z0, z1) -> if(zero(z0), z0, inc(z1)) if(true, z0, z1) -> p(z1) if(false, z0, z1) -> bitIter(half(z0), z1) Types: 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 INC :: 0':s -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> c3:c4 ZERO :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 P :: 0':s -> c7:c8 c7 :: c7:c8 c8 :: c7:c8 BITS :: 0':s -> c9 c9 :: c10:c11 -> c9 BITITER :: 0':s -> 0':s -> c10:c11 c10 :: c12:c13 -> c5:c6 -> c10:c11 IF :: true:false -> 0':s -> 0':s -> c12:c13 zero :: 0':s -> true:false inc :: 0':s -> 0':s c11 :: c12:c13 -> c3:c4 -> c10:c11 true :: true:false c12 :: c7:c8 -> c12:c13 false :: true:false c13 :: c10:c11 -> c:c1:c2 -> c12:c13 half :: 0':s -> 0':s p :: 0':s -> 0':s bits :: 0':s -> 0':s bitIter :: 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s hole_c:c1:c21_14 :: c:c1:c2 hole_0':s2_14 :: 0':s hole_c3:c43_14 :: c3:c4 hole_c5:c64_14 :: c5:c6 hole_c7:c85_14 :: c7:c8 hole_c96_14 :: c9 hole_c10:c117_14 :: c10:c11 hole_c12:c138_14 :: c12:c13 hole_true:false9_14 :: true:false gen_c:c1:c210_14 :: Nat -> c:c1:c2 gen_0':s11_14 :: Nat -> 0':s gen_c3:c412_14 :: Nat -> c3:c4 Generator Equations: gen_c:c1:c210_14(0) <=> c gen_c:c1:c210_14(+(x, 1)) <=> c2(gen_c:c1:c210_14(x)) gen_0':s11_14(0) <=> 0' gen_0':s11_14(+(x, 1)) <=> s(gen_0':s11_14(x)) gen_c3:c412_14(0) <=> c3 gen_c3:c412_14(+(x, 1)) <=> c4(gen_c3:c412_14(x)) The following defined symbols remain to be analysed: HALF, INC, BITITER, inc, half, bitIter They will be analysed ascendingly in the following order: HALF < BITITER INC < BITITER inc < BITITER half < BITITER inc < bitIter half < bitIter ---------------------------------------- (102) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (103) BOUNDS(n^1, INF) ---------------------------------------- (104) Obligation: Innermost TRS: Rules: HALF(0') -> c HALF(s(0')) -> c1 HALF(s(s(z0))) -> c2(HALF(z0)) INC(0') -> c3 INC(s(z0)) -> c4(INC(z0)) ZERO(0') -> c5 ZERO(s(z0)) -> c6 P(0') -> c7 P(s(z0)) -> c8 BITS(z0) -> c9(BITITER(z0, 0')) BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0)) BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1)) IF(true, z0, z1) -> c12(P(z1)) IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0)) half(0') -> 0' half(s(0')) -> 0' half(s(s(z0))) -> s(half(z0)) inc(0') -> 0' inc(s(z0)) -> s(inc(z0)) zero(0') -> true zero(s(z0)) -> false p(0') -> 0' p(s(z0)) -> z0 bits(z0) -> bitIter(z0, 0') bitIter(z0, z1) -> if(zero(z0), z0, inc(z1)) if(true, z0, z1) -> p(z1) if(false, z0, z1) -> bitIter(half(z0), z1) Types: 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 INC :: 0':s -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> c3:c4 ZERO :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 P :: 0':s -> c7:c8 c7 :: c7:c8 c8 :: c7:c8 BITS :: 0':s -> c9 c9 :: c10:c11 -> c9 BITITER :: 0':s -> 0':s -> c10:c11 c10 :: c12:c13 -> c5:c6 -> c10:c11 IF :: true:false -> 0':s -> 0':s -> c12:c13 zero :: 0':s -> true:false inc :: 0':s -> 0':s c11 :: c12:c13 -> c3:c4 -> c10:c11 true :: true:false c12 :: c7:c8 -> c12:c13 false :: true:false c13 :: c10:c11 -> c:c1:c2 -> c12:c13 half :: 0':s -> 0':s p :: 0':s -> 0':s bits :: 0':s -> 0':s bitIter :: 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s hole_c:c1:c21_14 :: c:c1:c2 hole_0':s2_14 :: 0':s hole_c3:c43_14 :: c3:c4 hole_c5:c64_14 :: c5:c6 hole_c7:c85_14 :: c7:c8 hole_c96_14 :: c9 hole_c10:c117_14 :: c10:c11 hole_c12:c138_14 :: c12:c13 hole_true:false9_14 :: true:false gen_c:c1:c210_14 :: Nat -> c:c1:c2 gen_0':s11_14 :: Nat -> 0':s gen_c3:c412_14 :: Nat -> c3:c4 Lemmas: HALF(gen_0':s11_14(*(2, n14_14))) -> gen_c:c1:c210_14(n14_14), rt in Omega(1 + n14_14) Generator Equations: gen_c:c1:c210_14(0) <=> c gen_c:c1:c210_14(+(x, 1)) <=> c2(gen_c:c1:c210_14(x)) gen_0':s11_14(0) <=> 0' gen_0':s11_14(+(x, 1)) <=> s(gen_0':s11_14(x)) gen_c3:c412_14(0) <=> c3 gen_c3:c412_14(+(x, 1)) <=> c4(gen_c3:c412_14(x)) The following defined symbols remain to be analysed: INC, BITITER, inc, half, bitIter They will be analysed ascendingly in the following order: INC < BITITER inc < BITITER half < BITITER inc < bitIter half < bitIter ---------------------------------------- (105) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: INC(gen_0':s11_14(n471_14)) -> gen_c3:c412_14(n471_14), rt in Omega(1 + n471_14) Induction Base: INC(gen_0':s11_14(0)) ->_R^Omega(1) c3 Induction Step: INC(gen_0':s11_14(+(n471_14, 1))) ->_R^Omega(1) c4(INC(gen_0':s11_14(n471_14))) ->_IH c4(gen_c3:c412_14(c472_14)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (106) Obligation: Innermost TRS: Rules: HALF(0') -> c HALF(s(0')) -> c1 HALF(s(s(z0))) -> c2(HALF(z0)) INC(0') -> c3 INC(s(z0)) -> c4(INC(z0)) ZERO(0') -> c5 ZERO(s(z0)) -> c6 P(0') -> c7 P(s(z0)) -> c8 BITS(z0) -> c9(BITITER(z0, 0')) BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0)) BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1)) IF(true, z0, z1) -> c12(P(z1)) IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0)) half(0') -> 0' half(s(0')) -> 0' half(s(s(z0))) -> s(half(z0)) inc(0') -> 0' inc(s(z0)) -> s(inc(z0)) zero(0') -> true zero(s(z0)) -> false p(0') -> 0' p(s(z0)) -> z0 bits(z0) -> bitIter(z0, 0') bitIter(z0, z1) -> if(zero(z0), z0, inc(z1)) if(true, z0, z1) -> p(z1) if(false, z0, z1) -> bitIter(half(z0), z1) Types: 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 INC :: 0':s -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> c3:c4 ZERO :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 P :: 0':s -> c7:c8 c7 :: c7:c8 c8 :: c7:c8 BITS :: 0':s -> c9 c9 :: c10:c11 -> c9 BITITER :: 0':s -> 0':s -> c10:c11 c10 :: c12:c13 -> c5:c6 -> c10:c11 IF :: true:false -> 0':s -> 0':s -> c12:c13 zero :: 0':s -> true:false inc :: 0':s -> 0':s c11 :: c12:c13 -> c3:c4 -> c10:c11 true :: true:false c12 :: c7:c8 -> c12:c13 false :: true:false c13 :: c10:c11 -> c:c1:c2 -> c12:c13 half :: 0':s -> 0':s p :: 0':s -> 0':s bits :: 0':s -> 0':s bitIter :: 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s hole_c:c1:c21_14 :: c:c1:c2 hole_0':s2_14 :: 0':s hole_c3:c43_14 :: c3:c4 hole_c5:c64_14 :: c5:c6 hole_c7:c85_14 :: c7:c8 hole_c96_14 :: c9 hole_c10:c117_14 :: c10:c11 hole_c12:c138_14 :: c12:c13 hole_true:false9_14 :: true:false gen_c:c1:c210_14 :: Nat -> c:c1:c2 gen_0':s11_14 :: Nat -> 0':s gen_c3:c412_14 :: Nat -> c3:c4 Lemmas: HALF(gen_0':s11_14(*(2, n14_14))) -> gen_c:c1:c210_14(n14_14), rt in Omega(1 + n14_14) INC(gen_0':s11_14(n471_14)) -> gen_c3:c412_14(n471_14), rt in Omega(1 + n471_14) Generator Equations: gen_c:c1:c210_14(0) <=> c gen_c:c1:c210_14(+(x, 1)) <=> c2(gen_c:c1:c210_14(x)) gen_0':s11_14(0) <=> 0' gen_0':s11_14(+(x, 1)) <=> s(gen_0':s11_14(x)) gen_c3:c412_14(0) <=> c3 gen_c3:c412_14(+(x, 1)) <=> c4(gen_c3:c412_14(x)) The following defined symbols remain to be analysed: inc, BITITER, half, bitIter They will be analysed ascendingly in the following order: inc < BITITER half < BITITER inc < bitIter half < bitIter ---------------------------------------- (107) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: inc(gen_0':s11_14(n848_14)) -> gen_0':s11_14(n848_14), rt in Omega(0) Induction Base: inc(gen_0':s11_14(0)) ->_R^Omega(0) 0' Induction Step: inc(gen_0':s11_14(+(n848_14, 1))) ->_R^Omega(0) s(inc(gen_0':s11_14(n848_14))) ->_IH s(gen_0':s11_14(c849_14)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (108) Obligation: Innermost TRS: Rules: HALF(0') -> c HALF(s(0')) -> c1 HALF(s(s(z0))) -> c2(HALF(z0)) INC(0') -> c3 INC(s(z0)) -> c4(INC(z0)) ZERO(0') -> c5 ZERO(s(z0)) -> c6 P(0') -> c7 P(s(z0)) -> c8 BITS(z0) -> c9(BITITER(z0, 0')) BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0)) BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1)) IF(true, z0, z1) -> c12(P(z1)) IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0)) half(0') -> 0' half(s(0')) -> 0' half(s(s(z0))) -> s(half(z0)) inc(0') -> 0' inc(s(z0)) -> s(inc(z0)) zero(0') -> true zero(s(z0)) -> false p(0') -> 0' p(s(z0)) -> z0 bits(z0) -> bitIter(z0, 0') bitIter(z0, z1) -> if(zero(z0), z0, inc(z1)) if(true, z0, z1) -> p(z1) if(false, z0, z1) -> bitIter(half(z0), z1) Types: 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 INC :: 0':s -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> c3:c4 ZERO :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 P :: 0':s -> c7:c8 c7 :: c7:c8 c8 :: c7:c8 BITS :: 0':s -> c9 c9 :: c10:c11 -> c9 BITITER :: 0':s -> 0':s -> c10:c11 c10 :: c12:c13 -> c5:c6 -> c10:c11 IF :: true:false -> 0':s -> 0':s -> c12:c13 zero :: 0':s -> true:false inc :: 0':s -> 0':s c11 :: c12:c13 -> c3:c4 -> c10:c11 true :: true:false c12 :: c7:c8 -> c12:c13 false :: true:false c13 :: c10:c11 -> c:c1:c2 -> c12:c13 half :: 0':s -> 0':s p :: 0':s -> 0':s bits :: 0':s -> 0':s bitIter :: 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s hole_c:c1:c21_14 :: c:c1:c2 hole_0':s2_14 :: 0':s hole_c3:c43_14 :: c3:c4 hole_c5:c64_14 :: c5:c6 hole_c7:c85_14 :: c7:c8 hole_c96_14 :: c9 hole_c10:c117_14 :: c10:c11 hole_c12:c138_14 :: c12:c13 hole_true:false9_14 :: true:false gen_c:c1:c210_14 :: Nat -> c:c1:c2 gen_0':s11_14 :: Nat -> 0':s gen_c3:c412_14 :: Nat -> c3:c4 Lemmas: HALF(gen_0':s11_14(*(2, n14_14))) -> gen_c:c1:c210_14(n14_14), rt in Omega(1 + n14_14) INC(gen_0':s11_14(n471_14)) -> gen_c3:c412_14(n471_14), rt in Omega(1 + n471_14) inc(gen_0':s11_14(n848_14)) -> gen_0':s11_14(n848_14), rt in Omega(0) Generator Equations: gen_c:c1:c210_14(0) <=> c gen_c:c1:c210_14(+(x, 1)) <=> c2(gen_c:c1:c210_14(x)) gen_0':s11_14(0) <=> 0' gen_0':s11_14(+(x, 1)) <=> s(gen_0':s11_14(x)) gen_c3:c412_14(0) <=> c3 gen_c3:c412_14(+(x, 1)) <=> c4(gen_c3:c412_14(x)) The following defined symbols remain to be analysed: half, BITITER, bitIter They will be analysed ascendingly in the following order: half < BITITER half < bitIter ---------------------------------------- (109) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: half(gen_0':s11_14(*(2, n1204_14))) -> gen_0':s11_14(n1204_14), rt in Omega(0) Induction Base: half(gen_0':s11_14(*(2, 0))) ->_R^Omega(0) 0' Induction Step: half(gen_0':s11_14(*(2, +(n1204_14, 1)))) ->_R^Omega(0) s(half(gen_0':s11_14(*(2, n1204_14)))) ->_IH s(gen_0':s11_14(c1205_14)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (110) Obligation: Innermost TRS: Rules: HALF(0') -> c HALF(s(0')) -> c1 HALF(s(s(z0))) -> c2(HALF(z0)) INC(0') -> c3 INC(s(z0)) -> c4(INC(z0)) ZERO(0') -> c5 ZERO(s(z0)) -> c6 P(0') -> c7 P(s(z0)) -> c8 BITS(z0) -> c9(BITITER(z0, 0')) BITITER(z0, z1) -> c10(IF(zero(z0), z0, inc(z1)), ZERO(z0)) BITITER(z0, z1) -> c11(IF(zero(z0), z0, inc(z1)), INC(z1)) IF(true, z0, z1) -> c12(P(z1)) IF(false, z0, z1) -> c13(BITITER(half(z0), z1), HALF(z0)) half(0') -> 0' half(s(0')) -> 0' half(s(s(z0))) -> s(half(z0)) inc(0') -> 0' inc(s(z0)) -> s(inc(z0)) zero(0') -> true zero(s(z0)) -> false p(0') -> 0' p(s(z0)) -> z0 bits(z0) -> bitIter(z0, 0') bitIter(z0, z1) -> if(zero(z0), z0, inc(z1)) if(true, z0, z1) -> p(z1) if(false, z0, z1) -> bitIter(half(z0), z1) Types: 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 INC :: 0':s -> c3:c4 c3 :: c3:c4 c4 :: c3:c4 -> c3:c4 ZERO :: 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 P :: 0':s -> c7:c8 c7 :: c7:c8 c8 :: c7:c8 BITS :: 0':s -> c9 c9 :: c10:c11 -> c9 BITITER :: 0':s -> 0':s -> c10:c11 c10 :: c12:c13 -> c5:c6 -> c10:c11 IF :: true:false -> 0':s -> 0':s -> c12:c13 zero :: 0':s -> true:false inc :: 0':s -> 0':s c11 :: c12:c13 -> c3:c4 -> c10:c11 true :: true:false c12 :: c7:c8 -> c12:c13 false :: true:false c13 :: c10:c11 -> c:c1:c2 -> c12:c13 half :: 0':s -> 0':s p :: 0':s -> 0':s bits :: 0':s -> 0':s bitIter :: 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s hole_c:c1:c21_14 :: c:c1:c2 hole_0':s2_14 :: 0':s hole_c3:c43_14 :: c3:c4 hole_c5:c64_14 :: c5:c6 hole_c7:c85_14 :: c7:c8 hole_c96_14 :: c9 hole_c10:c117_14 :: c10:c11 hole_c12:c138_14 :: c12:c13 hole_true:false9_14 :: true:false gen_c:c1:c210_14 :: Nat -> c:c1:c2 gen_0':s11_14 :: Nat -> 0':s gen_c3:c412_14 :: Nat -> c3:c4 Lemmas: HALF(gen_0':s11_14(*(2, n14_14))) -> gen_c:c1:c210_14(n14_14), rt in Omega(1 + n14_14) INC(gen_0':s11_14(n471_14)) -> gen_c3:c412_14(n471_14), rt in Omega(1 + n471_14) inc(gen_0':s11_14(n848_14)) -> gen_0':s11_14(n848_14), rt in Omega(0) half(gen_0':s11_14(*(2, n1204_14))) -> gen_0':s11_14(n1204_14), rt in Omega(0) Generator Equations: gen_c:c1:c210_14(0) <=> c gen_c:c1:c210_14(+(x, 1)) <=> c2(gen_c:c1:c210_14(x)) gen_0':s11_14(0) <=> 0' gen_0':s11_14(+(x, 1)) <=> s(gen_0':s11_14(x)) gen_c3:c412_14(0) <=> c3 gen_c3:c412_14(+(x, 1)) <=> c4(gen_c3:c412_14(x)) The following defined symbols remain to be analysed: BITITER, bitIter