/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 213 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), 41.4 s] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 11 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 1127 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), 79 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 25 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 77 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 150 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 156 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 96 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 1 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 209 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 53 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 199 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 83 ms] (56) CpxRNTS (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 5849 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 3977 ms] (62) CpxRNTS (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 707 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 903 ms] (68) CpxRNTS (69) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 648 ms] (72) CpxRNTS (73) IntTrsBoundProof [UPPER BOUND(ID), 208 ms] (74) CpxRNTS (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 178 ms] (78) CpxRNTS (79) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (80) CpxRNTS (81) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 86 ms] (84) CpxRNTS (85) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (86) CpxRNTS (87) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (88) CpxRNTS (89) IntTrsBoundProof [UPPER BOUND(ID), 187 ms] (90) CpxRNTS (91) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (92) CpxRNTS (93) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (94) CpxRNTS (95) IntTrsBoundProof [UPPER BOUND(ID), 117 ms] (96) CpxRNTS (97) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (98) CpxRNTS (99) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (100) CpxRNTS (101) IntTrsBoundProof [UPPER BOUND(ID), 197 ms] (102) CpxRNTS (103) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (104) CpxRNTS (105) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (106) CpxRNTS (107) IntTrsBoundProof [UPPER BOUND(ID), 207 ms] (108) CpxRNTS (109) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (110) CpxRNTS (111) FinalProof [FINISHED, 0 ms] (112) BOUNDS(1, n^2) (113) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (114) TRS for Loop Detection (115) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (116) BEST (117) proven lower bound (118) LowerBoundPropagationProof [FINISHED, 0 ms] (119) BOUNDS(n^1, INF) (120) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: cond1(true, x, y) -> cond2(gr(x, 0), x, y) cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) cond2(false, x, y) -> cond3(gr(y, 0), x, y) cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) gr(0, x) -> false gr(s(x), 0) -> true gr(s(x), s(y)) -> gr(x, y) or(false, false) -> false or(true, x) -> true or(x, true) -> true 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_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond3(x_1, x_2, x_3)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_cond3(x_1, x_2, x_3) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) 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^2). The TRS R consists of the following rules: cond1(true, x, y) -> cond2(gr(x, 0), x, y) cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) cond2(false, x, y) -> cond3(gr(y, 0), x, y) cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) gr(0, x) -> false gr(s(x), 0) -> true gr(s(x), s(y)) -> gr(x, y) or(false, false) -> false or(true, x) -> true or(x, true) -> true 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_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond3(x_1, x_2, x_3)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_cond3(x_1, x_2, x_3) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) 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^2). The TRS R consists of the following rules: cond1(true, x, y) -> cond2(gr(x, 0), x, y) cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) cond2(false, x, y) -> cond3(gr(y, 0), x, y) cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) gr(0, x) -> false gr(s(x), 0) -> true gr(s(x), s(y)) -> gr(x, y) or(false, false) -> false or(true, x) -> true or(x, true) -> true 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_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond3(x_1, x_2, x_3)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_cond3(x_1, x_2, x_3) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) 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^2). The TRS R consists of the following rules: cond1(true, x, y) -> cond2(gr(x, 0), x, y) [1] cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) [1] cond2(false, x, y) -> cond3(gr(y, 0), x, y) [1] cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) [1] cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) [1] gr(0, x) -> false [1] gr(s(x), 0) -> true [1] gr(s(x), s(y)) -> gr(x, y) [1] or(false, false) -> false [1] or(true, x) -> true [1] or(x, true) -> true [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_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_cond3(x_1, x_2, x_3)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) [0] encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0] encArg(cons_p(x_1)) -> p(encArg(x_1)) [0] encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_true -> true [0] encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0] encode_p(x_1) -> p(encArg(x_1)) [0] encode_false -> false [0] encode_cond3(x_1, x_2, x_3) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) [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: cond1(true, x, y) -> cond2(gr(x, 0), x, y) [1] cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) [1] cond2(false, x, y) -> cond3(gr(y, 0), x, y) [1] cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) [1] cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) [1] gr(0, x) -> false [1] gr(s(x), 0) -> true [1] gr(s(x), s(y)) -> gr(x, y) [1] or(false, false) -> false [1] or(true, x) -> true [1] or(x, true) -> true [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_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_cond3(x_1, x_2, x_3)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) [0] encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0] encArg(cons_p(x_1)) -> p(encArg(x_1)) [0] encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_true -> true [0] encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0] encode_p(x_1) -> p(encArg(x_1)) [0] encode_false -> false [0] encode_cond3(x_1, x_2, x_3) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_s(x_1) -> s(encArg(x_1)) [0] The TRS has the following type information: cond1 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p true :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p cond2 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p gr :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p 0 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p or :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p p :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p false :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p cond3 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p s :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p encArg :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p cons_cond1 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p cons_cond2 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p cons_cond3 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p cons_gr :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p cons_or :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p cons_p :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p encode_cond1 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p encode_true :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p encode_cond2 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p encode_gr :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p encode_0 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p encode_or :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p encode_p :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p encode_false :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p encode_cond3 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p encode_s :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or: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: or_2 gr_2 p_1 cond3_3 cond1_3 cond2_3 encArg_1 encode_cond1_3 encode_true encode_cond2_3 encode_gr_2 encode_0 encode_or_2 encode_p_1 encode_false encode_cond3_3 encode_s_1 Due to the following rules being added: encArg(v0) -> null_encArg [0] encode_cond1(v0, v1, v2) -> null_encode_cond1 [0] encode_true -> null_encode_true [0] encode_cond2(v0, v1, v2) -> null_encode_cond2 [0] encode_gr(v0, v1) -> null_encode_gr [0] encode_0 -> null_encode_0 [0] encode_or(v0, v1) -> null_encode_or [0] encode_p(v0) -> null_encode_p [0] encode_false -> null_encode_false [0] encode_cond3(v0, v1, v2) -> null_encode_cond3 [0] encode_s(v0) -> null_encode_s [0] or(v0, v1) -> null_or [0] gr(v0, v1) -> null_gr [0] p(v0) -> null_p [0] cond3(v0, v1, v2) -> null_cond3 [0] cond1(v0, v1, v2) -> null_cond1 [0] cond2(v0, v1, v2) -> null_cond2 [0] And the following fresh constants: null_encArg, null_encode_cond1, null_encode_true, null_encode_cond2, null_encode_gr, null_encode_0, null_encode_or, null_encode_p, null_encode_false, null_encode_cond3, null_encode_s, null_or, null_gr, null_p, null_cond3, null_cond1, null_cond2 ---------------------------------------- (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: cond1(true, x, y) -> cond2(gr(x, 0), x, y) [1] cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) [1] cond2(false, x, y) -> cond3(gr(y, 0), x, y) [1] cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) [1] cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) [1] gr(0, x) -> false [1] gr(s(x), 0) -> true [1] gr(s(x), s(y)) -> gr(x, y) [1] or(false, false) -> false [1] or(true, x) -> true [1] or(x, true) -> true [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_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_cond3(x_1, x_2, x_3)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) [0] encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0] encArg(cons_p(x_1)) -> p(encArg(x_1)) [0] encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_true -> true [0] encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0] encode_p(x_1) -> p(encArg(x_1)) [0] encode_false -> false [0] encode_cond3(x_1, x_2, x_3) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_s(x_1) -> s(encArg(x_1)) [0] encArg(v0) -> null_encArg [0] encode_cond1(v0, v1, v2) -> null_encode_cond1 [0] encode_true -> null_encode_true [0] encode_cond2(v0, v1, v2) -> null_encode_cond2 [0] encode_gr(v0, v1) -> null_encode_gr [0] encode_0 -> null_encode_0 [0] encode_or(v0, v1) -> null_encode_or [0] encode_p(v0) -> null_encode_p [0] encode_false -> null_encode_false [0] encode_cond3(v0, v1, v2) -> null_encode_cond3 [0] encode_s(v0) -> null_encode_s [0] or(v0, v1) -> null_or [0] gr(v0, v1) -> null_gr [0] p(v0) -> null_p [0] cond3(v0, v1, v2) -> null_cond3 [0] cond1(v0, v1, v2) -> null_cond1 [0] cond2(v0, v1, v2) -> null_cond2 [0] The TRS has the following type information: cond1 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 true :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cond2 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 gr :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 0 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 or :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 p :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 false :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cond3 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 s :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encArg :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cons_cond1 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cons_cond2 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cons_cond3 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cons_gr :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cons_or :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cons_p :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_cond1 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_true :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_cond2 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_gr :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_0 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_or :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_p :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_false :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_cond3 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_s :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encArg :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_cond1 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_true :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_cond2 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_gr :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_0 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_or :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_p :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_false :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_cond3 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_s :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_or :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_gr :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_p :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_cond3 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_cond1 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_cond2 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 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: cond1(true, 0, y) -> cond2(false, 0, y) [2] cond1(true, s(x'), y) -> cond2(true, s(x'), y) [2] cond1(true, x, y) -> cond2(null_gr, x, y) [1] cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) [1] cond2(false, x, 0) -> cond3(false, x, 0) [2] cond2(false, x, s(x7)) -> cond3(true, x, s(x7)) [2] cond2(false, x, y) -> cond3(null_gr, x, y) [1] cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) [1] cond3(false, 0, 0) -> cond1(or(false, false), 0, 0) [3] cond3(false, 0, s(x16)) -> cond1(or(false, true), 0, s(x16)) [3] cond3(false, 0, y) -> cond1(or(false, null_gr), 0, y) [2] cond3(false, s(x15), 0) -> cond1(or(true, false), s(x15), 0) [3] cond3(false, s(x15), s(x17)) -> cond1(or(true, true), s(x15), s(x17)) [3] cond3(false, s(x15), y) -> cond1(or(true, null_gr), s(x15), y) [2] cond3(false, x, 0) -> cond1(or(null_gr, false), x, 0) [2] cond3(false, x, s(x18)) -> cond1(or(null_gr, true), x, s(x18)) [2] cond3(false, x, y) -> cond1(or(null_gr, null_gr), x, y) [1] gr(0, x) -> false [1] gr(s(x), 0) -> true [1] gr(s(x), s(y)) -> gr(x, y) [1] or(false, false) -> false [1] or(true, x) -> true [1] or(x, true) -> true [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_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_cond3(x_1, x_2, x_3)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) [0] encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [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_12960))) -> p(s(encArg(x_12960))) [0] encArg(cons_p(cons_cond1(x_12961, x_22114, x_31268))) -> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) [0] encArg(cons_p(cons_cond2(x_12962, x_22115, x_31269))) -> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) [0] encArg(cons_p(cons_cond3(x_12963, x_22116, x_31270))) -> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) [0] encArg(cons_p(cons_gr(x_12964, x_22117))) -> p(gr(encArg(x_12964), encArg(x_22117))) [0] encArg(cons_p(cons_or(x_12965, x_22118))) -> p(or(encArg(x_12965), encArg(x_22118))) [0] encArg(cons_p(cons_p(x_12966))) -> p(p(encArg(x_12966))) [0] encArg(cons_p(x_1)) -> p(null_encArg) [0] encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_true -> true [0] encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0] encode_p(true) -> p(true) [0] encode_p(0) -> p(0) [0] encode_p(false) -> p(false) [0] encode_p(s(x_14997)) -> p(s(encArg(x_14997))) [0] encode_p(cons_cond1(x_14998, x_23569, x_32141)) -> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) [0] encode_p(cons_cond2(x_14999, x_23570, x_32142)) -> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) [0] encode_p(cons_cond3(x_15000, x_23571, x_32143)) -> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) [0] encode_p(cons_gr(x_15001, x_23572)) -> p(gr(encArg(x_15001), encArg(x_23572))) [0] encode_p(cons_or(x_15002, x_23573)) -> p(or(encArg(x_15002), encArg(x_23573))) [0] encode_p(cons_p(x_15003)) -> p(p(encArg(x_15003))) [0] encode_p(x_1) -> p(null_encArg) [0] encode_false -> false [0] encode_cond3(x_1, x_2, x_3) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_s(x_1) -> s(encArg(x_1)) [0] encArg(v0) -> null_encArg [0] encode_cond1(v0, v1, v2) -> null_encode_cond1 [0] encode_true -> null_encode_true [0] encode_cond2(v0, v1, v2) -> null_encode_cond2 [0] encode_gr(v0, v1) -> null_encode_gr [0] encode_0 -> null_encode_0 [0] encode_or(v0, v1) -> null_encode_or [0] encode_p(v0) -> null_encode_p [0] encode_false -> null_encode_false [0] encode_cond3(v0, v1, v2) -> null_encode_cond3 [0] encode_s(v0) -> null_encode_s [0] or(v0, v1) -> null_or [0] gr(v0, v1) -> null_gr [0] p(v0) -> null_p [0] cond3(v0, v1, v2) -> null_cond3 [0] cond1(v0, v1, v2) -> null_cond1 [0] cond2(v0, v1, v2) -> null_cond2 [0] The TRS has the following type information: cond1 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 true :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cond2 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 gr :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 0 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 or :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 p :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 false :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cond3 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 s :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encArg :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cons_cond1 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cons_cond2 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cons_cond3 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cons_gr :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cons_or :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 cons_p :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_cond1 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_true :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_cond2 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_gr :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_0 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_or :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_p :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_false :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_cond3 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 encode_s :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 -> true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encArg :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_cond1 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_true :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_cond2 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_gr :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_0 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_or :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_p :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_false :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_cond3 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_encode_s :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_or :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_gr :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_p :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_cond3 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_cond1 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 null_cond2 :: true:0:false:s:cons_cond1:cons_cond2:cons_cond3:cons_gr:cons_or:cons_p:null_encArg:null_encode_cond1:null_encode_true:null_encode_cond2:null_encode_gr:null_encode_0:null_encode_or:null_encode_p:null_encode_false:null_encode_cond3:null_encode_s:null_or:null_gr:null_p:null_cond3:null_cond1:null_cond2 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_cond1 => 0 null_encode_true => 0 null_encode_cond2 => 0 null_encode_gr => 0 null_encode_0 => 0 null_encode_or => 0 null_encode_p => 0 null_encode_false => 0 null_encode_cond3 => 0 null_encode_s => 0 null_or => 0 null_gr => 0 null_p => 0 null_cond3 => 0 null_cond1 => 0 null_cond2 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + x', y) :|: z = 2, z' = 1 + x', z'' = y, x' >= 0, y >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, y) :|: z = 2, z'' = y, y >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, x, y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, x, 1 + x7) :|: z' = x, z = 1, x7 >= 0, x >= 0, z'' = 1 + x7 cond2(z, z', z'') -{ 2 }-> cond3(1, x, 0) :|: z'' = 0, z' = x, z = 1, x >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 cond2(z, z', z'') -{ 1 }-> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 cond3(z, z', z'') -{ 1 }-> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 cond3(z, z', z'') -{ 3 }-> cond1(or(2, 2), 1 + x15, 1 + x17) :|: x17 >= 0, z'' = 1 + x17, z = 1, z' = 1 + x15, x15 >= 0 cond3(z, z', z'') -{ 3 }-> cond1(or(2, 1), 1 + x15, 0) :|: z'' = 0, z = 1, z' = 1 + x15, x15 >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(2, 0), 1 + x15, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + x15, x15 >= 0 cond3(z, z', z'') -{ 3 }-> cond1(or(1, 2), 0, 1 + x16) :|: z'' = 1 + x16, z = 1, x16 >= 0, z' = 0 cond3(z, z', z'') -{ 3 }-> cond1(or(1, 1), 0, 0) :|: z'' = 0, z = 1, z' = 0 cond3(z, z', z'') -{ 2 }-> cond1(or(1, 0), 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0 cond3(z, z', z'') -{ 2 }-> cond1(or(0, 2), x, 1 + x18) :|: z'' = 1 + x18, z' = x, z = 1, x >= 0, x18 >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(0, 1), x, 0) :|: z'' = 0, z' = x, z = 1, x >= 0 cond3(z, z', z'') -{ 1 }-> cond1(or(0, 0), x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 encArg(z) -{ 0 }-> p(p(encArg(x_12966))) :|: z = 1 + (1 + x_12966), x_12966 >= 0 encArg(z) -{ 0 }-> p(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 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_12960)) :|: x_12960 >= 0, z = 1 + (1 + x_12960) encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_gr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_or(z, z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_or(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_p(z) -{ 0 }-> p(p(encArg(x_15003))) :|: z = 1 + x_15003, x_15003 >= 0 encode_p(z) -{ 0 }-> p(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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_14997)) :|: x_14997 >= 0, z = 1 + x_14997 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 :|: gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x gr(z, z') -{ 1 }-> 2 :|: x >= 0, z = 1 + x, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' = x, x >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 or(z, z') -{ 1 }-> 2 :|: z = 2, z' = x, x >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, x >= 0, z = x or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 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: or(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 or(z, z') -{ 1 }-> 2 :|: z = 2, z' = x, x >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, x >= 0, z = x or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + x', y) :|: z = 2, z' = 1 + x', z'' = y, x' >= 0, y >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, y) :|: z = 2, z'' = y, y >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, x, y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, x, 1 + x7) :|: z' = x, z = 1, x7 >= 0, x >= 0, z'' = 1 + x7 cond2(z, z', z'') -{ 2 }-> cond3(1, x, 0) :|: z'' = 0, z' = x, z = 1, x >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(x, 0), gr(y, 0)), x', y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0, x' >= 0, x = 1 + x' cond2(z, z', z'') -{ 2 }-> cond1(or(gr(x, 0), gr(y, 0)), 0, y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0, x = 0 cond2(z, z', z'') -{ 1 }-> cond1(or(gr(x, 0), gr(y, 0)), 0, y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0, v0 >= 0, x = v0 cond2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(x, 0), gr(y, 0)), x, x') :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0, x' >= 0, y = 1 + x' cond3(z, z', z'') -{ 2 }-> cond1(or(gr(x, 0), gr(y, 0)), x, 0) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0, y = 0 cond3(z, z', z'') -{ 1 }-> cond1(or(gr(x, 0), gr(y, 0)), x, 0) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0, v0 >= 0, y = v0 cond3(z, z', z'') -{ 3 }-> cond1(2, x, 1 + x18) :|: z'' = 1 + x18, z' = x, z = 1, x >= 0, x18 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + x16) :|: z'' = 1 + x16, z = 1, x16 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + x15, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + x15, x15 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + x15, 0) :|: z'' = 0, z = 1, z' = 1 + x15, x15 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + x15, 1 + x17) :|: x17 >= 0, z'' = 1 + x17, z = 1, z' = 1 + x15, x15 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, x, 0) :|: z'' = 0, z' = x, z = 1, x >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, x, 1 + x18) :|: z'' = 1 + x18, z' = x, z = 1, x >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + x16) :|: z'' = 1 + x16, z = 1, x16 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + x15, y) :|: z'' = y, z = 1, y >= 0, z' = 1 + x15, x15 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + x15, 0) :|: z'' = 0, z = 1, z' = 1 + x15, x15 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + x15, 1 + x17) :|: x17 >= 0, z'' = 1 + x17, z = 1, z' = 1 + x15, x15 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 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(x_12966))) :|: z = 1 + (1 + x_12966), x_12966 >= 0 encArg(z) -{ 0 }-> p(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(x_12960)) :|: x_12960 >= 0, z = 1 + (1 + x_12960) encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_gr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_or(z, z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_or(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 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_15003))) :|: z = 1 + x_15003, x_15003 >= 0 encode_p(z) -{ 0 }-> p(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(x_14997)) :|: x_14997 >= 0, z = 1 + x_14997 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 :|: gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x gr(z, z') -{ 1 }-> 2 :|: x >= 0, z = 1 + x, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' = x, x >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 or(z, z') -{ 1 }-> 2 :|: z = 2, z' = x, x >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, x >= 0, z = x or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z' - 1, z'') :|: z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', z'' - 1) :|: z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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 } { gr } { encode_true } { p } { or } { cond1, cond2, cond3 } { encArg } { encode_p } { encode_or } { encode_s } { encode_cond3 } { encode_gr } { encode_cond2 } { encode_cond1 } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z' - 1, z'') :|: z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', z'' - 1) :|: z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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}, {gr}, {encode_true}, {p}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} ---------------------------------------- (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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z' - 1, z'') :|: z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', z'' - 1) :|: z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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}, {gr}, {encode_true}, {p}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} ---------------------------------------- (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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z' - 1, z'') :|: z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', z'' - 1) :|: z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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}, {gr}, {encode_true}, {p}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} 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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z' - 1, z'') :|: z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', z'' - 1) :|: z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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}, {gr}, {encode_true}, {p}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} 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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z' - 1, z'') :|: z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', z'' - 1) :|: z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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}, {gr}, {encode_true}, {p}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} 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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z' - 1, z'') :|: z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', z'' - 1) :|: z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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}, {gr}, {encode_true}, {p}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} 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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z' - 1, z'') :|: z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', z'' - 1) :|: z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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: {gr}, {encode_true}, {p}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} 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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z' - 1, z'') :|: z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', z'' - 1) :|: z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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: {gr}, {encode_true}, {p}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} 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 KoAT for: gr 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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z' - 1, z'') :|: z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', z'' - 1) :|: z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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: {gr}, {encode_true}, {p}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: ?, size: O(1) [2] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: gr after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), 0, z'') :|: z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z' - 1, z'') :|: z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 1 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', 0) :|: z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 2 }-> cond1(or(gr(z', 0), gr(z'', 0)), z', z'' - 1) :|: z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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}, {p}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], 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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s', s''), z' - 1, z'') :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s1, s2), 0, z'') :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 5 }-> cond1(or(s3, s4), 0, z'') :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s5, s6), z', z'' - 1) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s7, s8), z', 0) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 5 }-> cond1(or(s9, s10), z', 0) :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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}, {p}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] ---------------------------------------- (41) 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 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s', s''), z' - 1, z'') :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s1, s2), 0, z'') :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 5 }-> cond1(or(s3, s4), 0, z'') :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s5, s6), z', z'' - 1) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s7, s8), z', 0) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 5 }-> cond1(or(s9, s10), z', 0) :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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}, {p}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: ?, size: O(1) [2] ---------------------------------------- (43) 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 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s', s''), z' - 1, z'') :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s1, s2), 0, z'') :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 5 }-> cond1(or(s3, s4), 0, z'') :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s5, s6), z', z'' - 1) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s7, s8), z', 0) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 5 }-> cond1(or(s9, s10), z', 0) :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], 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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s', s''), z' - 1, z'') :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s1, s2), 0, z'') :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 5 }-> cond1(or(s3, s4), 0, z'') :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s5, s6), z', z'' - 1) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s7, s8), z', 0) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 5 }-> cond1(or(s9, s10), z', 0) :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], 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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s', s''), z' - 1, z'') :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s1, s2), 0, z'') :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 5 }-> cond1(or(s3, s4), 0, z'') :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s5, s6), z', z'' - 1) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s7, s8), z', 0) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 5 }-> cond1(or(s9, s10), z', 0) :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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}, {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], 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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s', s''), z' - 1, z'') :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s1, s2), 0, z'') :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 5 }-> cond1(or(s3, s4), 0, z'') :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s5, s6), z', z'' - 1) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s7, s8), z', 0) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 5 }-> cond1(or(s9, s10), z', 0) :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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: {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], 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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s', s''), z' - 1, z'') :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s1, s2), 0, z'') :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 5 }-> cond1(or(s3, s4), 0, z'') :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s5, s6), z', z'' - 1) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s7, s8), z', 0) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 5 }-> cond1(or(s9, s10), z', 0) :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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: {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], 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: or after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s', s''), z' - 1, z'') :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s1, s2), 0, z'') :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 5 }-> cond1(or(s3, s4), 0, z'') :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s5, s6), z', z'' - 1) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s7, s8), z', 0) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 5 }-> cond1(or(s9, s10), z', 0) :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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: {or}, {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: ?, size: O(1) [2] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: or after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s', s''), z' - 1, z'') :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 6 }-> cond1(or(s1, s2), 0, z'') :|: s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 5 }-> cond1(or(s3, s4), 0, z'') :|: s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s5, s6), z', z'' - 1) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 6 }-> cond1(or(s7, s8), z', 0) :|: s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 5 }-> cond1(or(s9, s10), z', 0) :|: s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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: {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (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: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 7 }-> cond1(s11, z' - 1, z'') :|: s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 7 }-> cond1(s12, 0, z'') :|: s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 6 }-> cond1(s13, 0, z'') :|: s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 7 }-> cond1(s14, z', z'' - 1) :|: s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 7 }-> cond1(s15, z', 0) :|: s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 6 }-> cond1(s16, z', 0) :|: s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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: {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: cond1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed SIZE bound using CoFloCo for: cond2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed SIZE bound using CoFloCo for: cond3 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 7 }-> cond1(s11, z' - 1, z'') :|: s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 7 }-> cond1(s12, 0, z'') :|: s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 6 }-> cond1(s13, 0, z'') :|: s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 7 }-> cond1(s14, z', z'' - 1) :|: s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 7 }-> cond1(s15, z', 0) :|: s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 6 }-> cond1(s16, z', 0) :|: s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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: {cond1,cond2,cond3}, {encArg}, {encode_p}, {encode_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: ?, size: O(1) [0] cond2: runtime: ?, size: O(1) [0] cond3: runtime: ?, size: O(1) [0] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: cond1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 156 + 77*z + 133*z' + 189*z'' Computed RUNTIME bound using CoFloCo for: cond2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 318 + 133*z' + 189*z'' Computed RUNTIME bound using CoFloCo for: cond3 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 317 + 133*z' + 189*z'' ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 2 }-> cond2(2, 1 + (z' - 1), z'') :|: z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 2 }-> cond2(1, 0, z'') :|: z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 1 }-> cond2(0, z', z'') :|: z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(2, z', 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 2 }-> cond3(1, z', 0) :|: z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 1 }-> cond3(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 7 }-> cond1(s11, z' - 1, z'') :|: s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 7 }-> cond1(s12, 0, z'') :|: s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 6 }-> cond1(s13, 0, z'') :|: s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 7 }-> cond1(s14, z', z'' - 1) :|: s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 7 }-> cond1(s15, z', 0) :|: s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 6 }-> cond1(s16, z', 0) :|: s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 3 }-> cond1(2, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 4 }-> cond1(2, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 3 }-> cond1(2, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(2, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 0) :|: z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 1 }-> cond1(0, z', z'') :|: z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 0) :|: z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 2 }-> cond1(0, 1 + (z' - 1), z'') :|: z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 3 }-> cond1(0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] ---------------------------------------- (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: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] ---------------------------------------- (65) 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 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (67) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encArg after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 513*z + 189*z^2 ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_12965), encArg(x_22118))) :|: x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 0 }-> p(gr(encArg(x_12964), encArg(x_22117))) :|: x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ 0 }-> p(cond3(encArg(x_12963), encArg(x_22116), encArg(x_31270))) :|: x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) encArg(z) -{ 0 }-> p(cond2(encArg(x_12962), encArg(x_22115), encArg(x_31269))) :|: x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 0 }-> p(cond1(encArg(x_12961), encArg(x_22114), encArg(x_31268))) :|: x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> gr(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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_cond1(z, z', z'') -{ 0 }-> cond1(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 0 }-> cond2(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 0 }-> cond3(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 0 }-> gr(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> or(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 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(or(encArg(x_15002), encArg(x_23573))) :|: z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 0 encode_p(z) -{ 0 }-> p(gr(encArg(x_15001), encArg(x_23572))) :|: z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ 0 }-> p(cond3(encArg(x_15000), encArg(x_23571), encArg(x_32143))) :|: x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 0 }-> p(cond2(encArg(x_14999), encArg(x_23570), encArg(x_32142))) :|: x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 0 }-> p(cond1(encArg(x_14998), encArg(x_23569), encArg(x_32141))) :|: x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 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 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], 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: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] ---------------------------------------- (71) 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 ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (73) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_p after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1860 + 7983*z + 2835*z^2 ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] ---------------------------------------- (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: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_or after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_or}, {encode_s}, {encode_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: ?, size: O(1) [2] ---------------------------------------- (79) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_or after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] ---------------------------------------- (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: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] ---------------------------------------- (83) 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 ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: ?, size: O(n^1) [2 + z] ---------------------------------------- (85) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_s after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 513*z + 189*z^2 ---------------------------------------- (86) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [2 + z] ---------------------------------------- (87) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [2 + z] ---------------------------------------- (89) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_cond3 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_cond3}, {encode_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [2 + z] encode_cond3: runtime: ?, size: O(1) [0] ---------------------------------------- (91) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_cond3 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 642 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2 ---------------------------------------- (92) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [2 + z] encode_cond3: runtime: O(n^2) [642 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] ---------------------------------------- (93) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (94) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [2 + z] encode_cond3: runtime: O(n^2) [642 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] ---------------------------------------- (95) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_gr after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (96) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_gr}, {encode_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [2 + z] encode_cond3: runtime: O(n^2) [642 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] encode_gr: runtime: ?, size: O(1) [2] ---------------------------------------- (97) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_gr after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 5 + 513*z + 189*z^2 + 514*z' + 189*z'^2 ---------------------------------------- (98) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [2 + z] encode_cond3: runtime: O(n^2) [642 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] encode_gr: runtime: O(n^2) [5 + 513*z + 189*z^2 + 514*z' + 189*z'^2], size: O(1) [2] ---------------------------------------- (99) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (100) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [2 + z] encode_cond3: runtime: O(n^2) [642 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] encode_gr: runtime: O(n^2) [5 + 513*z + 189*z^2 + 514*z' + 189*z'^2], size: O(1) [2] ---------------------------------------- (101) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_cond2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (102) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_cond2}, {encode_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [2 + z] encode_cond3: runtime: O(n^2) [642 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] encode_gr: runtime: O(n^2) [5 + 513*z + 189*z^2 + 514*z' + 189*z'^2], size: O(1) [2] encode_cond2: runtime: ?, size: O(1) [0] ---------------------------------------- (103) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_cond2 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 643 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2 ---------------------------------------- (104) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [2 + z] encode_cond3: runtime: O(n^2) [642 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] encode_gr: runtime: O(n^2) [5 + 513*z + 189*z^2 + 514*z' + 189*z'^2], size: O(1) [2] encode_cond2: runtime: O(n^2) [643 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] ---------------------------------------- (105) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (106) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [2 + z] encode_cond3: runtime: O(n^2) [642 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] encode_gr: runtime: O(n^2) [5 + 513*z + 189*z^2 + 514*z' + 189*z'^2], size: O(1) [2] encode_cond2: runtime: O(n^2) [643 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] ---------------------------------------- (107) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_cond1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (108) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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_cond1} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [1] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [2 + z] encode_cond3: runtime: O(n^2) [642 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] encode_gr: runtime: O(n^2) [5 + 513*z + 189*z^2 + 514*z' + 189*z'^2], size: O(1) [2] encode_cond2: runtime: O(n^2) [643 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] encode_cond1: runtime: ?, size: O(1) [0] ---------------------------------------- (109) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_cond1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 558 + 590*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2 ---------------------------------------- (110) Obligation: Complexity RNTS consisting of the following rules: cond1(z, z', z'') -{ 320 + 189*z'' }-> s17 :|: s17 >= 0, s17 <= 0, z = 2, z'' >= 0, z' = 0 cond1(z, z', z'') -{ 320 + 133*z' + 189*z'' }-> s18 :|: s18 >= 0, s18 <= 0, z = 2, z' - 1 >= 0, z'' >= 0 cond1(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s19 :|: s19 >= 0, s19 <= 0, z = 2, z' >= 0, z'' >= 0 cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 319 + 133*z' }-> s20 :|: s20 >= 0, s20 <= 0, z'' = 0, z = 1, z' >= 0 cond2(z, z', z'') -{ 319 + 133*z' + 189*z'' }-> s21 :|: s21 >= 0, s21 <= 0, z = 1, z'' - 1 >= 0, z' >= 0 cond2(z, z', z'') -{ 318 + 133*z' + 189*z'' }-> s22 :|: s22 >= 0, s22 <= 0, z = 1, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 30 + 77*s11 + 133*z' + 189*z'' }-> s38 :|: s38 >= 0, s38 <= 0, s11 >= 0, s11 <= 2, s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z = 2, z' >= 0, z'' >= 0, z' - 1 >= 0 cond2(z, z', z'') -{ 163 + 77*s12 + 189*z'' }-> s39 :|: s39 >= 0, s39 <= 0, s12 >= 0, s12 <= 2, s1 >= 0, s1 <= 2, s2 >= 0, s2 <= 2, z = 2, z' >= 0, z'' >= 0, z' = 0 cond2(z, z', z'') -{ 162 + 77*s13 + 189*z'' }-> s40 :|: s40 >= 0, s40 <= 0, s13 >= 0, s13 <= 2, s3 >= 0, s3 <= 2, s4 >= 0, s4 <= 2, z = 2, z' >= 0, z'' >= 0 cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 159 }-> s23 :|: s23 >= 0, s23 <= 0, z'' = 0, z = 1, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1 cond3(z, z', z'') -{ 237 }-> s24 :|: s24 >= 0, s24 <= 0, z'' = 0, z = 1, z' = 0, 1 = 1 cond3(z, z', z'') -{ 159 + 189*z'' }-> s25 :|: s25 >= 0, s25 <= 0, z = 1, z'' - 1 >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 189*z'' }-> s26 :|: s26 >= 0, s26 <= 0, z = 1, z'' - 1 >= 0, z' = 0, 2 = 2, x >= 0, 1 = x cond3(z, z', z'') -{ 158 + 189*z'' }-> s27 :|: s27 >= 0, s27 <= 0, z = 1, z'' >= 0, z' = 0, v0 >= 0, v1 >= 0, 1 = v0, 0 = v1 cond3(z, z', z'') -{ 159 + 133*z' }-> s28 :|: s28 >= 0, s28 <= 0, z'' = 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1 cond3(z, z', z'') -{ 314 + 133*z' }-> s29 :|: s29 >= 0, s29 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 2 = 2, 1 = x, x >= 0 cond3(z, z', z'') -{ 159 + 133*z' + 189*z'' }-> s30 :|: s30 >= 0, s30 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1 cond3(z, z', z'') -{ 314 + 133*z' + 189*z'' }-> s31 :|: s31 >= 0, s31 <= 0, z'' - 1 >= 0, z = 1, z' - 1 >= 0, 2 = 2, 2 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s32 :|: s32 >= 0, s32 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 0 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s33 :|: s33 >= 0, s33 <= 0, z = 1, z'' >= 0, z' - 1 >= 0, 2 = 2, 0 = x, x >= 0 cond3(z, z', z'') -{ 158 + 133*z' }-> s34 :|: s34 >= 0, s34 <= 0, z'' = 0, z = 1, z' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 1 = v1 cond3(z, z', z'') -{ 158 + 133*z' + 189*z'' }-> s35 :|: s35 >= 0, s35 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, v0 >= 0, v1 >= 0, 0 = v0, 2 = v1 cond3(z, z', z'') -{ 313 + 133*z' + 189*z'' }-> s36 :|: s36 >= 0, s36 <= 0, z = 1, z' >= 0, z'' - 1 >= 0, 2 = 2, x' >= 0, 0 = x' cond3(z, z', z'') -{ 157 + 133*z' + 189*z'' }-> s37 :|: s37 >= 0, s37 <= 0, z = 1, z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1 cond3(z, z', z'') -{ -26 + 77*s14 + 133*z' + 189*z'' }-> s41 :|: s41 >= 0, s41 <= 0, s14 >= 0, s14 <= 2, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 2, z' >= 0, z'' >= 0, z'' - 1 >= 0 cond3(z, z', z'') -{ 163 + 77*s15 + 133*z' }-> s42 :|: s42 >= 0, s42 <= 0, s15 >= 0, s15 <= 2, s7 >= 0, s7 <= 2, s8 >= 0, s8 <= 2, z = 2, z' >= 0, z'' >= 0, z'' = 0 cond3(z, z', z'') -{ 162 + 77*s16 + 133*z' }-> s43 :|: s43 >= 0, s43 <= 0, s16 >= 0, s16 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, z = 2, z' >= 0, z'' >= 0 cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encArg(z) -{ 5 + s100 + 513*x_12964 + 189*x_12964^2 + 513*x_22117 + 189*x_22117^2 }-> s102 :|: s99 >= 0, s99 <= x_12964 + 1, s100 >= 0, s100 <= x_22117 + 1, s101 >= 0, s101 <= 2, s102 >= 0, s102 <= s101, x_12964 >= 0, z = 1 + (1 + x_12964 + x_22117), x_22117 >= 0 encArg(z) -{ -267 + -243*z + 189*z^2 }-> s105 :|: s103 >= 0, s103 <= z - 2 + 1, s104 >= 0, s104 <= s103, s105 >= 0, s105 <= s104, z - 2 >= 0 encArg(z) -{ 4 + 513*x_12965 + 189*x_12965^2 + 513*x_22118 + 189*x_22118^2 }-> s133 :|: s130 >= 0, s130 <= x_12965 + 1, s131 >= 0, s131 <= x_22118 + 1, s132 >= 0, s132 <= 2, s133 >= 0, s133 <= s132, x_12965 >= 0, z = 1 + (1 + x_12965 + x_22118), x_22118 >= 0 encArg(z) -{ 159 + 77*s45 + 133*s46 + 189*s47 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s48 :|: s45 >= 0, s45 <= x_1 + 1, s46 >= 0, s46 <= x_2 + 1, s47 >= 0, s47 <= x_3 + 1, s48 >= 0, s48 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 321 + 133*s50 + 189*s51 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s52 :|: s49 >= 0, s49 <= x_1 + 1, s50 >= 0, s50 <= x_2 + 1, s51 >= 0, s51 <= x_3 + 1, s52 >= 0, s52 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 320 + 133*s54 + 189*s55 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 + 513*x_3 + 189*x_3^2 }-> s56 :|: s53 >= 0, s53 <= x_1 + 1, s54 >= 0, s54 <= x_2 + 1, s55 >= 0, s55 <= x_3 + 1, s56 >= 0, s56 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 4 + s58 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s59 :|: s57 >= 0, s57 <= x_1 + 1, s58 >= 0, s58 <= x_2 + 1, s59 >= 0, s59 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 3 + 513*x_1 + 189*x_1^2 + 513*x_2 + 189*x_2^2 }-> s78 :|: s76 >= 0, s76 <= x_1 + 1, s77 >= 0, s77 <= x_2 + 1, s78 >= 0, s78 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ -268 + -243*z + 189*z^2 }-> s83 :|: s82 >= 0, s82 <= z - 2 + 1, s83 >= 0, s83 <= 1 + s82, z - 2 >= 0 encArg(z) -{ 160 + 77*s84 + 133*s85 + 189*s86 + 513*x_12961 + 189*x_12961^2 + 513*x_22114 + 189*x_22114^2 + 513*x_31268 + 189*x_31268^2 }-> s88 :|: s84 >= 0, s84 <= x_12961 + 1, s85 >= 0, s85 <= x_22114 + 1, s86 >= 0, s86 <= x_31268 + 1, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= s87, x_12961 >= 0, x_31268 >= 0, z = 1 + (1 + x_12961 + x_22114 + x_31268), x_22114 >= 0 encArg(z) -{ 322 + 133*s90 + 189*s91 + 513*x_12962 + 189*x_12962^2 + 513*x_22115 + 189*x_22115^2 + 513*x_31269 + 189*x_31269^2 }-> s93 :|: s89 >= 0, s89 <= x_12962 + 1, s90 >= 0, s90 <= x_22115 + 1, s91 >= 0, s91 <= x_31269 + 1, s92 >= 0, s92 <= 0, s93 >= 0, s93 <= s92, x_22115 >= 0, x_31269 >= 0, x_12962 >= 0, z = 1 + (1 + x_12962 + x_22115 + x_31269) encArg(z) -{ 321 + 133*s95 + 189*s96 + 513*x_12963 + 189*x_12963^2 + 513*x_22116 + 189*x_22116^2 + 513*x_31270 + 189*x_31270^2 }-> s98 :|: s94 >= 0, s94 <= x_12963 + 1, s95 >= 0, s95 <= x_22116 + 1, s96 >= 0, s96 <= x_31270 + 1, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= s97, x_31270 >= 0, x_12963 >= 0, x_22116 >= 0, z = 1 + (1 + x_12963 + x_22116 + x_31270) 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) -{ -323 + 135*z + 189*z^2 }-> 1 + s44 :|: s44 >= 0, s44 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_cond1(z, z', z'') -{ 159 + 77*s60 + 133*s61 + 189*s62 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s63 :|: s60 >= 0, s60 <= z + 1, s61 >= 0, s61 <= z' + 1, s62 >= 0, s62 <= z'' + 1, s63 >= 0, s63 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond2(z, z', z'') -{ 321 + 133*s65 + 189*s66 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s67 :|: s64 >= 0, s64 <= z + 1, s65 >= 0, s65 <= z' + 1, s66 >= 0, s66 <= z'' + 1, s67 >= 0, s67 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_cond3(z, z', z'') -{ 320 + 133*s72 + 189*s73 + 513*z + 189*z^2 + 513*z' + 189*z'^2 + 513*z'' + 189*z''^2 }-> s74 :|: s71 >= 0, s71 <= z + 1, s72 >= 0, s72 <= z' + 1, s73 >= 0, s73 <= z'' + 1, s74 >= 0, s74 <= 0, z >= 0, z'' >= 0, z' >= 0 encode_cond3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gr(z, z') -{ 4 + s69 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s70 :|: s68 >= 0, s68 <= z + 1, s69 >= 0, s69 <= z' + 1, s70 >= 0, s70 <= 2, z >= 0, z' >= 0 encode_gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_or(z, z') -{ 3 + 513*z + 189*z^2 + 513*z' + 189*z'^2 }-> s81 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, s81 >= 0, s81 <= 2, z >= 0, z' >= 0 encode_or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -322 + 135*z + 189*z^2 }-> s107 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= 1 + s106, z - 1 >= 0 encode_p(z) -{ 160 + 77*s108 + 133*s109 + 189*s110 + 513*x_14998 + 189*x_14998^2 + 513*x_23569 + 189*x_23569^2 + 513*x_32141 + 189*x_32141^2 }-> s112 :|: s108 >= 0, s108 <= x_14998 + 1, s109 >= 0, s109 <= x_23569 + 1, s110 >= 0, s110 <= x_32141 + 1, s111 >= 0, s111 <= 0, s112 >= 0, s112 <= s111, x_23569 >= 0, x_14998 >= 0, z = 1 + x_14998 + x_23569 + x_32141, x_32141 >= 0 encode_p(z) -{ 322 + 133*s114 + 189*s115 + 513*x_14999 + 189*x_14999^2 + 513*x_23570 + 189*x_23570^2 + 513*x_32142 + 189*x_32142^2 }-> s117 :|: s113 >= 0, s113 <= x_14999 + 1, s114 >= 0, s114 <= x_23570 + 1, s115 >= 0, s115 <= x_32142 + 1, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= s116, x_14999 >= 0, x_32142 >= 0, z = 1 + x_14999 + x_23570 + x_32142, x_23570 >= 0 encode_p(z) -{ 321 + 133*s119 + 189*s120 + 513*x_15000 + 189*x_15000^2 + 513*x_23571 + 189*x_23571^2 + 513*x_32143 + 189*x_32143^2 }-> s122 :|: s118 >= 0, s118 <= x_15000 + 1, s119 >= 0, s119 <= x_23571 + 1, s120 >= 0, s120 <= x_32143 + 1, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= s121, x_23571 >= 0, x_32143 >= 0, z = 1 + x_15000 + x_23571 + x_32143, x_15000 >= 0 encode_p(z) -{ 5 + s124 + 513*x_15001 + 189*x_15001^2 + 513*x_23572 + 189*x_23572^2 }-> s126 :|: s123 >= 0, s123 <= x_15001 + 1, s124 >= 0, s124 <= x_23572 + 1, s125 >= 0, s125 <= 2, s126 >= 0, s126 <= s125, z = 1 + x_15001 + x_23572, x_15001 >= 0, x_23572 >= 0 encode_p(z) -{ -321 + 135*z + 189*z^2 }-> s129 :|: s127 >= 0, s127 <= z - 1 + 1, s128 >= 0, s128 <= s127, s129 >= 0, s129 <= s128, z - 1 >= 0 encode_p(z) -{ 4 + 513*x_15002 + 189*x_15002^2 + 513*x_23573 + 189*x_23573^2 }-> s137 :|: s134 >= 0, s134 <= x_15002 + 1, s135 >= 0, s135 <= x_23573 + 1, s136 >= 0, s136 <= 2, s137 >= 0, s137 <= s136, z = 1 + x_15002 + x_23573, x_23573 >= 0, x_15002 >= 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) -{ 1 + 513*z + 189*z^2 }-> 1 + s75 :|: s75 >= 0, s75 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gr(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 gr(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0 gr(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 gr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z = 2, z' >= 0 or(z, z') -{ 1 }-> 2 :|: z' = 2, z >= 0 or(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 or(z, z') -{ 0 }-> 0 :|: z >= 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] gr: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] or: runtime: O(1) [1], size: O(1) [2] cond1: runtime: O(n^1) [156 + 77*z + 133*z' + 189*z''], size: O(1) [0] cond2: runtime: O(n^1) [318 + 133*z' + 189*z''], size: O(1) [0] cond3: runtime: O(n^1) [317 + 133*z' + 189*z''], size: O(1) [0] encArg: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [1 + z] encode_p: runtime: O(n^2) [1860 + 7983*z + 2835*z^2], size: O(n^1) [1 + z] encode_or: runtime: O(n^2) [3 + 513*z + 189*z^2 + 513*z' + 189*z'^2], size: O(1) [2] encode_s: runtime: O(n^2) [1 + 513*z + 189*z^2], size: O(n^1) [2 + z] encode_cond3: runtime: O(n^2) [642 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] encode_gr: runtime: O(n^2) [5 + 513*z + 189*z^2 + 514*z' + 189*z'^2], size: O(1) [2] encode_cond2: runtime: O(n^2) [643 + 513*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] encode_cond1: runtime: O(n^2) [558 + 590*z + 189*z^2 + 646*z' + 189*z'^2 + 702*z'' + 189*z''^2], size: O(1) [0] ---------------------------------------- (111) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (112) BOUNDS(1, n^2) ---------------------------------------- (113) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (114) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: cond1(true, x, y) -> cond2(gr(x, 0), x, y) cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) cond2(false, x, y) -> cond3(gr(y, 0), x, y) cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) gr(0, x) -> false gr(s(x), 0) -> true gr(s(x), s(y)) -> gr(x, y) or(false, false) -> false or(true, x) -> true or(x, true) -> true 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_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond3(x_1, x_2, x_3)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_cond3(x_1, x_2, x_3) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (115) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence gr(s(x), s(y)) ->^+ gr(x, y) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [x / s(x), y / s(y)]. The result substitution is [ ]. ---------------------------------------- (116) Complex Obligation (BEST) ---------------------------------------- (117) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: cond1(true, x, y) -> cond2(gr(x, 0), x, y) cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) cond2(false, x, y) -> cond3(gr(y, 0), x, y) cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) gr(0, x) -> false gr(s(x), 0) -> true gr(s(x), s(y)) -> gr(x, y) or(false, false) -> false or(true, x) -> true or(x, true) -> true 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_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond3(x_1, x_2, x_3)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_cond3(x_1, x_2, x_3) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (118) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (119) BOUNDS(n^1, INF) ---------------------------------------- (120) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: cond1(true, x, y) -> cond2(gr(x, 0), x, y) cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) cond2(false, x, y) -> cond3(gr(y, 0), x, y) cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) gr(0, x) -> false gr(s(x), 0) -> true gr(s(x), s(y)) -> gr(x, y) or(false, false) -> false or(true, x) -> true or(x, true) -> true 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_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond3(x_1, x_2, x_3)) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gr(x_1, x_2)) -> gr(encArg(x_1), encArg(x_2)) encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gr(x_1, x_2) -> gr(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_false -> false encode_cond3(x_1, x_2, x_3) -> cond3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST