/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^3)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 194 ms] (4) CpxRelTRS (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 7 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 387 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 70 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 5 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 97 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 96 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 171 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 85 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 108 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 52 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 915 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 749 ms] (56) CpxRNTS (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 1087 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 362 ms] (62) CpxRNTS (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 419 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 58 ms] (68) CpxRNTS (69) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 295 ms] (72) CpxRNTS (73) IntTrsBoundProof [UPPER BOUND(ID), 66 ms] (74) CpxRNTS (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 110 ms] (78) CpxRNTS (79) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (80) CpxRNTS (81) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 117 ms] (84) CpxRNTS (85) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (86) CpxRNTS (87) FinalProof [FINISHED, 0 ms] (88) BOUNDS(1, n^3) (89) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (90) TRS for Loop Detection (91) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (92) BEST (93) proven lower bound (94) LowerBoundPropagationProof [FINISHED, 0 ms] (95) BOUNDS(n^1, INF) (96) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: cond(true, x) -> cond(odd(x), p(x)) odd(0) -> false odd(s(0)) -> true odd(s(s(x))) -> odd(x) p(0) -> 0 p(s(x)) -> x S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) encArg(cons_odd(x_1)) -> odd(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) encode_true -> true encode_odd(x_1) -> odd(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_false -> false encode_s(x_1) -> s(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: cond(true, x) -> cond(odd(x), p(x)) odd(0) -> false odd(s(0)) -> true odd(s(s(x))) -> odd(x) p(0) -> 0 p(s(x)) -> x The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) encArg(cons_odd(x_1)) -> odd(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) encode_true -> true encode_odd(x_1) -> odd(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_false -> false encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: cond(true, x) -> cond(odd(x), p(x)) odd(0) -> false odd(s(0)) -> true odd(s(s(x))) -> odd(x) p(0) -> 0 p(s(x)) -> x The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) encArg(cons_odd(x_1)) -> odd(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) encode_true -> true encode_odd(x_1) -> odd(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_false -> false encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: cond(true, x) -> cond(odd(x), p(x)) [1] odd(0) -> false [1] odd(s(0)) -> true [1] odd(s(s(x))) -> odd(x) [1] p(0) -> 0 [1] p(s(x)) -> x [1] encArg(true) -> true [0] encArg(0) -> 0 [0] encArg(false) -> false [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) [0] encArg(cons_odd(x_1)) -> odd(encArg(x_1)) [0] encArg(cons_p(x_1)) -> p(encArg(x_1)) [0] encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) [0] encode_true -> true [0] encode_odd(x_1) -> odd(encArg(x_1)) [0] encode_p(x_1) -> p(encArg(x_1)) [0] encode_0 -> 0 [0] encode_false -> false [0] encode_s(x_1) -> s(encArg(x_1)) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: cond(true, x) -> cond(odd(x), p(x)) [1] odd(0) -> false [1] odd(s(0)) -> true [1] odd(s(s(x))) -> odd(x) [1] p(0) -> 0 [1] p(s(x)) -> x [1] encArg(true) -> true [0] encArg(0) -> 0 [0] encArg(false) -> false [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) [0] encArg(cons_odd(x_1)) -> odd(encArg(x_1)) [0] encArg(cons_p(x_1)) -> p(encArg(x_1)) [0] encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) [0] encode_true -> true [0] encode_odd(x_1) -> odd(encArg(x_1)) [0] encode_p(x_1) -> p(encArg(x_1)) [0] encode_0 -> 0 [0] encode_false -> false [0] encode_s(x_1) -> s(encArg(x_1)) [0] The TRS has the following type information: cond :: true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p true :: true:0:false:s:cons_cond:cons_odd:cons_p odd :: true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p p :: true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p 0 :: true:0:false:s:cons_cond:cons_odd:cons_p false :: true:0:false:s:cons_cond:cons_odd:cons_p s :: true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p encArg :: true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p cons_cond :: true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p cons_odd :: true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p cons_p :: true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p encode_cond :: true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p encode_true :: true:0:false:s:cons_cond:cons_odd:cons_p encode_odd :: true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p encode_p :: true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p encode_0 :: true:0:false:s:cons_cond:cons_odd:cons_p encode_false :: true:0:false:s:cons_cond:cons_odd:cons_p encode_s :: true:0:false:s:cons_cond:cons_odd:cons_p -> true:0:false:s:cons_cond:cons_odd:cons_p Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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: none (c) The following functions are completely defined: odd_1 p_1 cond_2 encArg_1 encode_cond_2 encode_true encode_odd_1 encode_p_1 encode_0 encode_false encode_s_1 Due to the following rules being added: encArg(v0) -> null_encArg [0] encode_cond(v0, v1) -> null_encode_cond [0] encode_true -> null_encode_true [0] encode_odd(v0) -> null_encode_odd [0] encode_p(v0) -> null_encode_p [0] encode_0 -> null_encode_0 [0] encode_false -> null_encode_false [0] encode_s(v0) -> null_encode_s [0] odd(v0) -> null_odd [0] p(v0) -> null_p [0] cond(v0, v1) -> null_cond [0] And the following fresh constants: null_encArg, null_encode_cond, null_encode_true, null_encode_odd, null_encode_p, null_encode_0, null_encode_false, null_encode_s, null_odd, null_p, null_cond ---------------------------------------- (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: cond(true, x) -> cond(odd(x), p(x)) [1] odd(0) -> false [1] odd(s(0)) -> true [1] odd(s(s(x))) -> odd(x) [1] p(0) -> 0 [1] p(s(x)) -> x [1] encArg(true) -> true [0] encArg(0) -> 0 [0] encArg(false) -> false [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) [0] encArg(cons_odd(x_1)) -> odd(encArg(x_1)) [0] encArg(cons_p(x_1)) -> p(encArg(x_1)) [0] encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) [0] encode_true -> true [0] encode_odd(x_1) -> odd(encArg(x_1)) [0] encode_p(x_1) -> p(encArg(x_1)) [0] encode_0 -> 0 [0] encode_false -> false [0] encode_s(x_1) -> s(encArg(x_1)) [0] encArg(v0) -> null_encArg [0] encode_cond(v0, v1) -> null_encode_cond [0] encode_true -> null_encode_true [0] encode_odd(v0) -> null_encode_odd [0] encode_p(v0) -> null_encode_p [0] encode_0 -> null_encode_0 [0] encode_false -> null_encode_false [0] encode_s(v0) -> null_encode_s [0] odd(v0) -> null_odd [0] p(v0) -> null_p [0] cond(v0, v1) -> null_cond [0] The TRS has the following type information: cond :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond true :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond odd :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond p :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond 0 :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond false :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond s :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encArg :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond cons_cond :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond cons_odd :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond cons_p :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_cond :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_true :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_odd :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_p :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_0 :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_false :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_s :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encArg :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_cond :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_true :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_odd :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_p :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_0 :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_false :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_s :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_odd :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_p :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_cond :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) 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: cond(true, 0) -> cond(false, 0) [3] cond(true, 0) -> cond(false, null_p) [2] cond(true, s(0)) -> cond(true, 0) [3] cond(true, s(0)) -> cond(true, null_p) [2] cond(true, s(s(x'))) -> cond(odd(x'), s(x')) [3] cond(true, s(s(x'))) -> cond(odd(x'), null_p) [2] cond(true, 0) -> cond(null_odd, 0) [2] cond(true, s(x'')) -> cond(null_odd, x'') [2] cond(true, x) -> cond(null_odd, null_p) [1] odd(0) -> false [1] odd(s(0)) -> true [1] odd(s(s(x))) -> odd(x) [1] p(0) -> 0 [1] p(s(x)) -> x [1] encArg(true) -> true [0] encArg(0) -> 0 [0] encArg(false) -> false [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) [0] encArg(cons_odd(true)) -> odd(true) [0] encArg(cons_odd(0)) -> odd(0) [0] encArg(cons_odd(false)) -> odd(false) [0] encArg(cons_odd(s(x_135))) -> odd(s(encArg(x_135))) [0] encArg(cons_odd(cons_cond(x_136, x_28))) -> odd(cond(encArg(x_136), encArg(x_28))) [0] encArg(cons_odd(cons_odd(x_137))) -> odd(odd(encArg(x_137))) [0] encArg(cons_odd(cons_p(x_138))) -> odd(p(encArg(x_138))) [0] encArg(cons_odd(x_1)) -> odd(null_encArg) [0] encArg(cons_p(true)) -> p(true) [0] encArg(cons_p(0)) -> p(0) [0] encArg(cons_p(false)) -> p(false) [0] encArg(cons_p(s(x_139))) -> p(s(encArg(x_139))) [0] encArg(cons_p(cons_cond(x_140, x_29))) -> p(cond(encArg(x_140), encArg(x_29))) [0] encArg(cons_p(cons_odd(x_141))) -> p(odd(encArg(x_141))) [0] encArg(cons_p(cons_p(x_142))) -> p(p(encArg(x_142))) [0] encArg(cons_p(x_1)) -> p(null_encArg) [0] encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) [0] encode_true -> true [0] encode_odd(true) -> odd(true) [0] encode_odd(0) -> odd(0) [0] encode_odd(false) -> odd(false) [0] encode_odd(s(x_179)) -> odd(s(encArg(x_179))) [0] encode_odd(cons_cond(x_180, x_219)) -> odd(cond(encArg(x_180), encArg(x_219))) [0] encode_odd(cons_odd(x_181)) -> odd(odd(encArg(x_181))) [0] encode_odd(cons_p(x_182)) -> odd(p(encArg(x_182))) [0] encode_odd(x_1) -> odd(null_encArg) [0] encode_p(true) -> p(true) [0] encode_p(0) -> p(0) [0] encode_p(false) -> p(false) [0] encode_p(s(x_183)) -> p(s(encArg(x_183))) [0] encode_p(cons_cond(x_184, x_220)) -> p(cond(encArg(x_184), encArg(x_220))) [0] encode_p(cons_odd(x_185)) -> p(odd(encArg(x_185))) [0] encode_p(cons_p(x_186)) -> p(p(encArg(x_186))) [0] encode_p(x_1) -> p(null_encArg) [0] encode_0 -> 0 [0] encode_false -> false [0] encode_s(x_1) -> s(encArg(x_1)) [0] encArg(v0) -> null_encArg [0] encode_cond(v0, v1) -> null_encode_cond [0] encode_true -> null_encode_true [0] encode_odd(v0) -> null_encode_odd [0] encode_p(v0) -> null_encode_p [0] encode_0 -> null_encode_0 [0] encode_false -> null_encode_false [0] encode_s(v0) -> null_encode_s [0] odd(v0) -> null_odd [0] p(v0) -> null_p [0] cond(v0, v1) -> null_cond [0] The TRS has the following type information: cond :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond true :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond odd :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond p :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond 0 :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond false :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond s :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encArg :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond cons_cond :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond cons_odd :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond cons_p :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_cond :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_true :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_odd :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_p :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_0 :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_false :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond encode_s :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond -> true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encArg :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_cond :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_true :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_odd :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_p :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_0 :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_false :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_encode_s :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_odd :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_p :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond null_cond :: true:0:false:s:cons_cond:cons_odd:cons_p:null_encArg:null_encode_cond:null_encode_true:null_encode_odd:null_encode_p:null_encode_0:null_encode_false:null_encode_s:null_odd:null_p:null_cond Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: true => 2 0 => 0 false => 1 null_encArg => 0 null_encode_cond => 0 null_encode_true => 0 null_encode_odd => 0 null_encode_p => 0 null_encode_0 => 0 null_encode_false => 0 null_encode_s => 0 null_odd => 0 null_p => 0 null_cond => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(x'), 0) :|: z = 2, z' = 1 + (1 + x'), x' >= 0 cond(z, z') -{ 3 }-> cond(odd(x'), 1 + x') :|: z = 2, z' = 1 + (1 + x'), x' >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, x'') :|: z = 2, z' = 1 + x'', x'' >= 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' = x, x >= 0 cond(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encArg(z) -{ 0 }-> p(p(encArg(x_142))) :|: z = 1 + (1 + x_142), x_142 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(x_141))) :|: z = 1 + (1 + x_141), x_141 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> p(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> p(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> p(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(x_139)) :|: z = 1 + (1 + x_139), x_139 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(x_138))) :|: x_138 >= 0, z = 1 + (1 + x_138) encArg(z) -{ 0 }-> odd(odd(encArg(x_137))) :|: x_137 >= 0, z = 1 + (1 + x_137) encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(x_135)) :|: z = 1 + (1 + x_135), x_135 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_cond(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(x_182))) :|: x_182 >= 0, z = 1 + x_182 encode_odd(z) -{ 0 }-> odd(odd(encArg(x_181))) :|: z = 1 + x_181, x_181 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: x_1 >= 0, z = x_1 encode_odd(z) -{ 0 }-> odd(1 + encArg(x_179)) :|: z = 1 + x_179, x_179 >= 0 encode_odd(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_p(z) -{ 0 }-> p(p(encArg(x_186))) :|: z = 1 + x_186, x_186 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(x_185))) :|: z = 1 + x_185, x_185 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(2) :|: z = 2 encode_p(z) -{ 0 }-> p(1) :|: z = 1 encode_p(z) -{ 0 }-> p(0) :|: z = 0 encode_p(z) -{ 0 }-> p(0) :|: x_1 >= 0, z = x_1 encode_p(z) -{ 0 }-> p(1 + encArg(x_183)) :|: z = 1 + x_183, x_183 >= 0 encode_p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(x) :|: x >= 0, z = 1 + (1 + x) odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(x'), 0) :|: z = 2, z' = 1 + (1 + x'), x' >= 0 cond(z, z') -{ 3 }-> cond(odd(x'), 1 + x') :|: z = 2, z' = 1 + (1 + x'), x' >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, x'') :|: z = 2, z' = 1 + x'', x'' >= 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' = x, x >= 0 cond(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(x_142))) :|: z = 1 + (1 + x_142), x_142 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(x_141))) :|: z = 1 + (1 + x_141), x_141 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(x_139)) :|: z = 1 + (1 + x_139), x_139 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(x_138))) :|: x_138 >= 0, z = 1 + (1 + x_138) encArg(z) -{ 0 }-> odd(odd(encArg(x_137))) :|: x_137 >= 0, z = 1 + (1 + x_137) encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(x_135)) :|: z = 1 + (1 + x_135), x_135 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + x_1, x_1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + x_1, x_1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_cond(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(x_182))) :|: x_182 >= 0, z = 1 + x_182 encode_odd(z) -{ 0 }-> odd(odd(encArg(x_181))) :|: z = 1 + x_181, x_181 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: x_1 >= 0, z = x_1 encode_odd(z) -{ 0 }-> odd(1 + encArg(x_179)) :|: z = 1 + x_179, x_179 >= 0 encode_odd(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(x_186))) :|: z = 1 + x_186, x_186 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(x_185))) :|: z = 1 + x_185, x_185 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(x_183)) :|: z = 1 + x_183, x_183 >= 0 encode_p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: x_1 >= 0, z = x_1, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: x_1 >= 0, z = x_1, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(x) :|: x >= 0, z = 1 + (1 + x) odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { encode_0 } { encode_false } { encode_true } { odd } { p } { cond } { encArg } { encode_p } { encode_odd } { encode_s } { encode_cond } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_0}, {encode_false}, {encode_true}, {odd}, {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_0}, {encode_false}, {encode_true}, {odd}, {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_0 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_0}, {encode_false}, {encode_true}, {odd}, {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: ?, size: O(1) [0] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_0 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_false}, {encode_true}, {odd}, {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_false}, {encode_true}, {odd}, {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_false 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: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_false}, {encode_true}, {odd}, {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: ?, size: O(1) [1] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_false after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_true}, {odd}, {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_true}, {odd}, {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_true after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_true}, {odd}, {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: ?, size: O(1) [2] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_true after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {odd}, {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {odd}, {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: odd after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {odd}, {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: ?, size: O(1) [2] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: odd after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 2 }-> cond(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> odd(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> odd(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> odd(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(2) :|: z = 2 encode_odd(z) -{ 0 }-> odd(1) :|: z = 1 encode_odd(z) -{ 0 }-> odd(0) :|: z = 0 encode_odd(z) -{ 0 }-> odd(0) :|: z >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 3 + z' }-> cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 2 + z' }-> cond(s', 0) :|: s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] ---------------------------------------- (47) 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 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 3 + z' }-> cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 2 + z' }-> cond(s', 0) :|: s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (49) 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 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 3 + z' }-> cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 2 + z' }-> cond(s', 0) :|: s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (51) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 3 + z' }-> cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 2 + z' }-> cond(s', 0) :|: s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: cond after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 3 + z' }-> cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 2 + z' }-> cond(s', 0) :|: s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {cond}, {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: ?, size: O(1) [0] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: cond after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 9 + 5*z' + z'^2 ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 3 + z' }-> cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 2 + z' }-> cond(s', 0) :|: s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 3 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 2 }-> cond(2, 0) :|: z = 2, z' = 1 + 0 cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(1, 0) :|: z = 2, z' = 0 cond(z, z') -{ 2 }-> cond(0, 0) :|: z = 2, z' = 0 cond(z, z') -{ 1 }-> cond(0, 0) :|: z = 2, z' >= 0 cond(z, z') -{ 2 }-> cond(0, z' - 1) :|: z = 2, z' - 1 >= 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] ---------------------------------------- (57) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: encArg after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encArg}, {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encArg after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 73 + 169*z + 49*z^2 + 6*z^3 ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(cond(encArg(x_140), encArg(x_29))) :|: z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(odd(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> odd(cond(encArg(x_136), encArg(x_28))) :|: z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 0 }-> odd(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> cond(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 0 }-> cond(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> odd(p(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> odd(cond(encArg(x_180), encArg(x_219))) :|: z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ 0 }-> odd(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(odd(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(cond(encArg(x_184), encArg(x_220))) :|: z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [1 + z] ---------------------------------------- (63) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 155 + 5*s20 + s20^2 + 169*x_1 + 49*x_1^2 + 6*x_1^3 + 169*x_2 + 49*x_2^2 + 6*x_2^3 }-> s21 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -114 + s22 + 45*z + 13*z^2 + 6*z^3 }-> s23 :|: s22 >= 0, s22 <= z - 2 + 1, s23 >= 0, s23 <= 2, z - 2 >= 0 encArg(z) -{ 157 + 5*s25 + s25^2 + s26 + 169*x_136 + 49*x_136^2 + 6*x_136^3 + 169*x_28 + 49*x_28^2 + 6*x_28^3 }-> s27 :|: s24 >= 0, s24 <= x_136 + 1, s25 >= 0, s25 <= x_28 + 1, s26 >= 0, s26 <= 0, s27 >= 0, s27 <= 2, z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ -113 + s28 + s29 + 45*z + 13*z^2 + 6*z^3 }-> s30 :|: s28 >= 0, s28 <= z - 2 + 1, s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, z - 2 >= 0 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ -114 + s45 + 45*z + 13*z^2 + 6*z^3 }-> s46 :|: s44 >= 0, s44 <= z - 2 + 1, s45 >= 0, s45 <= s44, s46 >= 0, s46 <= 2, z - 2 >= 0 encArg(z) -{ -116 + 45*z + 13*z^2 + 6*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= 1 + s47, z - 2 >= 0 encArg(z) -{ 156 + 5*s50 + s50^2 + 169*x_140 + 49*x_140^2 + 6*x_140^3 + 169*x_29 + 49*x_29^2 + 6*x_29^3 }-> s52 :|: s49 >= 0, s49 <= x_140 + 1, s50 >= 0, s50 <= x_29 + 1, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= s51, z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ -114 + s53 + 45*z + 13*z^2 + 6*z^3 }-> s55 :|: s53 >= 0, s53 <= z - 2 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, z - 2 >= 0 encArg(z) -{ -115 + 45*z + 13*z^2 + 6*z^3 }-> s58 :|: s56 >= 0, s56 <= z - 2 + 1, s57 >= 0, s57 <= s56, s58 >= 0, s58 <= s57, z - 2 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -53 + 89*z + 31*z^2 + 6*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 155 + 5*s32 + s32^2 + 169*z + 49*z^2 + 6*z^3 + 169*z' + 49*z'^2 + 6*z'^3 }-> s33 :|: s31 >= 0, s31 <= z + 1, s32 >= 0, s32 <= z' + 1, s33 >= 0, s33 <= 0, z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ -50 + s34 + 89*z + 31*z^2 + 6*z^3 }-> s35 :|: s34 >= 0, s34 <= z - 1 + 1, s35 >= 0, s35 <= 2, z - 1 >= 0 encode_odd(z) -{ 157 + 5*s37 + s37^2 + s38 + 169*x_180 + 49*x_180^2 + 6*x_180^3 + 169*x_219 + 49*x_219^2 + 6*x_219^3 }-> s39 :|: s36 >= 0, s36 <= x_180 + 1, s37 >= 0, s37 <= x_219 + 1, s38 >= 0, s38 <= 0, s39 >= 0, s39 <= 2, z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ -49 + s40 + s41 + 89*z + 31*z^2 + 6*z^3 }-> s42 :|: s40 >= 0, s40 <= z - 1 + 1, s41 >= 0, s41 <= 2, s42 >= 0, s42 <= 2, z - 1 >= 0 encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ -50 + s60 + 89*z + 31*z^2 + 6*z^3 }-> s61 :|: s59 >= 0, s59 <= z - 1 + 1, s60 >= 0, s60 <= s59, s61 >= 0, s61 <= 2, z - 1 >= 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ -52 + 89*z + 31*z^2 + 6*z^3 }-> s63 :|: s62 >= 0, s62 <= z - 1 + 1, s63 >= 0, s63 <= 1 + s62, z - 1 >= 0 encode_p(z) -{ 156 + 5*s65 + s65^2 + 169*x_184 + 49*x_184^2 + 6*x_184^3 + 169*x_220 + 49*x_220^2 + 6*x_220^3 }-> s67 :|: s64 >= 0, s64 <= x_184 + 1, s65 >= 0, s65 <= x_220 + 1, s66 >= 0, s66 <= 0, s67 >= 0, s67 <= s66, z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ -50 + s68 + 89*z + 31*z^2 + 6*z^3 }-> s70 :|: s68 >= 0, s68 <= z - 1 + 1, s69 >= 0, s69 <= 2, s70 >= 0, s70 <= s69, z - 1 >= 0 encode_p(z) -{ -51 + 89*z + 31*z^2 + 6*z^3 }-> s73 :|: s71 >= 0, s71 <= z - 1 + 1, s72 >= 0, s72 <= s71, s73 >= 0, s73 <= s72, z - 1 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 73 + 169*z + 49*z^2 + 6*z^3 }-> 1 + s43 :|: s43 >= 0, s43 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [1 + z] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 155 + 5*s20 + s20^2 + 169*x_1 + 49*x_1^2 + 6*x_1^3 + 169*x_2 + 49*x_2^2 + 6*x_2^3 }-> s21 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -114 + s22 + 45*z + 13*z^2 + 6*z^3 }-> s23 :|: s22 >= 0, s22 <= z - 2 + 1, s23 >= 0, s23 <= 2, z - 2 >= 0 encArg(z) -{ 157 + 5*s25 + s25^2 + s26 + 169*x_136 + 49*x_136^2 + 6*x_136^3 + 169*x_28 + 49*x_28^2 + 6*x_28^3 }-> s27 :|: s24 >= 0, s24 <= x_136 + 1, s25 >= 0, s25 <= x_28 + 1, s26 >= 0, s26 <= 0, s27 >= 0, s27 <= 2, z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ -113 + s28 + s29 + 45*z + 13*z^2 + 6*z^3 }-> s30 :|: s28 >= 0, s28 <= z - 2 + 1, s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, z - 2 >= 0 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ -114 + s45 + 45*z + 13*z^2 + 6*z^3 }-> s46 :|: s44 >= 0, s44 <= z - 2 + 1, s45 >= 0, s45 <= s44, s46 >= 0, s46 <= 2, z - 2 >= 0 encArg(z) -{ -116 + 45*z + 13*z^2 + 6*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= 1 + s47, z - 2 >= 0 encArg(z) -{ 156 + 5*s50 + s50^2 + 169*x_140 + 49*x_140^2 + 6*x_140^3 + 169*x_29 + 49*x_29^2 + 6*x_29^3 }-> s52 :|: s49 >= 0, s49 <= x_140 + 1, s50 >= 0, s50 <= x_29 + 1, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= s51, z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ -114 + s53 + 45*z + 13*z^2 + 6*z^3 }-> s55 :|: s53 >= 0, s53 <= z - 2 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, z - 2 >= 0 encArg(z) -{ -115 + 45*z + 13*z^2 + 6*z^3 }-> s58 :|: s56 >= 0, s56 <= z - 2 + 1, s57 >= 0, s57 <= s56, s58 >= 0, s58 <= s57, z - 2 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -53 + 89*z + 31*z^2 + 6*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 155 + 5*s32 + s32^2 + 169*z + 49*z^2 + 6*z^3 + 169*z' + 49*z'^2 + 6*z'^3 }-> s33 :|: s31 >= 0, s31 <= z + 1, s32 >= 0, s32 <= z' + 1, s33 >= 0, s33 <= 0, z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ -50 + s34 + 89*z + 31*z^2 + 6*z^3 }-> s35 :|: s34 >= 0, s34 <= z - 1 + 1, s35 >= 0, s35 <= 2, z - 1 >= 0 encode_odd(z) -{ 157 + 5*s37 + s37^2 + s38 + 169*x_180 + 49*x_180^2 + 6*x_180^3 + 169*x_219 + 49*x_219^2 + 6*x_219^3 }-> s39 :|: s36 >= 0, s36 <= x_180 + 1, s37 >= 0, s37 <= x_219 + 1, s38 >= 0, s38 <= 0, s39 >= 0, s39 <= 2, z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ -49 + s40 + s41 + 89*z + 31*z^2 + 6*z^3 }-> s42 :|: s40 >= 0, s40 <= z - 1 + 1, s41 >= 0, s41 <= 2, s42 >= 0, s42 <= 2, z - 1 >= 0 encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ -50 + s60 + 89*z + 31*z^2 + 6*z^3 }-> s61 :|: s59 >= 0, s59 <= z - 1 + 1, s60 >= 0, s60 <= s59, s61 >= 0, s61 <= 2, z - 1 >= 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ -52 + 89*z + 31*z^2 + 6*z^3 }-> s63 :|: s62 >= 0, s62 <= z - 1 + 1, s63 >= 0, s63 <= 1 + s62, z - 1 >= 0 encode_p(z) -{ 156 + 5*s65 + s65^2 + 169*x_184 + 49*x_184^2 + 6*x_184^3 + 169*x_220 + 49*x_220^2 + 6*x_220^3 }-> s67 :|: s64 >= 0, s64 <= x_184 + 1, s65 >= 0, s65 <= x_220 + 1, s66 >= 0, s66 <= 0, s67 >= 0, s67 <= s66, z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ -50 + s68 + 89*z + 31*z^2 + 6*z^3 }-> s70 :|: s68 >= 0, s68 <= z - 1 + 1, s69 >= 0, s69 <= 2, s70 >= 0, s70 <= s69, z - 1 >= 0 encode_p(z) -{ -51 + 89*z + 31*z^2 + 6*z^3 }-> s73 :|: s71 >= 0, s71 <= z - 1 + 1, s72 >= 0, s72 <= s71, s73 >= 0, s73 <= s72, z - 1 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 73 + 169*z + 49*z^2 + 6*z^3 }-> 1 + s43 :|: s43 >= 0, s43 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_p}, {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [1 + z] encode_p: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (67) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_p after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 166 + 613*z + 192*z^2 + 30*z^3 ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 155 + 5*s20 + s20^2 + 169*x_1 + 49*x_1^2 + 6*x_1^3 + 169*x_2 + 49*x_2^2 + 6*x_2^3 }-> s21 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -114 + s22 + 45*z + 13*z^2 + 6*z^3 }-> s23 :|: s22 >= 0, s22 <= z - 2 + 1, s23 >= 0, s23 <= 2, z - 2 >= 0 encArg(z) -{ 157 + 5*s25 + s25^2 + s26 + 169*x_136 + 49*x_136^2 + 6*x_136^3 + 169*x_28 + 49*x_28^2 + 6*x_28^3 }-> s27 :|: s24 >= 0, s24 <= x_136 + 1, s25 >= 0, s25 <= x_28 + 1, s26 >= 0, s26 <= 0, s27 >= 0, s27 <= 2, z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ -113 + s28 + s29 + 45*z + 13*z^2 + 6*z^3 }-> s30 :|: s28 >= 0, s28 <= z - 2 + 1, s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, z - 2 >= 0 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ -114 + s45 + 45*z + 13*z^2 + 6*z^3 }-> s46 :|: s44 >= 0, s44 <= z - 2 + 1, s45 >= 0, s45 <= s44, s46 >= 0, s46 <= 2, z - 2 >= 0 encArg(z) -{ -116 + 45*z + 13*z^2 + 6*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= 1 + s47, z - 2 >= 0 encArg(z) -{ 156 + 5*s50 + s50^2 + 169*x_140 + 49*x_140^2 + 6*x_140^3 + 169*x_29 + 49*x_29^2 + 6*x_29^3 }-> s52 :|: s49 >= 0, s49 <= x_140 + 1, s50 >= 0, s50 <= x_29 + 1, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= s51, z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ -114 + s53 + 45*z + 13*z^2 + 6*z^3 }-> s55 :|: s53 >= 0, s53 <= z - 2 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, z - 2 >= 0 encArg(z) -{ -115 + 45*z + 13*z^2 + 6*z^3 }-> s58 :|: s56 >= 0, s56 <= z - 2 + 1, s57 >= 0, s57 <= s56, s58 >= 0, s58 <= s57, z - 2 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -53 + 89*z + 31*z^2 + 6*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 155 + 5*s32 + s32^2 + 169*z + 49*z^2 + 6*z^3 + 169*z' + 49*z'^2 + 6*z'^3 }-> s33 :|: s31 >= 0, s31 <= z + 1, s32 >= 0, s32 <= z' + 1, s33 >= 0, s33 <= 0, z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ -50 + s34 + 89*z + 31*z^2 + 6*z^3 }-> s35 :|: s34 >= 0, s34 <= z - 1 + 1, s35 >= 0, s35 <= 2, z - 1 >= 0 encode_odd(z) -{ 157 + 5*s37 + s37^2 + s38 + 169*x_180 + 49*x_180^2 + 6*x_180^3 + 169*x_219 + 49*x_219^2 + 6*x_219^3 }-> s39 :|: s36 >= 0, s36 <= x_180 + 1, s37 >= 0, s37 <= x_219 + 1, s38 >= 0, s38 <= 0, s39 >= 0, s39 <= 2, z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ -49 + s40 + s41 + 89*z + 31*z^2 + 6*z^3 }-> s42 :|: s40 >= 0, s40 <= z - 1 + 1, s41 >= 0, s41 <= 2, s42 >= 0, s42 <= 2, z - 1 >= 0 encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ -50 + s60 + 89*z + 31*z^2 + 6*z^3 }-> s61 :|: s59 >= 0, s59 <= z - 1 + 1, s60 >= 0, s60 <= s59, s61 >= 0, s61 <= 2, z - 1 >= 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ -52 + 89*z + 31*z^2 + 6*z^3 }-> s63 :|: s62 >= 0, s62 <= z - 1 + 1, s63 >= 0, s63 <= 1 + s62, z - 1 >= 0 encode_p(z) -{ 156 + 5*s65 + s65^2 + 169*x_184 + 49*x_184^2 + 6*x_184^3 + 169*x_220 + 49*x_220^2 + 6*x_220^3 }-> s67 :|: s64 >= 0, s64 <= x_184 + 1, s65 >= 0, s65 <= x_220 + 1, s66 >= 0, s66 <= 0, s67 >= 0, s67 <= s66, z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ -50 + s68 + 89*z + 31*z^2 + 6*z^3 }-> s70 :|: s68 >= 0, s68 <= z - 1 + 1, s69 >= 0, s69 <= 2, s70 >= 0, s70 <= s69, z - 1 >= 0 encode_p(z) -{ -51 + 89*z + 31*z^2 + 6*z^3 }-> s73 :|: s71 >= 0, s71 <= z - 1 + 1, s72 >= 0, s72 <= s71, s73 >= 0, s73 <= s72, z - 1 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 73 + 169*z + 49*z^2 + 6*z^3 }-> 1 + s43 :|: s43 >= 0, s43 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [166 + 613*z + 192*z^2 + 30*z^3], size: O(n^1) [1 + z] ---------------------------------------- (69) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 155 + 5*s20 + s20^2 + 169*x_1 + 49*x_1^2 + 6*x_1^3 + 169*x_2 + 49*x_2^2 + 6*x_2^3 }-> s21 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -114 + s22 + 45*z + 13*z^2 + 6*z^3 }-> s23 :|: s22 >= 0, s22 <= z - 2 + 1, s23 >= 0, s23 <= 2, z - 2 >= 0 encArg(z) -{ 157 + 5*s25 + s25^2 + s26 + 169*x_136 + 49*x_136^2 + 6*x_136^3 + 169*x_28 + 49*x_28^2 + 6*x_28^3 }-> s27 :|: s24 >= 0, s24 <= x_136 + 1, s25 >= 0, s25 <= x_28 + 1, s26 >= 0, s26 <= 0, s27 >= 0, s27 <= 2, z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ -113 + s28 + s29 + 45*z + 13*z^2 + 6*z^3 }-> s30 :|: s28 >= 0, s28 <= z - 2 + 1, s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, z - 2 >= 0 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ -114 + s45 + 45*z + 13*z^2 + 6*z^3 }-> s46 :|: s44 >= 0, s44 <= z - 2 + 1, s45 >= 0, s45 <= s44, s46 >= 0, s46 <= 2, z - 2 >= 0 encArg(z) -{ -116 + 45*z + 13*z^2 + 6*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= 1 + s47, z - 2 >= 0 encArg(z) -{ 156 + 5*s50 + s50^2 + 169*x_140 + 49*x_140^2 + 6*x_140^3 + 169*x_29 + 49*x_29^2 + 6*x_29^3 }-> s52 :|: s49 >= 0, s49 <= x_140 + 1, s50 >= 0, s50 <= x_29 + 1, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= s51, z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ -114 + s53 + 45*z + 13*z^2 + 6*z^3 }-> s55 :|: s53 >= 0, s53 <= z - 2 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, z - 2 >= 0 encArg(z) -{ -115 + 45*z + 13*z^2 + 6*z^3 }-> s58 :|: s56 >= 0, s56 <= z - 2 + 1, s57 >= 0, s57 <= s56, s58 >= 0, s58 <= s57, z - 2 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -53 + 89*z + 31*z^2 + 6*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 155 + 5*s32 + s32^2 + 169*z + 49*z^2 + 6*z^3 + 169*z' + 49*z'^2 + 6*z'^3 }-> s33 :|: s31 >= 0, s31 <= z + 1, s32 >= 0, s32 <= z' + 1, s33 >= 0, s33 <= 0, z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ -50 + s34 + 89*z + 31*z^2 + 6*z^3 }-> s35 :|: s34 >= 0, s34 <= z - 1 + 1, s35 >= 0, s35 <= 2, z - 1 >= 0 encode_odd(z) -{ 157 + 5*s37 + s37^2 + s38 + 169*x_180 + 49*x_180^2 + 6*x_180^3 + 169*x_219 + 49*x_219^2 + 6*x_219^3 }-> s39 :|: s36 >= 0, s36 <= x_180 + 1, s37 >= 0, s37 <= x_219 + 1, s38 >= 0, s38 <= 0, s39 >= 0, s39 <= 2, z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ -49 + s40 + s41 + 89*z + 31*z^2 + 6*z^3 }-> s42 :|: s40 >= 0, s40 <= z - 1 + 1, s41 >= 0, s41 <= 2, s42 >= 0, s42 <= 2, z - 1 >= 0 encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ -50 + s60 + 89*z + 31*z^2 + 6*z^3 }-> s61 :|: s59 >= 0, s59 <= z - 1 + 1, s60 >= 0, s60 <= s59, s61 >= 0, s61 <= 2, z - 1 >= 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ -52 + 89*z + 31*z^2 + 6*z^3 }-> s63 :|: s62 >= 0, s62 <= z - 1 + 1, s63 >= 0, s63 <= 1 + s62, z - 1 >= 0 encode_p(z) -{ 156 + 5*s65 + s65^2 + 169*x_184 + 49*x_184^2 + 6*x_184^3 + 169*x_220 + 49*x_220^2 + 6*x_220^3 }-> s67 :|: s64 >= 0, s64 <= x_184 + 1, s65 >= 0, s65 <= x_220 + 1, s66 >= 0, s66 <= 0, s67 >= 0, s67 <= s66, z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ -50 + s68 + 89*z + 31*z^2 + 6*z^3 }-> s70 :|: s68 >= 0, s68 <= z - 1 + 1, s69 >= 0, s69 <= 2, s70 >= 0, s70 <= s69, z - 1 >= 0 encode_p(z) -{ -51 + 89*z + 31*z^2 + 6*z^3 }-> s73 :|: s71 >= 0, s71 <= z - 1 + 1, s72 >= 0, s72 <= s71, s73 >= 0, s73 <= s72, z - 1 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 73 + 169*z + 49*z^2 + 6*z^3 }-> 1 + s43 :|: s43 >= 0, s43 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [166 + 613*z + 192*z^2 + 30*z^3], size: O(n^1) [1 + z] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_odd after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 155 + 5*s20 + s20^2 + 169*x_1 + 49*x_1^2 + 6*x_1^3 + 169*x_2 + 49*x_2^2 + 6*x_2^3 }-> s21 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -114 + s22 + 45*z + 13*z^2 + 6*z^3 }-> s23 :|: s22 >= 0, s22 <= z - 2 + 1, s23 >= 0, s23 <= 2, z - 2 >= 0 encArg(z) -{ 157 + 5*s25 + s25^2 + s26 + 169*x_136 + 49*x_136^2 + 6*x_136^3 + 169*x_28 + 49*x_28^2 + 6*x_28^3 }-> s27 :|: s24 >= 0, s24 <= x_136 + 1, s25 >= 0, s25 <= x_28 + 1, s26 >= 0, s26 <= 0, s27 >= 0, s27 <= 2, z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ -113 + s28 + s29 + 45*z + 13*z^2 + 6*z^3 }-> s30 :|: s28 >= 0, s28 <= z - 2 + 1, s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, z - 2 >= 0 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ -114 + s45 + 45*z + 13*z^2 + 6*z^3 }-> s46 :|: s44 >= 0, s44 <= z - 2 + 1, s45 >= 0, s45 <= s44, s46 >= 0, s46 <= 2, z - 2 >= 0 encArg(z) -{ -116 + 45*z + 13*z^2 + 6*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= 1 + s47, z - 2 >= 0 encArg(z) -{ 156 + 5*s50 + s50^2 + 169*x_140 + 49*x_140^2 + 6*x_140^3 + 169*x_29 + 49*x_29^2 + 6*x_29^3 }-> s52 :|: s49 >= 0, s49 <= x_140 + 1, s50 >= 0, s50 <= x_29 + 1, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= s51, z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ -114 + s53 + 45*z + 13*z^2 + 6*z^3 }-> s55 :|: s53 >= 0, s53 <= z - 2 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, z - 2 >= 0 encArg(z) -{ -115 + 45*z + 13*z^2 + 6*z^3 }-> s58 :|: s56 >= 0, s56 <= z - 2 + 1, s57 >= 0, s57 <= s56, s58 >= 0, s58 <= s57, z - 2 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -53 + 89*z + 31*z^2 + 6*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 155 + 5*s32 + s32^2 + 169*z + 49*z^2 + 6*z^3 + 169*z' + 49*z'^2 + 6*z'^3 }-> s33 :|: s31 >= 0, s31 <= z + 1, s32 >= 0, s32 <= z' + 1, s33 >= 0, s33 <= 0, z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ -50 + s34 + 89*z + 31*z^2 + 6*z^3 }-> s35 :|: s34 >= 0, s34 <= z - 1 + 1, s35 >= 0, s35 <= 2, z - 1 >= 0 encode_odd(z) -{ 157 + 5*s37 + s37^2 + s38 + 169*x_180 + 49*x_180^2 + 6*x_180^3 + 169*x_219 + 49*x_219^2 + 6*x_219^3 }-> s39 :|: s36 >= 0, s36 <= x_180 + 1, s37 >= 0, s37 <= x_219 + 1, s38 >= 0, s38 <= 0, s39 >= 0, s39 <= 2, z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ -49 + s40 + s41 + 89*z + 31*z^2 + 6*z^3 }-> s42 :|: s40 >= 0, s40 <= z - 1 + 1, s41 >= 0, s41 <= 2, s42 >= 0, s42 <= 2, z - 1 >= 0 encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ -50 + s60 + 89*z + 31*z^2 + 6*z^3 }-> s61 :|: s59 >= 0, s59 <= z - 1 + 1, s60 >= 0, s60 <= s59, s61 >= 0, s61 <= 2, z - 1 >= 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ -52 + 89*z + 31*z^2 + 6*z^3 }-> s63 :|: s62 >= 0, s62 <= z - 1 + 1, s63 >= 0, s63 <= 1 + s62, z - 1 >= 0 encode_p(z) -{ 156 + 5*s65 + s65^2 + 169*x_184 + 49*x_184^2 + 6*x_184^3 + 169*x_220 + 49*x_220^2 + 6*x_220^3 }-> s67 :|: s64 >= 0, s64 <= x_184 + 1, s65 >= 0, s65 <= x_220 + 1, s66 >= 0, s66 <= 0, s67 >= 0, s67 <= s66, z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ -50 + s68 + 89*z + 31*z^2 + 6*z^3 }-> s70 :|: s68 >= 0, s68 <= z - 1 + 1, s69 >= 0, s69 <= 2, s70 >= 0, s70 <= s69, z - 1 >= 0 encode_p(z) -{ -51 + 89*z + 31*z^2 + 6*z^3 }-> s73 :|: s71 >= 0, s71 <= z - 1 + 1, s72 >= 0, s72 <= s71, s73 >= 0, s73 <= s72, z - 1 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 73 + 169*z + 49*z^2 + 6*z^3 }-> 1 + s43 :|: s43 >= 0, s43 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_odd}, {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [166 + 613*z + 192*z^2 + 30*z^3], size: O(n^1) [1 + z] encode_odd: runtime: ?, size: O(1) [2] ---------------------------------------- (73) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_odd after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 174 + 615*z + 192*z^2 + 30*z^3 ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 155 + 5*s20 + s20^2 + 169*x_1 + 49*x_1^2 + 6*x_1^3 + 169*x_2 + 49*x_2^2 + 6*x_2^3 }-> s21 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -114 + s22 + 45*z + 13*z^2 + 6*z^3 }-> s23 :|: s22 >= 0, s22 <= z - 2 + 1, s23 >= 0, s23 <= 2, z - 2 >= 0 encArg(z) -{ 157 + 5*s25 + s25^2 + s26 + 169*x_136 + 49*x_136^2 + 6*x_136^3 + 169*x_28 + 49*x_28^2 + 6*x_28^3 }-> s27 :|: s24 >= 0, s24 <= x_136 + 1, s25 >= 0, s25 <= x_28 + 1, s26 >= 0, s26 <= 0, s27 >= 0, s27 <= 2, z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ -113 + s28 + s29 + 45*z + 13*z^2 + 6*z^3 }-> s30 :|: s28 >= 0, s28 <= z - 2 + 1, s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, z - 2 >= 0 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ -114 + s45 + 45*z + 13*z^2 + 6*z^3 }-> s46 :|: s44 >= 0, s44 <= z - 2 + 1, s45 >= 0, s45 <= s44, s46 >= 0, s46 <= 2, z - 2 >= 0 encArg(z) -{ -116 + 45*z + 13*z^2 + 6*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= 1 + s47, z - 2 >= 0 encArg(z) -{ 156 + 5*s50 + s50^2 + 169*x_140 + 49*x_140^2 + 6*x_140^3 + 169*x_29 + 49*x_29^2 + 6*x_29^3 }-> s52 :|: s49 >= 0, s49 <= x_140 + 1, s50 >= 0, s50 <= x_29 + 1, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= s51, z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ -114 + s53 + 45*z + 13*z^2 + 6*z^3 }-> s55 :|: s53 >= 0, s53 <= z - 2 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, z - 2 >= 0 encArg(z) -{ -115 + 45*z + 13*z^2 + 6*z^3 }-> s58 :|: s56 >= 0, s56 <= z - 2 + 1, s57 >= 0, s57 <= s56, s58 >= 0, s58 <= s57, z - 2 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -53 + 89*z + 31*z^2 + 6*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 155 + 5*s32 + s32^2 + 169*z + 49*z^2 + 6*z^3 + 169*z' + 49*z'^2 + 6*z'^3 }-> s33 :|: s31 >= 0, s31 <= z + 1, s32 >= 0, s32 <= z' + 1, s33 >= 0, s33 <= 0, z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ -50 + s34 + 89*z + 31*z^2 + 6*z^3 }-> s35 :|: s34 >= 0, s34 <= z - 1 + 1, s35 >= 0, s35 <= 2, z - 1 >= 0 encode_odd(z) -{ 157 + 5*s37 + s37^2 + s38 + 169*x_180 + 49*x_180^2 + 6*x_180^3 + 169*x_219 + 49*x_219^2 + 6*x_219^3 }-> s39 :|: s36 >= 0, s36 <= x_180 + 1, s37 >= 0, s37 <= x_219 + 1, s38 >= 0, s38 <= 0, s39 >= 0, s39 <= 2, z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ -49 + s40 + s41 + 89*z + 31*z^2 + 6*z^3 }-> s42 :|: s40 >= 0, s40 <= z - 1 + 1, s41 >= 0, s41 <= 2, s42 >= 0, s42 <= 2, z - 1 >= 0 encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ -50 + s60 + 89*z + 31*z^2 + 6*z^3 }-> s61 :|: s59 >= 0, s59 <= z - 1 + 1, s60 >= 0, s60 <= s59, s61 >= 0, s61 <= 2, z - 1 >= 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ -52 + 89*z + 31*z^2 + 6*z^3 }-> s63 :|: s62 >= 0, s62 <= z - 1 + 1, s63 >= 0, s63 <= 1 + s62, z - 1 >= 0 encode_p(z) -{ 156 + 5*s65 + s65^2 + 169*x_184 + 49*x_184^2 + 6*x_184^3 + 169*x_220 + 49*x_220^2 + 6*x_220^3 }-> s67 :|: s64 >= 0, s64 <= x_184 + 1, s65 >= 0, s65 <= x_220 + 1, s66 >= 0, s66 <= 0, s67 >= 0, s67 <= s66, z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ -50 + s68 + 89*z + 31*z^2 + 6*z^3 }-> s70 :|: s68 >= 0, s68 <= z - 1 + 1, s69 >= 0, s69 <= 2, s70 >= 0, s70 <= s69, z - 1 >= 0 encode_p(z) -{ -51 + 89*z + 31*z^2 + 6*z^3 }-> s73 :|: s71 >= 0, s71 <= z - 1 + 1, s72 >= 0, s72 <= s71, s73 >= 0, s73 <= s72, z - 1 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 73 + 169*z + 49*z^2 + 6*z^3 }-> 1 + s43 :|: s43 >= 0, s43 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [166 + 613*z + 192*z^2 + 30*z^3], size: O(n^1) [1 + z] encode_odd: runtime: O(n^3) [174 + 615*z + 192*z^2 + 30*z^3], size: O(1) [2] ---------------------------------------- (75) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 155 + 5*s20 + s20^2 + 169*x_1 + 49*x_1^2 + 6*x_1^3 + 169*x_2 + 49*x_2^2 + 6*x_2^3 }-> s21 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -114 + s22 + 45*z + 13*z^2 + 6*z^3 }-> s23 :|: s22 >= 0, s22 <= z - 2 + 1, s23 >= 0, s23 <= 2, z - 2 >= 0 encArg(z) -{ 157 + 5*s25 + s25^2 + s26 + 169*x_136 + 49*x_136^2 + 6*x_136^3 + 169*x_28 + 49*x_28^2 + 6*x_28^3 }-> s27 :|: s24 >= 0, s24 <= x_136 + 1, s25 >= 0, s25 <= x_28 + 1, s26 >= 0, s26 <= 0, s27 >= 0, s27 <= 2, z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ -113 + s28 + s29 + 45*z + 13*z^2 + 6*z^3 }-> s30 :|: s28 >= 0, s28 <= z - 2 + 1, s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, z - 2 >= 0 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ -114 + s45 + 45*z + 13*z^2 + 6*z^3 }-> s46 :|: s44 >= 0, s44 <= z - 2 + 1, s45 >= 0, s45 <= s44, s46 >= 0, s46 <= 2, z - 2 >= 0 encArg(z) -{ -116 + 45*z + 13*z^2 + 6*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= 1 + s47, z - 2 >= 0 encArg(z) -{ 156 + 5*s50 + s50^2 + 169*x_140 + 49*x_140^2 + 6*x_140^3 + 169*x_29 + 49*x_29^2 + 6*x_29^3 }-> s52 :|: s49 >= 0, s49 <= x_140 + 1, s50 >= 0, s50 <= x_29 + 1, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= s51, z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ -114 + s53 + 45*z + 13*z^2 + 6*z^3 }-> s55 :|: s53 >= 0, s53 <= z - 2 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, z - 2 >= 0 encArg(z) -{ -115 + 45*z + 13*z^2 + 6*z^3 }-> s58 :|: s56 >= 0, s56 <= z - 2 + 1, s57 >= 0, s57 <= s56, s58 >= 0, s58 <= s57, z - 2 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -53 + 89*z + 31*z^2 + 6*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 155 + 5*s32 + s32^2 + 169*z + 49*z^2 + 6*z^3 + 169*z' + 49*z'^2 + 6*z'^3 }-> s33 :|: s31 >= 0, s31 <= z + 1, s32 >= 0, s32 <= z' + 1, s33 >= 0, s33 <= 0, z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ -50 + s34 + 89*z + 31*z^2 + 6*z^3 }-> s35 :|: s34 >= 0, s34 <= z - 1 + 1, s35 >= 0, s35 <= 2, z - 1 >= 0 encode_odd(z) -{ 157 + 5*s37 + s37^2 + s38 + 169*x_180 + 49*x_180^2 + 6*x_180^3 + 169*x_219 + 49*x_219^2 + 6*x_219^3 }-> s39 :|: s36 >= 0, s36 <= x_180 + 1, s37 >= 0, s37 <= x_219 + 1, s38 >= 0, s38 <= 0, s39 >= 0, s39 <= 2, z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ -49 + s40 + s41 + 89*z + 31*z^2 + 6*z^3 }-> s42 :|: s40 >= 0, s40 <= z - 1 + 1, s41 >= 0, s41 <= 2, s42 >= 0, s42 <= 2, z - 1 >= 0 encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ -50 + s60 + 89*z + 31*z^2 + 6*z^3 }-> s61 :|: s59 >= 0, s59 <= z - 1 + 1, s60 >= 0, s60 <= s59, s61 >= 0, s61 <= 2, z - 1 >= 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ -52 + 89*z + 31*z^2 + 6*z^3 }-> s63 :|: s62 >= 0, s62 <= z - 1 + 1, s63 >= 0, s63 <= 1 + s62, z - 1 >= 0 encode_p(z) -{ 156 + 5*s65 + s65^2 + 169*x_184 + 49*x_184^2 + 6*x_184^3 + 169*x_220 + 49*x_220^2 + 6*x_220^3 }-> s67 :|: s64 >= 0, s64 <= x_184 + 1, s65 >= 0, s65 <= x_220 + 1, s66 >= 0, s66 <= 0, s67 >= 0, s67 <= s66, z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ -50 + s68 + 89*z + 31*z^2 + 6*z^3 }-> s70 :|: s68 >= 0, s68 <= z - 1 + 1, s69 >= 0, s69 <= 2, s70 >= 0, s70 <= s69, z - 1 >= 0 encode_p(z) -{ -51 + 89*z + 31*z^2 + 6*z^3 }-> s73 :|: s71 >= 0, s71 <= z - 1 + 1, s72 >= 0, s72 <= s71, s73 >= 0, s73 <= s72, z - 1 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 73 + 169*z + 49*z^2 + 6*z^3 }-> 1 + s43 :|: s43 >= 0, s43 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [166 + 613*z + 192*z^2 + 30*z^3], size: O(n^1) [1 + z] encode_odd: runtime: O(n^3) [174 + 615*z + 192*z^2 + 30*z^3], size: O(1) [2] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_s after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 155 + 5*s20 + s20^2 + 169*x_1 + 49*x_1^2 + 6*x_1^3 + 169*x_2 + 49*x_2^2 + 6*x_2^3 }-> s21 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -114 + s22 + 45*z + 13*z^2 + 6*z^3 }-> s23 :|: s22 >= 0, s22 <= z - 2 + 1, s23 >= 0, s23 <= 2, z - 2 >= 0 encArg(z) -{ 157 + 5*s25 + s25^2 + s26 + 169*x_136 + 49*x_136^2 + 6*x_136^3 + 169*x_28 + 49*x_28^2 + 6*x_28^3 }-> s27 :|: s24 >= 0, s24 <= x_136 + 1, s25 >= 0, s25 <= x_28 + 1, s26 >= 0, s26 <= 0, s27 >= 0, s27 <= 2, z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ -113 + s28 + s29 + 45*z + 13*z^2 + 6*z^3 }-> s30 :|: s28 >= 0, s28 <= z - 2 + 1, s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, z - 2 >= 0 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ -114 + s45 + 45*z + 13*z^2 + 6*z^3 }-> s46 :|: s44 >= 0, s44 <= z - 2 + 1, s45 >= 0, s45 <= s44, s46 >= 0, s46 <= 2, z - 2 >= 0 encArg(z) -{ -116 + 45*z + 13*z^2 + 6*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= 1 + s47, z - 2 >= 0 encArg(z) -{ 156 + 5*s50 + s50^2 + 169*x_140 + 49*x_140^2 + 6*x_140^3 + 169*x_29 + 49*x_29^2 + 6*x_29^3 }-> s52 :|: s49 >= 0, s49 <= x_140 + 1, s50 >= 0, s50 <= x_29 + 1, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= s51, z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ -114 + s53 + 45*z + 13*z^2 + 6*z^3 }-> s55 :|: s53 >= 0, s53 <= z - 2 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, z - 2 >= 0 encArg(z) -{ -115 + 45*z + 13*z^2 + 6*z^3 }-> s58 :|: s56 >= 0, s56 <= z - 2 + 1, s57 >= 0, s57 <= s56, s58 >= 0, s58 <= s57, z - 2 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -53 + 89*z + 31*z^2 + 6*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 155 + 5*s32 + s32^2 + 169*z + 49*z^2 + 6*z^3 + 169*z' + 49*z'^2 + 6*z'^3 }-> s33 :|: s31 >= 0, s31 <= z + 1, s32 >= 0, s32 <= z' + 1, s33 >= 0, s33 <= 0, z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ -50 + s34 + 89*z + 31*z^2 + 6*z^3 }-> s35 :|: s34 >= 0, s34 <= z - 1 + 1, s35 >= 0, s35 <= 2, z - 1 >= 0 encode_odd(z) -{ 157 + 5*s37 + s37^2 + s38 + 169*x_180 + 49*x_180^2 + 6*x_180^3 + 169*x_219 + 49*x_219^2 + 6*x_219^3 }-> s39 :|: s36 >= 0, s36 <= x_180 + 1, s37 >= 0, s37 <= x_219 + 1, s38 >= 0, s38 <= 0, s39 >= 0, s39 <= 2, z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ -49 + s40 + s41 + 89*z + 31*z^2 + 6*z^3 }-> s42 :|: s40 >= 0, s40 <= z - 1 + 1, s41 >= 0, s41 <= 2, s42 >= 0, s42 <= 2, z - 1 >= 0 encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ -50 + s60 + 89*z + 31*z^2 + 6*z^3 }-> s61 :|: s59 >= 0, s59 <= z - 1 + 1, s60 >= 0, s60 <= s59, s61 >= 0, s61 <= 2, z - 1 >= 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ -52 + 89*z + 31*z^2 + 6*z^3 }-> s63 :|: s62 >= 0, s62 <= z - 1 + 1, s63 >= 0, s63 <= 1 + s62, z - 1 >= 0 encode_p(z) -{ 156 + 5*s65 + s65^2 + 169*x_184 + 49*x_184^2 + 6*x_184^3 + 169*x_220 + 49*x_220^2 + 6*x_220^3 }-> s67 :|: s64 >= 0, s64 <= x_184 + 1, s65 >= 0, s65 <= x_220 + 1, s66 >= 0, s66 <= 0, s67 >= 0, s67 <= s66, z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ -50 + s68 + 89*z + 31*z^2 + 6*z^3 }-> s70 :|: s68 >= 0, s68 <= z - 1 + 1, s69 >= 0, s69 <= 2, s70 >= 0, s70 <= s69, z - 1 >= 0 encode_p(z) -{ -51 + 89*z + 31*z^2 + 6*z^3 }-> s73 :|: s71 >= 0, s71 <= z - 1 + 1, s72 >= 0, s72 <= s71, s73 >= 0, s73 <= s72, z - 1 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 73 + 169*z + 49*z^2 + 6*z^3 }-> 1 + s43 :|: s43 >= 0, s43 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_s}, {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [166 + 613*z + 192*z^2 + 30*z^3], size: O(n^1) [1 + z] encode_odd: runtime: O(n^3) [174 + 615*z + 192*z^2 + 30*z^3], size: O(1) [2] encode_s: runtime: ?, size: O(n^1) [2 + z] ---------------------------------------- (79) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_s after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 73 + 169*z + 49*z^2 + 6*z^3 ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 155 + 5*s20 + s20^2 + 169*x_1 + 49*x_1^2 + 6*x_1^3 + 169*x_2 + 49*x_2^2 + 6*x_2^3 }-> s21 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -114 + s22 + 45*z + 13*z^2 + 6*z^3 }-> s23 :|: s22 >= 0, s22 <= z - 2 + 1, s23 >= 0, s23 <= 2, z - 2 >= 0 encArg(z) -{ 157 + 5*s25 + s25^2 + s26 + 169*x_136 + 49*x_136^2 + 6*x_136^3 + 169*x_28 + 49*x_28^2 + 6*x_28^3 }-> s27 :|: s24 >= 0, s24 <= x_136 + 1, s25 >= 0, s25 <= x_28 + 1, s26 >= 0, s26 <= 0, s27 >= 0, s27 <= 2, z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ -113 + s28 + s29 + 45*z + 13*z^2 + 6*z^3 }-> s30 :|: s28 >= 0, s28 <= z - 2 + 1, s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, z - 2 >= 0 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ -114 + s45 + 45*z + 13*z^2 + 6*z^3 }-> s46 :|: s44 >= 0, s44 <= z - 2 + 1, s45 >= 0, s45 <= s44, s46 >= 0, s46 <= 2, z - 2 >= 0 encArg(z) -{ -116 + 45*z + 13*z^2 + 6*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= 1 + s47, z - 2 >= 0 encArg(z) -{ 156 + 5*s50 + s50^2 + 169*x_140 + 49*x_140^2 + 6*x_140^3 + 169*x_29 + 49*x_29^2 + 6*x_29^3 }-> s52 :|: s49 >= 0, s49 <= x_140 + 1, s50 >= 0, s50 <= x_29 + 1, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= s51, z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ -114 + s53 + 45*z + 13*z^2 + 6*z^3 }-> s55 :|: s53 >= 0, s53 <= z - 2 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, z - 2 >= 0 encArg(z) -{ -115 + 45*z + 13*z^2 + 6*z^3 }-> s58 :|: s56 >= 0, s56 <= z - 2 + 1, s57 >= 0, s57 <= s56, s58 >= 0, s58 <= s57, z - 2 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -53 + 89*z + 31*z^2 + 6*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 155 + 5*s32 + s32^2 + 169*z + 49*z^2 + 6*z^3 + 169*z' + 49*z'^2 + 6*z'^3 }-> s33 :|: s31 >= 0, s31 <= z + 1, s32 >= 0, s32 <= z' + 1, s33 >= 0, s33 <= 0, z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ -50 + s34 + 89*z + 31*z^2 + 6*z^3 }-> s35 :|: s34 >= 0, s34 <= z - 1 + 1, s35 >= 0, s35 <= 2, z - 1 >= 0 encode_odd(z) -{ 157 + 5*s37 + s37^2 + s38 + 169*x_180 + 49*x_180^2 + 6*x_180^3 + 169*x_219 + 49*x_219^2 + 6*x_219^3 }-> s39 :|: s36 >= 0, s36 <= x_180 + 1, s37 >= 0, s37 <= x_219 + 1, s38 >= 0, s38 <= 0, s39 >= 0, s39 <= 2, z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ -49 + s40 + s41 + 89*z + 31*z^2 + 6*z^3 }-> s42 :|: s40 >= 0, s40 <= z - 1 + 1, s41 >= 0, s41 <= 2, s42 >= 0, s42 <= 2, z - 1 >= 0 encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ -50 + s60 + 89*z + 31*z^2 + 6*z^3 }-> s61 :|: s59 >= 0, s59 <= z - 1 + 1, s60 >= 0, s60 <= s59, s61 >= 0, s61 <= 2, z - 1 >= 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ -52 + 89*z + 31*z^2 + 6*z^3 }-> s63 :|: s62 >= 0, s62 <= z - 1 + 1, s63 >= 0, s63 <= 1 + s62, z - 1 >= 0 encode_p(z) -{ 156 + 5*s65 + s65^2 + 169*x_184 + 49*x_184^2 + 6*x_184^3 + 169*x_220 + 49*x_220^2 + 6*x_220^3 }-> s67 :|: s64 >= 0, s64 <= x_184 + 1, s65 >= 0, s65 <= x_220 + 1, s66 >= 0, s66 <= 0, s67 >= 0, s67 <= s66, z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ -50 + s68 + 89*z + 31*z^2 + 6*z^3 }-> s70 :|: s68 >= 0, s68 <= z - 1 + 1, s69 >= 0, s69 <= 2, s70 >= 0, s70 <= s69, z - 1 >= 0 encode_p(z) -{ -51 + 89*z + 31*z^2 + 6*z^3 }-> s73 :|: s71 >= 0, s71 <= z - 1 + 1, s72 >= 0, s72 <= s71, s73 >= 0, s73 <= s72, z - 1 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 73 + 169*z + 49*z^2 + 6*z^3 }-> 1 + s43 :|: s43 >= 0, s43 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [166 + 613*z + 192*z^2 + 30*z^3], size: O(n^1) [1 + z] encode_odd: runtime: O(n^3) [174 + 615*z + 192*z^2 + 30*z^3], size: O(1) [2] encode_s: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [2 + z] ---------------------------------------- (81) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 155 + 5*s20 + s20^2 + 169*x_1 + 49*x_1^2 + 6*x_1^3 + 169*x_2 + 49*x_2^2 + 6*x_2^3 }-> s21 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -114 + s22 + 45*z + 13*z^2 + 6*z^3 }-> s23 :|: s22 >= 0, s22 <= z - 2 + 1, s23 >= 0, s23 <= 2, z - 2 >= 0 encArg(z) -{ 157 + 5*s25 + s25^2 + s26 + 169*x_136 + 49*x_136^2 + 6*x_136^3 + 169*x_28 + 49*x_28^2 + 6*x_28^3 }-> s27 :|: s24 >= 0, s24 <= x_136 + 1, s25 >= 0, s25 <= x_28 + 1, s26 >= 0, s26 <= 0, s27 >= 0, s27 <= 2, z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ -113 + s28 + s29 + 45*z + 13*z^2 + 6*z^3 }-> s30 :|: s28 >= 0, s28 <= z - 2 + 1, s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, z - 2 >= 0 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ -114 + s45 + 45*z + 13*z^2 + 6*z^3 }-> s46 :|: s44 >= 0, s44 <= z - 2 + 1, s45 >= 0, s45 <= s44, s46 >= 0, s46 <= 2, z - 2 >= 0 encArg(z) -{ -116 + 45*z + 13*z^2 + 6*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= 1 + s47, z - 2 >= 0 encArg(z) -{ 156 + 5*s50 + s50^2 + 169*x_140 + 49*x_140^2 + 6*x_140^3 + 169*x_29 + 49*x_29^2 + 6*x_29^3 }-> s52 :|: s49 >= 0, s49 <= x_140 + 1, s50 >= 0, s50 <= x_29 + 1, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= s51, z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ -114 + s53 + 45*z + 13*z^2 + 6*z^3 }-> s55 :|: s53 >= 0, s53 <= z - 2 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, z - 2 >= 0 encArg(z) -{ -115 + 45*z + 13*z^2 + 6*z^3 }-> s58 :|: s56 >= 0, s56 <= z - 2 + 1, s57 >= 0, s57 <= s56, s58 >= 0, s58 <= s57, z - 2 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -53 + 89*z + 31*z^2 + 6*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 155 + 5*s32 + s32^2 + 169*z + 49*z^2 + 6*z^3 + 169*z' + 49*z'^2 + 6*z'^3 }-> s33 :|: s31 >= 0, s31 <= z + 1, s32 >= 0, s32 <= z' + 1, s33 >= 0, s33 <= 0, z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ -50 + s34 + 89*z + 31*z^2 + 6*z^3 }-> s35 :|: s34 >= 0, s34 <= z - 1 + 1, s35 >= 0, s35 <= 2, z - 1 >= 0 encode_odd(z) -{ 157 + 5*s37 + s37^2 + s38 + 169*x_180 + 49*x_180^2 + 6*x_180^3 + 169*x_219 + 49*x_219^2 + 6*x_219^3 }-> s39 :|: s36 >= 0, s36 <= x_180 + 1, s37 >= 0, s37 <= x_219 + 1, s38 >= 0, s38 <= 0, s39 >= 0, s39 <= 2, z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ -49 + s40 + s41 + 89*z + 31*z^2 + 6*z^3 }-> s42 :|: s40 >= 0, s40 <= z - 1 + 1, s41 >= 0, s41 <= 2, s42 >= 0, s42 <= 2, z - 1 >= 0 encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ -50 + s60 + 89*z + 31*z^2 + 6*z^3 }-> s61 :|: s59 >= 0, s59 <= z - 1 + 1, s60 >= 0, s60 <= s59, s61 >= 0, s61 <= 2, z - 1 >= 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ -52 + 89*z + 31*z^2 + 6*z^3 }-> s63 :|: s62 >= 0, s62 <= z - 1 + 1, s63 >= 0, s63 <= 1 + s62, z - 1 >= 0 encode_p(z) -{ 156 + 5*s65 + s65^2 + 169*x_184 + 49*x_184^2 + 6*x_184^3 + 169*x_220 + 49*x_220^2 + 6*x_220^3 }-> s67 :|: s64 >= 0, s64 <= x_184 + 1, s65 >= 0, s65 <= x_220 + 1, s66 >= 0, s66 <= 0, s67 >= 0, s67 <= s66, z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ -50 + s68 + 89*z + 31*z^2 + 6*z^3 }-> s70 :|: s68 >= 0, s68 <= z - 1 + 1, s69 >= 0, s69 <= 2, s70 >= 0, s70 <= s69, z - 1 >= 0 encode_p(z) -{ -51 + 89*z + 31*z^2 + 6*z^3 }-> s73 :|: s71 >= 0, s71 <= z - 1 + 1, s72 >= 0, s72 <= s71, s73 >= 0, s73 <= s72, z - 1 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 73 + 169*z + 49*z^2 + 6*z^3 }-> 1 + s43 :|: s43 >= 0, s43 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [166 + 613*z + 192*z^2 + 30*z^3], size: O(n^1) [1 + z] encode_odd: runtime: O(n^3) [174 + 615*z + 192*z^2 + 30*z^3], size: O(1) [2] encode_s: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [2 + z] ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_cond after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 155 + 5*s20 + s20^2 + 169*x_1 + 49*x_1^2 + 6*x_1^3 + 169*x_2 + 49*x_2^2 + 6*x_2^3 }-> s21 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -114 + s22 + 45*z + 13*z^2 + 6*z^3 }-> s23 :|: s22 >= 0, s22 <= z - 2 + 1, s23 >= 0, s23 <= 2, z - 2 >= 0 encArg(z) -{ 157 + 5*s25 + s25^2 + s26 + 169*x_136 + 49*x_136^2 + 6*x_136^3 + 169*x_28 + 49*x_28^2 + 6*x_28^3 }-> s27 :|: s24 >= 0, s24 <= x_136 + 1, s25 >= 0, s25 <= x_28 + 1, s26 >= 0, s26 <= 0, s27 >= 0, s27 <= 2, z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ -113 + s28 + s29 + 45*z + 13*z^2 + 6*z^3 }-> s30 :|: s28 >= 0, s28 <= z - 2 + 1, s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, z - 2 >= 0 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ -114 + s45 + 45*z + 13*z^2 + 6*z^3 }-> s46 :|: s44 >= 0, s44 <= z - 2 + 1, s45 >= 0, s45 <= s44, s46 >= 0, s46 <= 2, z - 2 >= 0 encArg(z) -{ -116 + 45*z + 13*z^2 + 6*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= 1 + s47, z - 2 >= 0 encArg(z) -{ 156 + 5*s50 + s50^2 + 169*x_140 + 49*x_140^2 + 6*x_140^3 + 169*x_29 + 49*x_29^2 + 6*x_29^3 }-> s52 :|: s49 >= 0, s49 <= x_140 + 1, s50 >= 0, s50 <= x_29 + 1, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= s51, z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ -114 + s53 + 45*z + 13*z^2 + 6*z^3 }-> s55 :|: s53 >= 0, s53 <= z - 2 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, z - 2 >= 0 encArg(z) -{ -115 + 45*z + 13*z^2 + 6*z^3 }-> s58 :|: s56 >= 0, s56 <= z - 2 + 1, s57 >= 0, s57 <= s56, s58 >= 0, s58 <= s57, z - 2 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -53 + 89*z + 31*z^2 + 6*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 155 + 5*s32 + s32^2 + 169*z + 49*z^2 + 6*z^3 + 169*z' + 49*z'^2 + 6*z'^3 }-> s33 :|: s31 >= 0, s31 <= z + 1, s32 >= 0, s32 <= z' + 1, s33 >= 0, s33 <= 0, z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ -50 + s34 + 89*z + 31*z^2 + 6*z^3 }-> s35 :|: s34 >= 0, s34 <= z - 1 + 1, s35 >= 0, s35 <= 2, z - 1 >= 0 encode_odd(z) -{ 157 + 5*s37 + s37^2 + s38 + 169*x_180 + 49*x_180^2 + 6*x_180^3 + 169*x_219 + 49*x_219^2 + 6*x_219^3 }-> s39 :|: s36 >= 0, s36 <= x_180 + 1, s37 >= 0, s37 <= x_219 + 1, s38 >= 0, s38 <= 0, s39 >= 0, s39 <= 2, z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ -49 + s40 + s41 + 89*z + 31*z^2 + 6*z^3 }-> s42 :|: s40 >= 0, s40 <= z - 1 + 1, s41 >= 0, s41 <= 2, s42 >= 0, s42 <= 2, z - 1 >= 0 encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ -50 + s60 + 89*z + 31*z^2 + 6*z^3 }-> s61 :|: s59 >= 0, s59 <= z - 1 + 1, s60 >= 0, s60 <= s59, s61 >= 0, s61 <= 2, z - 1 >= 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ -52 + 89*z + 31*z^2 + 6*z^3 }-> s63 :|: s62 >= 0, s62 <= z - 1 + 1, s63 >= 0, s63 <= 1 + s62, z - 1 >= 0 encode_p(z) -{ 156 + 5*s65 + s65^2 + 169*x_184 + 49*x_184^2 + 6*x_184^3 + 169*x_220 + 49*x_220^2 + 6*x_220^3 }-> s67 :|: s64 >= 0, s64 <= x_184 + 1, s65 >= 0, s65 <= x_220 + 1, s66 >= 0, s66 <= 0, s67 >= 0, s67 <= s66, z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ -50 + s68 + 89*z + 31*z^2 + 6*z^3 }-> s70 :|: s68 >= 0, s68 <= z - 1 + 1, s69 >= 0, s69 <= 2, s70 >= 0, s70 <= s69, z - 1 >= 0 encode_p(z) -{ -51 + 89*z + 31*z^2 + 6*z^3 }-> s73 :|: s71 >= 0, s71 <= z - 1 + 1, s72 >= 0, s72 <= s71, s73 >= 0, s73 <= s72, z - 1 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 73 + 169*z + 49*z^2 + 6*z^3 }-> 1 + s43 :|: s43 >= 0, s43 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_cond} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [166 + 613*z + 192*z^2 + 30*z^3], size: O(n^1) [1 + z] encode_odd: runtime: O(n^3) [174 + 615*z + 192*z^2 + 30*z^3], size: O(1) [2] encode_s: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [2 + z] encode_cond: runtime: ?, size: O(1) [0] ---------------------------------------- (85) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_cond after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 161 + 169*z + 49*z^2 + 6*z^3 + 176*z' + 50*z'^2 + 6*z'^3 ---------------------------------------- (86) Obligation: Complexity RNTS consisting of the following rules: cond(z, z') -{ 11 }-> s10 :|: s10 >= 0, s10 <= 0, z = 2, z' = 0 cond(z, z') -{ 12 }-> s11 :|: s11 >= 0, s11 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 11 }-> s12 :|: s12 >= 0, s12 <= 0, z = 2, z' = 1 + 0 cond(z, z') -{ 8 + 4*z' + z'^2 }-> s13 :|: s13 >= 0, s13 <= 0, s >= 0, s <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 + z' }-> s14 :|: s14 >= 0, s14 <= 0, s' >= 0, s' <= 2, z = 2, z' - 2 >= 0 cond(z, z') -{ 11 }-> s15 :|: s15 >= 0, s15 <= 0, z = 2, z' = 0 cond(z, z') -{ 7 + 3*z' + z'^2 }-> s16 :|: s16 >= 0, s16 <= 0, z = 2, z' - 1 >= 0 cond(z, z') -{ 10 }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z' >= 0 cond(z, z') -{ 12 }-> s9 :|: s9 >= 0, s9 <= 0, z = 2, z' = 0 cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encArg(z) -{ 4 }-> s1 :|: s1 >= 0, s1 <= 2, z = 1 + 2 encArg(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2, z = 1 + 0 encArg(z) -{ 155 + 5*s20 + s20^2 + 169*x_1 + 49*x_1^2 + 6*x_1^3 + 169*x_2 + 49*x_2^2 + 6*x_2^3 }-> s21 :|: s19 >= 0, s19 <= x_1 + 1, s20 >= 0, s20 <= x_2 + 1, s21 >= 0, s21 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -114 + s22 + 45*z + 13*z^2 + 6*z^3 }-> s23 :|: s22 >= 0, s22 <= z - 2 + 1, s23 >= 0, s23 <= 2, z - 2 >= 0 encArg(z) -{ 157 + 5*s25 + s25^2 + s26 + 169*x_136 + 49*x_136^2 + 6*x_136^3 + 169*x_28 + 49*x_28^2 + 6*x_28^3 }-> s27 :|: s24 >= 0, s24 <= x_136 + 1, s25 >= 0, s25 <= x_28 + 1, s26 >= 0, s26 <= 0, s27 >= 0, s27 <= 2, z = 1 + (1 + x_136 + x_28), x_136 >= 0, x_28 >= 0 encArg(z) -{ 3 }-> s3 :|: s3 >= 0, s3 <= 2, z = 1 + 1 encArg(z) -{ -113 + s28 + s29 + 45*z + 13*z^2 + 6*z^3 }-> s30 :|: s28 >= 0, s28 <= z - 2 + 1, s29 >= 0, s29 <= 2, s30 >= 0, s30 <= 2, z - 2 >= 0 encArg(z) -{ 2 }-> s4 :|: s4 >= 0, s4 <= 2, z - 1 >= 0 encArg(z) -{ -114 + s45 + 45*z + 13*z^2 + 6*z^3 }-> s46 :|: s44 >= 0, s44 <= z - 2 + 1, s45 >= 0, s45 <= s44, s46 >= 0, s46 <= 2, z - 2 >= 0 encArg(z) -{ -116 + 45*z + 13*z^2 + 6*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= 1 + s47, z - 2 >= 0 encArg(z) -{ 156 + 5*s50 + s50^2 + 169*x_140 + 49*x_140^2 + 6*x_140^3 + 169*x_29 + 49*x_29^2 + 6*x_29^3 }-> s52 :|: s49 >= 0, s49 <= x_140 + 1, s50 >= 0, s50 <= x_29 + 1, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= s51, z = 1 + (1 + x_140 + x_29), x_140 >= 0, x_29 >= 0 encArg(z) -{ -114 + s53 + 45*z + 13*z^2 + 6*z^3 }-> s55 :|: s53 >= 0, s53 <= z - 2 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, z - 2 >= 0 encArg(z) -{ -115 + 45*z + 13*z^2 + 6*z^3 }-> s58 :|: s56 >= 0, s56 <= z - 2 + 1, s57 >= 0, s57 <= s56, s58 >= 0, s58 <= s57, z - 2 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -53 + 89*z + 31*z^2 + 6*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond(z, z') -{ 155 + 5*s32 + s32^2 + 169*z + 49*z^2 + 6*z^3 + 169*z' + 49*z'^2 + 6*z'^3 }-> s33 :|: s31 >= 0, s31 <= z + 1, s32 >= 0, s32 <= z' + 1, s33 >= 0, s33 <= 0, z >= 0, z' >= 0 encode_cond(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_odd(z) -{ -50 + s34 + 89*z + 31*z^2 + 6*z^3 }-> s35 :|: s34 >= 0, s34 <= z - 1 + 1, s35 >= 0, s35 <= 2, z - 1 >= 0 encode_odd(z) -{ 157 + 5*s37 + s37^2 + s38 + 169*x_180 + 49*x_180^2 + 6*x_180^3 + 169*x_219 + 49*x_219^2 + 6*x_219^3 }-> s39 :|: s36 >= 0, s36 <= x_180 + 1, s37 >= 0, s37 <= x_219 + 1, s38 >= 0, s38 <= 0, s39 >= 0, s39 <= 2, z = 1 + x_180 + x_219, x_219 >= 0, x_180 >= 0 encode_odd(z) -{ -49 + s40 + s41 + 89*z + 31*z^2 + 6*z^3 }-> s42 :|: s40 >= 0, s40 <= z - 1 + 1, s41 >= 0, s41 <= 2, s42 >= 0, s42 <= 2, z - 1 >= 0 encode_odd(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 2, z = 2 encode_odd(z) -{ 2 }-> s6 :|: s6 >= 0, s6 <= 2, z = 0 encode_odd(z) -{ -50 + s60 + 89*z + 31*z^2 + 6*z^3 }-> s61 :|: s59 >= 0, s59 <= z - 1 + 1, s60 >= 0, s60 <= s59, s61 >= 0, s61 <= 2, z - 1 >= 0 encode_odd(z) -{ 3 }-> s7 :|: s7 >= 0, s7 <= 2, z = 1 encode_odd(z) -{ 2 }-> s8 :|: s8 >= 0, s8 <= 2, z >= 0 encode_odd(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ -52 + 89*z + 31*z^2 + 6*z^3 }-> s63 :|: s62 >= 0, s62 <= z - 1 + 1, s63 >= 0, s63 <= 1 + s62, z - 1 >= 0 encode_p(z) -{ 156 + 5*s65 + s65^2 + 169*x_184 + 49*x_184^2 + 6*x_184^3 + 169*x_220 + 49*x_220^2 + 6*x_220^3 }-> s67 :|: s64 >= 0, s64 <= x_184 + 1, s65 >= 0, s65 <= x_220 + 1, s66 >= 0, s66 <= 0, s67 >= 0, s67 <= s66, z = 1 + x_184 + x_220, x_184 >= 0, x_220 >= 0 encode_p(z) -{ -50 + s68 + 89*z + 31*z^2 + 6*z^3 }-> s70 :|: s68 >= 0, s68 <= z - 1 + 1, s69 >= 0, s69 <= 2, s70 >= 0, s70 <= s69, z - 1 >= 0 encode_p(z) -{ -51 + 89*z + 31*z^2 + 6*z^3 }-> s73 :|: s71 >= 0, s71 <= z - 1 + 1, s72 >= 0, s72 <= s71, s73 >= 0, s73 <= s72, z - 1 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 73 + 169*z + 49*z^2 + 6*z^3 }-> 1 + s43 :|: s43 >= 0, s43 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: odd(z) -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= 2, z - 2 >= 0 odd(z) -{ 1 }-> 2 :|: z = 1 + 0 odd(z) -{ 1 }-> 1 :|: z = 0 odd(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] odd: runtime: O(n^1) [2 + z], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] cond: runtime: O(n^2) [9 + 5*z' + z'^2], size: O(1) [0] encArg: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [166 + 613*z + 192*z^2 + 30*z^3], size: O(n^1) [1 + z] encode_odd: runtime: O(n^3) [174 + 615*z + 192*z^2 + 30*z^3], size: O(1) [2] encode_s: runtime: O(n^3) [73 + 169*z + 49*z^2 + 6*z^3], size: O(n^1) [2 + z] encode_cond: runtime: O(n^3) [161 + 169*z + 49*z^2 + 6*z^3 + 176*z' + 50*z'^2 + 6*z'^3], size: O(1) [0] ---------------------------------------- (87) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (88) BOUNDS(1, n^3) ---------------------------------------- (89) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (90) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: cond(true, x) -> cond(odd(x), p(x)) odd(0) -> false odd(s(0)) -> true odd(s(s(x))) -> odd(x) p(0) -> 0 p(s(x)) -> x The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) encArg(cons_odd(x_1)) -> odd(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) encode_true -> true encode_odd(x_1) -> odd(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_false -> false encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (91) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence odd(s(s(x))) ->^+ odd(x) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [x / s(s(x))]. The result substitution is [ ]. ---------------------------------------- (92) Complex Obligation (BEST) ---------------------------------------- (93) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: cond(true, x) -> cond(odd(x), p(x)) odd(0) -> false odd(s(0)) -> true odd(s(s(x))) -> odd(x) p(0) -> 0 p(s(x)) -> x The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) encArg(cons_odd(x_1)) -> odd(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) encode_true -> true encode_odd(x_1) -> odd(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_false -> false encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (94) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (95) BOUNDS(n^1, INF) ---------------------------------------- (96) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: cond(true, x) -> cond(odd(x), p(x)) odd(0) -> false odd(s(0)) -> true odd(s(s(x))) -> odd(x) p(0) -> 0 p(s(x)) -> x The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_cond(x_1, x_2)) -> cond(encArg(x_1), encArg(x_2)) encArg(cons_odd(x_1)) -> odd(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond(x_1, x_2) -> cond(encArg(x_1), encArg(x_2)) encode_true -> true encode_odd(x_1) -> odd(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_false -> false encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST