/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^3)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 340 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), 14.1 s] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 322 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 137 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 15 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 1 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 86 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 21 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 270 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 165 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 115 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 21 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 35 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 1897 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 680 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 652 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 366 ms] (66) CpxRNTS (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 219 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 13 ms] (72) CpxRNTS (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 328 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (78) CpxRNTS (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 248 ms] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (84) CpxRNTS (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (86) CpxRNTS (87) IntTrsBoundProof [UPPER BOUND(ID), 217 ms] (88) CpxRNTS (89) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (90) CpxRNTS (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (92) CpxRNTS (93) IntTrsBoundProof [UPPER BOUND(ID), 96 ms] (94) CpxRNTS (95) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (96) CpxRNTS (97) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (98) CpxRNTS (99) IntTrsBoundProof [UPPER BOUND(ID), 126 ms] (100) CpxRNTS (101) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (102) CpxRNTS (103) FinalProof [FINISHED, 0 ms] (104) BOUNDS(1, n^3) (105) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CpxRelTRS (107) TypeInferenceProof [BOTH BOUNDS(ID, ID), 2 ms] (108) typed CpxTrs (109) OrderProof [LOWER BOUND(ID), 0 ms] (110) typed CpxTrs (111) RewriteLemmaProof [LOWER BOUND(ID), 268 ms] (112) BEST (113) proven lower bound (114) LowerBoundPropagationProof [FINISHED, 0 ms] (115) BOUNDS(n^1, INF) (116) typed CpxTrs (117) RewriteLemmaProof [LOWER BOUND(ID), 72 ms] (118) typed CpxTrs (119) RewriteLemmaProof [LOWER BOUND(ID), 462 ms] (120) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) div(x, y) -> ify(ge(y, s(0)), x, y) ify(false, x, y) -> divByZeroError ify(true, x, y) -> if(ge(x, y), x, y) if(false, x, y) -> 0 if(true, x, y) -> s(div(minus(x, y), y)) 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(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(divByZeroError) -> divByZeroError encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_ify(x_1, x_2, x_3)) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_ify(x_1, x_2, x_3) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encode_divByZeroError -> divByZeroError encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) div(x, y) -> ify(ge(y, s(0)), x, y) ify(false, x, y) -> divByZeroError ify(true, x, y) -> if(ge(x, y), x, y) if(false, x, y) -> 0 if(true, x, y) -> s(div(minus(x, y), y)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(divByZeroError) -> divByZeroError encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_ify(x_1, x_2, x_3)) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_ify(x_1, x_2, x_3) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encode_divByZeroError -> divByZeroError encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) div(x, y) -> ify(ge(y, s(0)), x, y) ify(false, x, y) -> divByZeroError ify(true, x, y) -> if(ge(x, y), x, y) if(false, x, y) -> 0 if(true, x, y) -> s(div(minus(x, y), y)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(divByZeroError) -> divByZeroError encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_ify(x_1, x_2, x_3)) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_ify(x_1, x_2, x_3) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encode_divByZeroError -> divByZeroError encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] div(x, y) -> ify(ge(y, s(0)), x, y) [1] ify(false, x, y) -> divByZeroError [1] ify(true, x, y) -> if(ge(x, y), x, y) [1] if(false, x, y) -> 0 [1] if(true, x, y) -> s(div(minus(x, y), y)) [1] encArg(0) -> 0 [0] encArg(true) -> true [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(false) -> false [0] encArg(divByZeroError) -> divByZeroError [0] encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) [0] encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0] encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) [0] encArg(cons_ify(x_1, x_2, x_3)) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_true -> true [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_false -> false [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0] encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) [0] encode_ify(x_1, x_2, x_3) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_divByZeroError -> divByZeroError [0] encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [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: ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] div(x, y) -> ify(ge(y, s(0)), x, y) [1] ify(false, x, y) -> divByZeroError [1] ify(true, x, y) -> if(ge(x, y), x, y) [1] if(false, x, y) -> 0 [1] if(true, x, y) -> s(div(minus(x, y), y)) [1] encArg(0) -> 0 [0] encArg(true) -> true [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(false) -> false [0] encArg(divByZeroError) -> divByZeroError [0] encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) [0] encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0] encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) [0] encArg(cons_ify(x_1, x_2, x_3)) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_true -> true [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_false -> false [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0] encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) [0] encode_ify(x_1, x_2, x_3) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_divByZeroError -> divByZeroError [0] encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] The TRS has the following type information: ge :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if 0 :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if true :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if s :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if false :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if minus :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if div :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if ify :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if divByZeroError :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if if :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encArg :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_ge :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_minus :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_div :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_ify :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_if :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_ge :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_0 :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_true :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_s :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_false :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_minus :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_div :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_ify :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_divByZeroError :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_if :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if 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: ge_2 minus_2 div_2 ify_3 if_3 encArg_1 encode_ge_2 encode_0 encode_true encode_s_1 encode_false encode_minus_2 encode_div_2 encode_ify_3 encode_divByZeroError encode_if_3 Due to the following rules being added: encArg(v0) -> null_encArg [0] encode_ge(v0, v1) -> null_encode_ge [0] encode_0 -> null_encode_0 [0] encode_true -> null_encode_true [0] encode_s(v0) -> null_encode_s [0] encode_false -> null_encode_false [0] encode_minus(v0, v1) -> null_encode_minus [0] encode_div(v0, v1) -> null_encode_div [0] encode_ify(v0, v1, v2) -> null_encode_ify [0] encode_divByZeroError -> null_encode_divByZeroError [0] encode_if(v0, v1, v2) -> null_encode_if [0] ge(v0, v1) -> null_ge [0] minus(v0, v1) -> null_minus [0] ify(v0, v1, v2) -> null_ify [0] if(v0, v1, v2) -> null_if [0] And the following fresh constants: null_encArg, null_encode_ge, null_encode_0, null_encode_true, null_encode_s, null_encode_false, null_encode_minus, null_encode_div, null_encode_ify, null_encode_divByZeroError, null_encode_if, null_ge, null_minus, null_ify, null_if ---------------------------------------- (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: ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] div(x, y) -> ify(ge(y, s(0)), x, y) [1] ify(false, x, y) -> divByZeroError [1] ify(true, x, y) -> if(ge(x, y), x, y) [1] if(false, x, y) -> 0 [1] if(true, x, y) -> s(div(minus(x, y), y)) [1] encArg(0) -> 0 [0] encArg(true) -> true [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(false) -> false [0] encArg(divByZeroError) -> divByZeroError [0] encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) [0] encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0] encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) [0] encArg(cons_ify(x_1, x_2, x_3)) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_true -> true [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_false -> false [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0] encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) [0] encode_ify(x_1, x_2, x_3) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_divByZeroError -> divByZeroError [0] encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(v0) -> null_encArg [0] encode_ge(v0, v1) -> null_encode_ge [0] encode_0 -> null_encode_0 [0] encode_true -> null_encode_true [0] encode_s(v0) -> null_encode_s [0] encode_false -> null_encode_false [0] encode_minus(v0, v1) -> null_encode_minus [0] encode_div(v0, v1) -> null_encode_div [0] encode_ify(v0, v1, v2) -> null_encode_ify [0] encode_divByZeroError -> null_encode_divByZeroError [0] encode_if(v0, v1, v2) -> null_encode_if [0] ge(v0, v1) -> null_ge [0] minus(v0, v1) -> null_minus [0] ify(v0, v1, v2) -> null_ify [0] if(v0, v1, v2) -> null_if [0] The TRS has the following type information: ge :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if 0 :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if true :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if s :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if false :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if minus :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if div :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if ify :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if divByZeroError :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if if :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if encArg :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if cons_ge :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if cons_minus :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if cons_div :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if cons_ify :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if cons_if :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if encode_ge :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if encode_0 :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if encode_true :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if encode_s :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if encode_false :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if encode_minus :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 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0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if encode_if :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_encArg :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_encode_ge :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_encode_0 :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_encode_true :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_encode_s :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_encode_false :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_encode_minus :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_encode_div :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_encode_ify :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_encode_divByZeroError :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_encode_if :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_ge :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_minus :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_ify :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_if :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if 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: ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] div(x, 0) -> ify(false, x, 0) [2] div(x, s(x')) -> ify(ge(x', 0), x, s(x')) [2] div(x, y) -> ify(null_ge, x, y) [1] ify(false, x, y) -> divByZeroError [1] ify(true, x, 0) -> if(true, x, 0) [2] ify(true, 0, s(x'')) -> if(false, 0, s(x'')) [2] ify(true, s(x1), s(y')) -> if(ge(x1, y'), s(x1), s(y')) [2] ify(true, x, y) -> if(null_ge, x, y) [1] if(false, x, y) -> 0 [1] if(true, x, 0) -> s(div(x, 0)) [2] if(true, s(x2), s(y'')) -> s(div(minus(x2, y''), s(y''))) [2] if(true, x, y) -> s(div(null_minus, y)) [1] encArg(0) -> 0 [0] encArg(true) -> true [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(false) -> false [0] encArg(divByZeroError) -> divByZeroError [0] encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) [0] encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0] encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) [0] encArg(cons_ify(x_1, x_2, x_3)) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_true -> true [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_false -> false [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0] encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) [0] encode_ify(x_1, x_2, x_3) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_divByZeroError -> divByZeroError [0] encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(v0) -> null_encArg [0] encode_ge(v0, v1) -> null_encode_ge [0] encode_0 -> null_encode_0 [0] encode_true -> null_encode_true [0] encode_s(v0) -> null_encode_s [0] encode_false -> null_encode_false [0] encode_minus(v0, v1) -> null_encode_minus [0] encode_div(v0, v1) -> null_encode_div [0] encode_ify(v0, v1, v2) -> null_encode_ify [0] encode_divByZeroError -> null_encode_divByZeroError [0] encode_if(v0, v1, v2) -> null_encode_if [0] ge(v0, v1) -> null_ge [0] minus(v0, v1) -> null_minus [0] ify(v0, v1, v2) -> null_ify [0] if(v0, v1, v2) -> null_if [0] The TRS has the following type information: ge :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if -> 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if 0 :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if true :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if s :: 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0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_ify :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if null_if :: 0:true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if:null_encArg:null_encode_ge:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_div:null_encode_ify:null_encode_divByZeroError:null_encode_if:null_ge:null_minus:null_ify:null_if 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: 0 => 0 true => 3 false => 2 divByZeroError => 1 null_encArg => 0 null_encode_ge => 0 null_encode_0 => 0 null_encode_true => 0 null_encode_s => 0 null_encode_false => 0 null_encode_minus => 0 null_encode_div => 0 null_encode_ify => 0 null_encode_divByZeroError => 0 null_encode_if => 0 null_ge => 0 null_minus => 0 null_ify => 0 null_if => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(x', 0), x, 1 + x') :|: z' = 1 + x', x >= 0, x' >= 0, z = x div(z, z') -{ 2 }-> ify(2, x, 0) :|: x >= 0, z = x, z' = 0 div(z, z') -{ 1 }-> ify(0, x, y) :|: x >= 0, y >= 0, z = x, z' = y encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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_div(z, z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_if(z, z', z'') -{ 0 }-> if(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_if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(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_ify(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 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 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x ge(z, z') -{ 1 }-> 3 :|: x >= 0, z = x, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' = 1 + x, x >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(x, 0) :|: z'' = 0, z = 3, z' = x, x >= 0 if(z, z', z'') -{ 2 }-> 1 + div(minus(x2, y''), 1 + y'') :|: z = 3, z' = 1 + x2, y'' >= 0, x2 >= 0, z'' = 1 + y'' if(z, z', z'') -{ 1 }-> 1 + div(0, y) :|: z = 3, z' = x, z'' = y, x >= 0, y >= 0 ify(z, z', z'') -{ 2 }-> if(ge(x1, y'), 1 + x1, 1 + y') :|: z = 3, x1 >= 0, y' >= 0, z' = 1 + x1, z'' = 1 + y' ify(z, z', z'') -{ 2 }-> if(3, x, 0) :|: z'' = 0, z = 3, z' = x, x >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + x'') :|: z = 3, z'' = 1 + x'', x'' >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, x, y) :|: z = 3, z' = x, z'' = y, x >= 0, y >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 2 }-> 1 + div(minus(z' - 1, z'' - 1), 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { minus } { encode_0 } { encode_false } { ge } { encode_divByZeroError } { encode_true } { div, if, ify } { encArg } { encode_if } { encode_ify } { encode_div } { encode_minus } { encode_ge } { encode_s } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 2 }-> 1 + div(minus(z' - 1, z'' - 1), 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {encode_0}, {encode_false}, {ge}, {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 2 }-> 1 + div(minus(z' - 1, z'' - 1), 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {encode_0}, {encode_false}, {ge}, {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 2 }-> 1 + div(minus(z' - 1, z'' - 1), 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {encode_0}, {encode_false}, {ge}, {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 2 }-> 1 + div(minus(z' - 1, z'' - 1), 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_0}, {encode_false}, {ge}, {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_0}, {encode_false}, {ge}, {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (27) 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 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_0}, {encode_false}, {ge}, {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: ?, size: O(1) [0] ---------------------------------------- (29) 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 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_false}, {ge}, {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_false}, {ge}, {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (33) 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: 2 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_false}, {ge}, {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: ?, size: O(1) [2] ---------------------------------------- (35) 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 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {ge}, {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {ge}, {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: ge after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {ge}, {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: ?, size: O(1) [3] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: ge after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 }-> ify(ge(z' - 1, 0), z, 1 + (z' - 1)) :|: z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 4 }-> ify(s1, z, 1 + (z' - 1)) :|: s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 3 + z'' }-> if(s2, 1 + (z' - 1), 1 + (z'' - 1)) :|: s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_divByZeroError after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 4 }-> ify(s1, z, 1 + (z' - 1)) :|: s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 3 + z'' }-> if(s2, 1 + (z' - 1), 1 + (z'' - 1)) :|: s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_divByZeroError}, {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: ?, size: O(1) [1] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_divByZeroError after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 4 }-> ify(s1, z, 1 + (z' - 1)) :|: s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 3 + z'' }-> if(s2, 1 + (z' - 1), 1 + (z'' - 1)) :|: s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 4 }-> ify(s1, z, 1 + (z' - 1)) :|: s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 3 + z'' }-> if(s2, 1 + (z' - 1), 1 + (z'' - 1)) :|: s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] ---------------------------------------- (51) 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: 3 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 4 }-> ify(s1, z, 1 + (z' - 1)) :|: s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 3 + z'' }-> if(s2, 1 + (z' - 1), 1 + (z'' - 1)) :|: s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_true}, {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: ?, size: O(1) [3] ---------------------------------------- (53) 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 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 4 }-> ify(s1, z, 1 + (z' - 1)) :|: s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 3 + z'' }-> if(s2, 1 + (z' - 1), 1 + (z'' - 1)) :|: s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 4 }-> ify(s1, z, 1 + (z' - 1)) :|: s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 3 + z'' }-> if(s2, 1 + (z' - 1), 1 + (z'' - 1)) :|: s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: div after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' Computed SIZE bound using CoFloCo for: ify after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 4 }-> ify(s1, z, 1 + (z' - 1)) :|: s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 3 + z'' }-> if(s2, 1 + (z' - 1), 1 + (z'' - 1)) :|: s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {div,if,ify}, {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: ?, size: O(n^1) [1 + z] if: runtime: ?, size: O(n^1) [2 + z'] ify: runtime: ?, size: O(n^1) [2 + z'] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: div after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 24 + 9*z + 2*z*z' + 3*z' Computed RUNTIME bound using KoAT for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 69 + 18*z' + 2*z'*z'' + 5*z'' Computed RUNTIME bound using KoAT for: ify after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 285 + 54*z' + 4*z'*z'' + 16*z'' ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 4 }-> ify(s1, z, 1 + (z' - 1)) :|: s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 2 }-> ify(2, z, 0) :|: z >= 0, z' = 0 div(z, z') -{ 1 }-> ify(0, z, z') :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 + z'' }-> 1 + div(s', 1 + (z'' - 1)) :|: s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 2 }-> 1 + div(z', 0) :|: z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 1 }-> 1 + div(0, z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 3 + z'' }-> if(s2, 1 + (z' - 1), 1 + (z'' - 1)) :|: s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 2 }-> if(3, z', 0) :|: z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 2 }-> if(2, 0, 1 + (z'' - 1)) :|: z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 1 }-> if(0, z', z'') :|: z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] ---------------------------------------- (61) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] ---------------------------------------- (63) 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 + 2*z ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encArg}, {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: ?, size: O(n^1) [1 + 2*z] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encArg after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 496 + 1734*z + 770*z^2 + 96*z^3 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> ify(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 }-> if(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 }-> ge(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 0 }-> ge(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> if(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 0 }-> ify(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] ---------------------------------------- (67) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z' ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_if}, {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: ?, size: O(n^1) [3 + 2*z'] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_if after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3 ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] ---------------------------------------- (73) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_ify after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z' ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_ify}, {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: ?, size: O(n^1) [3 + 2*z'] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_ify after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3 ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: O(n^3) [1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] ---------------------------------------- (79) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: O(n^3) [1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] ---------------------------------------- (81) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_div after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 2*z ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_div}, {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: O(n^3) [1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_div: runtime: ?, size: O(n^1) [2 + 2*z] ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_div after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 1030 + 1756*z + 8*z*z' + 770*z^2 + 96*z^3 + 1744*z' + 770*z'^2 + 96*z'^3 ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: O(n^3) [1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_div: runtime: O(n^3) [1030 + 1756*z + 8*z*z' + 770*z^2 + 96*z^3 + 1744*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [2 + 2*z] ---------------------------------------- (85) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (86) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: O(n^3) [1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_div: runtime: O(n^3) [1030 + 1756*z + 8*z*z' + 770*z^2 + 96*z^3 + 1744*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [2 + 2*z] ---------------------------------------- (87) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 2*z ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_minus}, {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: O(n^3) [1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_div: runtime: O(n^3) [1030 + 1756*z + 8*z*z' + 770*z^2 + 96*z^3 + 1744*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [2 + 2*z] encode_minus: runtime: ?, size: O(n^1) [1 + 2*z] ---------------------------------------- (89) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_minus after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 994 + 1734*z + 770*z^2 + 96*z^3 + 1736*z' + 770*z'^2 + 96*z'^3 ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: O(n^3) [1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_div: runtime: O(n^3) [1030 + 1756*z + 8*z*z' + 770*z^2 + 96*z^3 + 1744*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [2 + 2*z] encode_minus: runtime: O(n^3) [994 + 1734*z + 770*z^2 + 96*z^3 + 1736*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [1 + 2*z] ---------------------------------------- (91) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (92) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: O(n^3) [1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_div: runtime: O(n^3) [1030 + 1756*z + 8*z*z' + 770*z^2 + 96*z^3 + 1744*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [2 + 2*z] encode_minus: runtime: O(n^3) [994 + 1734*z + 770*z^2 + 96*z^3 + 1736*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [1 + 2*z] ---------------------------------------- (93) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_ge after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (94) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_ge}, {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: O(n^3) [1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_div: runtime: O(n^3) [1030 + 1756*z + 8*z*z' + 770*z^2 + 96*z^3 + 1744*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [2 + 2*z] encode_minus: runtime: O(n^3) [994 + 1734*z + 770*z^2 + 96*z^3 + 1736*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [1 + 2*z] encode_ge: runtime: ?, size: O(1) [3] ---------------------------------------- (95) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_ge after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 995 + 1734*z + 770*z^2 + 96*z^3 + 1736*z' + 770*z'^2 + 96*z'^3 ---------------------------------------- (96) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: O(n^3) [1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_div: runtime: O(n^3) [1030 + 1756*z + 8*z*z' + 770*z^2 + 96*z^3 + 1744*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [2 + 2*z] encode_minus: runtime: O(n^3) [994 + 1734*z + 770*z^2 + 96*z^3 + 1736*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [1 + 2*z] encode_ge: runtime: O(n^3) [995 + 1734*z + 770*z^2 + 96*z^3 + 1736*z' + 770*z'^2 + 96*z'^3], size: O(1) [3] ---------------------------------------- (97) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (98) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: O(n^3) [1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_div: runtime: O(n^3) [1030 + 1756*z + 8*z*z' + 770*z^2 + 96*z^3 + 1744*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [2 + 2*z] encode_minus: runtime: O(n^3) [994 + 1734*z + 770*z^2 + 96*z^3 + 1736*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [1 + 2*z] encode_ge: runtime: O(n^3) [995 + 1734*z + 770*z^2 + 96*z^3 + 1736*z' + 770*z'^2 + 96*z'^3], size: O(1) [3] ---------------------------------------- (99) 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 + 2*z ---------------------------------------- (100) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_s} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: O(n^3) [1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_div: runtime: O(n^3) [1030 + 1756*z + 8*z*z' + 770*z^2 + 96*z^3 + 1744*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [2 + 2*z] encode_minus: runtime: O(n^3) [994 + 1734*z + 770*z^2 + 96*z^3 + 1736*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [1 + 2*z] encode_ge: runtime: O(n^3) [995 + 1734*z + 770*z^2 + 96*z^3 + 1736*z' + 770*z'^2 + 96*z'^3], size: O(1) [3] encode_s: runtime: ?, size: O(n^1) [2 + 2*z] ---------------------------------------- (101) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_s after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 496 + 1734*z + 770*z^2 + 96*z^3 ---------------------------------------- (102) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 287 + 54*z }-> s3 :|: s3 >= 0, s3 <= z + 2, z >= 0, z' = 0 div(z, z') -{ 289 + 54*z + 4*z*z' + 16*z' }-> s4 :|: s4 >= 0, s4 <= z + 2, s1 >= 0, s1 <= 3, z >= 0, z' - 1 >= 0 div(z, z') -{ 286 + 54*z + 4*z*z' + 16*z' }-> s5 :|: s5 >= 0, s5 <= z + 2, z >= 0, z' >= 0 encArg(z) -{ 994 + s15 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s16 :|: s14 >= 0, s14 <= 2 * x_1 + 1, s15 >= 0, s15 <= 2 * x_2 + 1, s16 >= 0, s16 <= 3, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 993 + s18 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s19 :|: s17 >= 0, s17 <= 2 * x_1 + 1, s18 >= 0, s18 <= 2 * x_2 + 1, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1016 + 9*s20 + 2*s20*s21 + 3*s21 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 1, s21 >= 0, s21 <= 2 * x_2 + 1, s22 >= 0, s22 <= s20 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 1773 + 54*s24 + 4*s24*s25 + 16*s25 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s26 :|: s23 >= 0, s23 <= 2 * x_1 + 1, s24 >= 0, s24 <= 2 * x_2 + 1, s25 >= 0, s25 <= 2 * x_3 + 1, s26 >= 0, s26 <= s24 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 1557 + 18*s28 + 2*s28*s29 + 5*s29 + 1734*x_1 + 770*x_1^2 + 96*x_1^3 + 1734*x_2 + 770*x_2^2 + 96*x_2^3 + 1734*x_3 + 770*x_3^2 + 96*x_3^3 }-> s30 :|: s27 >= 0, s27 <= 2 * x_1 + 1, s28 >= 0, s28 <= 2 * x_2 + 1, s29 >= 0, s29 <= 2 * x_3 + 1, s30 >= 0, s30 <= s28 + 2, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> 3 :|: z = 3 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) -{ -564 + 482*z + 482*z^2 + 96*z^3 }-> 1 + s13 :|: s13 >= 0, s13 <= 2 * (z - 1) + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_div(z, z') -{ 1016 + 9*s38 + 2*s38*s39 + 3*s39 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s40 :|: s38 >= 0, s38 <= 2 * z + 1, s39 >= 0, s39 <= 2 * z' + 1, s40 >= 0, s40 <= s38 + 1, z >= 0, z' >= 0 encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_divByZeroError -{ 0 }-> 1 :|: encode_divByZeroError -{ 0 }-> 0 :|: encode_false -{ 0 }-> 2 :|: encode_false -{ 0 }-> 0 :|: encode_ge(z, z') -{ 994 + s32 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s33 :|: s31 >= 0, s31 <= 2 * z + 1, s32 >= 0, s32 <= 2 * z' + 1, s33 >= 0, s33 <= 3, z >= 0, z' >= 0 encode_ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if(z, z', z'') -{ 1557 + 18*s46 + 2*s46*s47 + 5*s47 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s48 :|: s45 >= 0, s45 <= 2 * z + 1, s46 >= 0, s46 <= 2 * z' + 1, s47 >= 0, s47 <= 2 * z'' + 1, s48 >= 0, s48 <= s46 + 2, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_ify(z, z', z'') -{ 1773 + 54*s42 + 4*s42*s43 + 16*s43 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 + 1734*z'' + 770*z''^2 + 96*z''^3 }-> s44 :|: s41 >= 0, s41 <= 2 * z + 1, s42 >= 0, s42 <= 2 * z' + 1, s43 >= 0, s43 <= 2 * z'' + 1, s44 >= 0, s44 <= s42 + 2, z >= 0, z'' >= 0, z' >= 0 encode_ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_minus(z, z') -{ 993 + s36 + 1734*z + 770*z^2 + 96*z^3 + 1734*z' + 770*z'^2 + 96*z'^3 }-> s37 :|: s35 >= 0, s35 <= 2 * z + 1, s36 >= 0, s36 <= 2 * z' + 1, s37 >= 0, s37 <= s35, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 496 + 1734*z + 770*z^2 + 96*z^3 }-> 1 + s34 :|: s34 >= 0, s34 <= 2 * z + 1, z >= 0 encode_true -{ 0 }-> 3 :|: encode_true -{ 0 }-> 0 :|: ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 3 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' >= 0, z'' >= 0 if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if(z, z', z'') -{ 26 + 9*z' }-> 1 + s10 :|: s10 >= 0, s10 <= z' + 1, z'' = 0, z = 3, z' >= 0 if(z, z', z'') -{ 26 + 9*s' + 2*s'*z'' + 4*z'' }-> 1 + s11 :|: s11 >= 0, s11 <= s' + 1, s' >= 0, s' <= z' - 1, z = 3, z'' - 1 >= 0, z' - 1 >= 0 if(z, z', z'') -{ 25 + 3*z'' }-> 1 + s12 :|: s12 >= 0, s12 <= 0 + 1, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 71 + 18*z' }-> s6 :|: s6 >= 0, s6 <= z' + 2, z'' = 0, z = 3, z' >= 0 ify(z, z', z'') -{ 71 + 5*z'' }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 3, z'' - 1 >= 0, z' = 0 ify(z, z', z'') -{ 72 + 18*z' + 2*z'*z'' + 6*z'' }-> s8 :|: s8 >= 0, s8 <= 1 + (z' - 1) + 2, s2 >= 0, s2 <= 3, z = 3, z' - 1 >= 0, z'' - 1 >= 0 ify(z, z', z'') -{ 70 + 18*z' + 2*z'*z'' + 5*z'' }-> s9 :|: s9 >= 0, s9 <= z' + 2, z = 3, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 1 }-> 1 :|: z = 2, z' >= 0, z'' >= 0 ify(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] encode_0: runtime: O(1) [0], size: O(1) [0] encode_false: runtime: O(1) [0], size: O(1) [2] ge: runtime: O(n^1) [2 + z'], size: O(1) [3] encode_divByZeroError: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [3] div: runtime: O(n^2) [24 + 9*z + 2*z*z' + 3*z'], size: O(n^1) [1 + z] if: runtime: O(n^2) [69 + 18*z' + 2*z'*z'' + 5*z''], size: O(n^1) [2 + z'] ify: runtime: O(n^2) [285 + 54*z' + 4*z'*z'' + 16*z''], size: O(n^1) [2 + z'] encArg: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [1 + 2*z] encode_if: runtime: O(n^3) [1582 + 1734*z + 770*z^2 + 96*z^3 + 1774*z' + 8*z'*z'' + 770*z'^2 + 96*z'^3 + 1748*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_ify: runtime: O(n^3) [1847 + 1734*z + 770*z^2 + 96*z^3 + 1850*z' + 16*z'*z'' + 770*z'^2 + 96*z'^3 + 1774*z'' + 770*z''^2 + 96*z''^3], size: O(n^1) [3 + 2*z'] encode_div: runtime: O(n^3) [1030 + 1756*z + 8*z*z' + 770*z^2 + 96*z^3 + 1744*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [2 + 2*z] encode_minus: runtime: O(n^3) [994 + 1734*z + 770*z^2 + 96*z^3 + 1736*z' + 770*z'^2 + 96*z'^3], size: O(n^1) [1 + 2*z] encode_ge: runtime: O(n^3) [995 + 1734*z + 770*z^2 + 96*z^3 + 1736*z' + 770*z'^2 + 96*z'^3], size: O(1) [3] encode_s: runtime: O(n^3) [496 + 1734*z + 770*z^2 + 96*z^3], size: O(n^1) [2 + 2*z] ---------------------------------------- (103) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (104) BOUNDS(1, n^3) ---------------------------------------- (105) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (106) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) div(x, y) -> ify(ge(y, s(0')), x, y) ify(false, x, y) -> divByZeroError ify(true, x, y) -> if(ge(x, y), x, y) if(false, x, y) -> 0' if(true, x, y) -> s(div(minus(x, y), y)) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(divByZeroError) -> divByZeroError encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_ify(x_1, x_2, x_3)) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_ify(x_1, x_2, x_3) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encode_divByZeroError -> divByZeroError encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (107) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (108) Obligation: Innermost TRS: Rules: ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) div(x, y) -> ify(ge(y, s(0')), x, y) ify(false, x, y) -> divByZeroError ify(true, x, y) -> if(ge(x, y), x, y) if(false, x, y) -> 0' if(true, x, y) -> s(div(minus(x, y), y)) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(divByZeroError) -> divByZeroError encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_ify(x_1, x_2, x_3)) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_ify(x_1, x_2, x_3) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encode_divByZeroError -> divByZeroError encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if 0' :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if true :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if s :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if false :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if divByZeroError :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encArg :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_0 :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_true :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_s :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_false :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_divByZeroError :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if hole_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if1_4 :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4 :: Nat -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if ---------------------------------------- (109) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: ge, minus, div, ify, if, encArg They will be analysed ascendingly in the following order: ge < div ge < ify ge < encArg minus < if minus < encArg div = ify div = if div < encArg ify = if ify < encArg if < encArg ---------------------------------------- (110) Obligation: Innermost TRS: Rules: ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) div(x, y) -> ify(ge(y, s(0')), x, y) ify(false, x, y) -> divByZeroError ify(true, x, y) -> if(ge(x, y), x, y) if(false, x, y) -> 0' if(true, x, y) -> s(div(minus(x, y), y)) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(divByZeroError) -> divByZeroError encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_ify(x_1, x_2, x_3)) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_ify(x_1, x_2, x_3) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encode_divByZeroError -> divByZeroError encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if 0' :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if true :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if s :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if false :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if divByZeroError :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encArg :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_0 :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_true :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_s :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_false :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_divByZeroError :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if hole_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if1_4 :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4 :: Nat -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if Generator Equations: gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(0) <=> 0' gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(+(x, 1)) <=> s(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(x)) The following defined symbols remain to be analysed: ge, minus, div, ify, if, encArg They will be analysed ascendingly in the following order: ge < div ge < ify ge < encArg minus < if minus < encArg div = ify div = if div < encArg ify = if ify < encArg if < encArg ---------------------------------------- (111) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: ge(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n4_4), gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n4_4)) -> true, rt in Omega(1 + n4_4) Induction Base: ge(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(0), gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(0)) ->_R^Omega(1) true Induction Step: ge(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(+(n4_4, 1)), gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(+(n4_4, 1))) ->_R^Omega(1) ge(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n4_4), gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n4_4)) ->_IH true We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (112) Complex Obligation (BEST) ---------------------------------------- (113) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) div(x, y) -> ify(ge(y, s(0')), x, y) ify(false, x, y) -> divByZeroError ify(true, x, y) -> if(ge(x, y), x, y) if(false, x, y) -> 0' if(true, x, y) -> s(div(minus(x, y), y)) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(divByZeroError) -> divByZeroError encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_ify(x_1, x_2, x_3)) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_ify(x_1, x_2, x_3) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encode_divByZeroError -> divByZeroError encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if 0' :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if true :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if s :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if false :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if divByZeroError :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encArg :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_0 :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_true :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_s :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_false :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_divByZeroError :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if hole_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if1_4 :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4 :: Nat -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if Generator Equations: gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(0) <=> 0' gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(+(x, 1)) <=> s(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(x)) The following defined symbols remain to be analysed: ge, minus, div, ify, if, encArg They will be analysed ascendingly in the following order: ge < div ge < ify ge < encArg minus < if minus < encArg div = ify div = if div < encArg ify = if ify < encArg if < encArg ---------------------------------------- (114) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (115) BOUNDS(n^1, INF) ---------------------------------------- (116) Obligation: Innermost TRS: Rules: ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) div(x, y) -> ify(ge(y, s(0')), x, y) ify(false, x, y) -> divByZeroError ify(true, x, y) -> if(ge(x, y), x, y) if(false, x, y) -> 0' if(true, x, y) -> s(div(minus(x, y), y)) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(divByZeroError) -> divByZeroError encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_ify(x_1, x_2, x_3)) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_ify(x_1, x_2, x_3) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encode_divByZeroError -> divByZeroError encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if 0' :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if true :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if s :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if false :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if divByZeroError :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encArg :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_0 :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_true :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_s :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_false :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_divByZeroError :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if hole_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if1_4 :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4 :: Nat -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if Lemmas: ge(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n4_4), gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n4_4)) -> true, rt in Omega(1 + n4_4) Generator Equations: gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(0) <=> 0' gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(+(x, 1)) <=> s(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(x)) The following defined symbols remain to be analysed: minus, div, ify, if, encArg They will be analysed ascendingly in the following order: minus < if minus < encArg div = ify div = if div < encArg ify = if ify < encArg if < encArg ---------------------------------------- (117) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: minus(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n550_4), gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n550_4)) -> gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(0), rt in Omega(1 + n550_4) Induction Base: minus(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(0), gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(0)) ->_R^Omega(1) gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(0) Induction Step: minus(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(+(n550_4, 1)), gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(+(n550_4, 1))) ->_R^Omega(1) minus(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n550_4), gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n550_4)) ->_IH gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (118) Obligation: Innermost TRS: Rules: ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) div(x, y) -> ify(ge(y, s(0')), x, y) ify(false, x, y) -> divByZeroError ify(true, x, y) -> if(ge(x, y), x, y) if(false, x, y) -> 0' if(true, x, y) -> s(div(minus(x, y), y)) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(divByZeroError) -> divByZeroError encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_ify(x_1, x_2, x_3)) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_ify(x_1, x_2, x_3) -> ify(encArg(x_1), encArg(x_2), encArg(x_3)) encode_divByZeroError -> divByZeroError encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if 0' :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if true :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if s :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if false :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if divByZeroError :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encArg :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if cons_if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_ge :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_0 :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_true :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_s :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_false :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_minus :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_div :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_ify :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_divByZeroError :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if encode_if :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if hole_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if1_4 :: 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4 :: Nat -> 0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if Lemmas: ge(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n4_4), gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n4_4)) -> true, rt in Omega(1 + n4_4) minus(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n550_4), gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n550_4)) -> gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(0), rt in Omega(1 + n550_4) Generator Equations: gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(0) <=> 0' gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(+(x, 1)) <=> s(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(x)) The following defined symbols remain to be analysed: ify, div, if, encArg They will be analysed ascendingly in the following order: div = ify div = if div < encArg ify = if ify < encArg if < encArg ---------------------------------------- (119) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n1336_4)) -> gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n1336_4), rt in Omega(0) Induction Base: encArg(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(+(n1336_4, 1))) ->_R^Omega(0) s(encArg(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(n1336_4))) ->_IH s(gen_0':true:s:false:divByZeroError:cons_ge:cons_minus:cons_div:cons_ify:cons_if2_4(c1337_4)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (120) BOUNDS(1, INF)